[{"data":1,"prerenderedAt":17801},["ShallowReactive",2],{"uc-internet-shopping":3},{"en":4,"de":9159},{"id":5,"title":6,"body":7,"description":9150,"extension":9151,"faq":9152,"meta":9153,"navigation":7921,"noindex":9154,"path":9155,"seo":9156,"stem":9157,"__hash__":9158},"content\u002Fuse-cases\u002Finternet-shopping.md","Internet-shopping optimization problem",{"type":8,"value":9,"toc":9141},"minimark",[10,35,52,65,70,75,78,1262,1266,1285,2587,3124,3128,3240,5921,5924,7861,7865,7875,8138,8143,8147,8150,8209,8341,8535,8538,9010,9013,9060,9063,9066,9118,9121,9137],[11,12,13,14,18,19,22,23,30,31,34],"p",{},"Let us take a deep dive into the so-called \"internet-shopping optimization problem\", also known as \"supplier selection and order allocation\". This is a very common problem that occurs basically whenever a large number of items needs to be ordered from a large number of suppliers. As it involves ",[15,16,17],"em",{},"integer"," quantities, it is usually formulated as a ",[15,20,21],{},"mixed-integer linear problem"," (MILP). Unfortunately, MILPs are known to be ",[24,25,29],"a",{"href":26,"rel":27},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FNP-hardness",[28],"nofollow","NP-hard"," - which more or less says that they can be ",[15,32,33],{},"very"," hard to solve.",[11,36,37,38,43,44,47,48,51],{},"You can replicate all results by running ",[24,39,42],{"href":40,"rel":41},"https:\u002F\u002Fgithub.com\u002FQuicopt\u002FTutorials\u002Fblob\u002Fmain\u002Fuse-cases\u002Finternet-shopping.ipynb",[28],"this notebook",". Note that this will execute the open-source solver CP-SAT ",[15,45,46],{},"on your machine"," while Quicopt runs on our ",[15,49,50],{},"API server",". This means the hardware is not identical, which it should be for a strictly rigorous comparison.",[11,53,54,55,59,60,64],{},"Even if you're running CP-SAT on CPUs faster than our test server, you should still notice that ",[56,57,58],"strong",{},"Quicopt is a lot faster!"," On top of that, CP-SAT uses 8 cores while Quicopt uses just one! (CP-SAT's number of cores can be controlled by adjusting ",[61,62,63],"code",{},"WORKERS",", see below. Finally, be mindful of the fact that CP-SAT operates non-deterministically, i.e. runtimes will fluctuate quite a lot, as will solution quality.)",[66,67,69],"h2",{"id":68},"the-problem-as-a-milp","The problem as a MILP",[71,72,74],"h3",{"id":73},"variables","Variables",[11,76,77],{},"The variables of our optimization problems are given in the following table:",[79,80,81,100],"table",{},[82,83,84],"thead",{},[85,86,87,91,94,97],"tr",{},[88,89,90],"th",{},"symbol",[88,92,93],{},"values",[88,95,96],{},"meaning",[88,98,99],{},"cost it 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ships for free no\u002Fyes",[106,1159,1160,1261],{},[109,1161,1163,1187],{"className":1162},[112],[109,1164,1166],{"className":1165},[116],[118,1167,1168],{"xmlns":120},[122,1169,1170,1184],{},[125,1171,1172,1175],{},[652,1173,1174],{},"−",[267,1176,1177,1179,1181],{},[131,1178,271],{},[131,1180,862],{},[275,1182,1183],{},"ship",[138,1185,1186],{"encoding":140},"-C^{\\text{ship}}_k",[109,1188,1190],{"className":1189,"ariaHidden":146},[145],[109,1191,1193,1197,1200],{"className":1192},[150],[109,1194],{"className":1195,"style":1196},[154],"height:1.2683em;vertical-align:-0.3013em;",[109,1198,1174],{"className":1199},[159],[109,1201,1203,1206],{"className":1202},[159],[109,1204,271],{"className":1205,"style":297},[159,163],[109,1207,1209],{"className":1208},[168],[109,1210,1212,1252],{"className":1211},[172,173],[109,1213,1215,1249],{"className":1214},[177],[109,1216,1219,1231],{"className":1217,"style":1218},[181],"height:0.967em;",[109,1220,1222,1225],{"style":1221},"top:-2.3987em;margin-left:-0.0715em;margin-right:0.05em;",[109,1223],{"className":1224,"style":190},[189],[109,1226,1228],{"className":1227},[194,195,196,197],[109,1229,862],{"className":1230,"style":905},[159,163,197],[109,1232,1234,1237],{"style":1233},"top:-3.1809em;margin-right:0.05em;",[109,1235],{"className":1236,"style":190},[189],[109,1238,1240],{"className":1239},[194,195,196,197],[109,1241,1243],{"className":1242},[159,197],[109,1244,1246],{"className":1245},[159,338,197],[109,1247,1183],{"className":1248},[159,197],[109,1250,206],{"className":1251},[205],[109,1253,1255],{"className":1254},[177],[109,1256,1259],{"className":1257,"style":1258},[181],"height:0.3013em;",[109,1260],{}," (a saving)",[71,1263,1265],{"id":1264},"objective-function","Objective function",[11,1267,1268,1269,1272,1273,1276,1277,1280,1281,1284],{},"The problem has four \"levels\": ",[56,1270,1271],{},"items"," to buy, ",[56,1274,1275],{},"variants"," of each item, ",[56,1278,1279],{},"offers"," for each variant, and finally the ",[56,1282,1283],{},"suppliers"," offering the offers. The cost function has two main terms:",[109,1286,1289],{"className":1287},[1288],"katex-display",[109,1290,1292,1522],{"className":1291},[112],[109,1293,1295],{"className":1294},[116],[118,1296,1298],{"xmlns":120,"display":1297},"block",[122,1299,1300,1519],{},[125,1301,1302,1334,1337,1354,1371,1375,1383,1386,1392,1394,1397,1399,1416,1419,1427,1429,1435,1437,1445,1447,1453,1456,1458,1460,1462,1464,1477,1479,1487,1489,1495,1497,1505,1507,1513,1515],{},[1303,1304,1305,1313],"munder",{},[125,1306,1307,1310],{},[131,1308,1309],{},"min",[652,1311,1312],{},"⁡",[125,1314,1315,1318,1320,1322,1324,1326,1328,1330,1332],{},[131,1316,133],{"mathvariant":1317},"bold-italic",[652,1319,654],{"separator":146},[131,1321,374],{"mathvariant":1317},[652,1323,654],{"separator":146},[131,1325,578],{"mathvariant":1317},[652,1327,654],{"separator":146},[131,1329,859],{"mathvariant":1317},[652,1331,654],{"separator":146},[131,1333,862],{"mathvariant":1317},[275,1335,1336],{},"  ",[1303,1338,1339,1342],{},[652,1340,1341],{},"∑",[125,1343,1344,1347,1350],{},[131,1345,1346],{},"i",[652,1348,1349],{},"∈",[131,1351,1353],{"mathvariant":1352},"script","I",[1303,1355,1356,1358],{},[652,1357,1341],{},[125,1359,1360,1362,1364],{},[131,1361,136],{},[652,1363,1349],{},[128,1365,1366,1369],{},[131,1367,1368],{"mathvariant":1352},"V",[131,1370,1346],{},[652,1372,1374],{"fence":146,"stretchy":146,"minsize":1373,"maxsize":1373},"1.8em","(",[267,1376,1377,1379,1381],{},[131,1378,271],{},[131,1380,136],{},[275,1382,277],{},[275,1384,1385],{}," ",[128,1387,1388,1390],{},[131,1389,133],{},[131,1391,136],{},[275,1393,1336],{},[652,1395,1396],{},"+",[275,1398,1336],{},[1303,1400,1401,1403],{},[652,1402,1341],{},[125,1404,1405,1407,1409],{},[131,1406,377],{},[652,1408,1349],{},[128,1410,1411,1414],{},[131,1412,1413],{"mathvariant":1352},"O",[131,1415,136],{},[652,1417,1374],{"fence":146,"stretchy":146,"minsize":1418,"maxsize":1418},"1.2em",[267,1420,1421,1423,1425],{},[131,1422,271],{},[131,1424,377],{},[275,1426,768],{},[275,1428,1385],{},[128,1430,1431,1433],{},[131,1432,578],{},[131,1434,377],{},[652,1436,1396],{},[267,1438,1439,1441,1443],{},[131,1440,271],{},[131,1442,377],{},[275,1444,486],{},[275,1446,1385],{},[128,1448,1449,1451],{},[131,1450,374],{},[131,1452,377],{},[652,1454,1455],{"fence":146,"stretchy":146,"minsize":1418,"maxsize":1418},")",[652,1457,1455],{"fence":146,"stretchy":146,"minsize":1373,"maxsize":1373},[275,1459,1336],{},[652,1461,1396],{},[275,1463,1336],{},[1303,1465,1466,1468],{},[652,1467,1341],{},[125,1469,1470,1472,1474],{},[131,1471,862],{},[652,1473,1349],{},[131,1475,1476],{"mathvariant":1352},"K",[652,1478,1374],{"fence":146,"stretchy":146,"minsize":1418,"maxsize":1418},[267,1480,1481,1483,1485],{},[131,1482,271],{},[131,1484,862],{},[275,1486,974],{},[275,1488,1385],{},[128,1490,1491,1493],{},[131,1492,859],{},[131,1494,862],{},[652,1496,1174],{},[267,1498,1499,1501,1503],{},[131,1500,271],{},[131,1502,862],{},[275,1504,1183],{},[275,1506,1385],{},[128,1508,1509,1511],{},[131,1510,1067],{},[131,1512,862],{},[652,1514,1455],{"fence":146,"stretchy":146,"minsize":1418,"maxsize":1418},[131,1516,1518],{"mathvariant":1517},"normal",".",[138,1520,1521],{"encoding":140},"\\min_{\\boldsymbol{z}, \\boldsymbol{x}, \\boldsymbol{n}, \\boldsymbol{u}, \\boldsymbol{k}}\\; \\sum_{i \\in \\mathcal{I}}\\sum_{v \\in \\mathcal{V}_i}\\Bigl(C^{\\text{choice}}_v\\, z_v \\;+\\; \\sum_{o \\in \\mathcal{O}_v}\\bigl(C^{\\text{pack}}_o\\, n_o + C^{\\text{use}}_o\\, x_o\\bigr)\\Bigr) \\;+\\; \\sum_{k \\in \\mathcal{K}}\\bigl(C^{\\text{fee}}_k\\, u_k - C^{\\text{ship}}_k\\, f_k\\bigr).",[109,1523,1525,1942,2157,2293,2471],{"className":1524,"ariaHidden":146},[145],[109,1526,1528,1532,1643,1647,1650,1713,1716,1814,1823,1882,1885,1925,1928,1932,1936,1939],{"className":1527},[150],[109,1529],{"className":1530,"style":1531},[154],"height:2.5444em;vertical-align:-1.3944em;",[109,1533,1537],{"className":1534},[1535,1536],"mop","op-limits",[109,1538,1540,1634],{"className":1539},[172,173],[109,1541,1543,1631],{"className":1542},[177],[109,1544,1547,1620],{"className":1545,"style":1546},[181],"height:0.6679em;",[109,1548,1550,1554],{"style":1549},"top:-2.3479em;margin-left:0em;",[109,1551],{"className":1552,"style":1553},[189],"height:3em;",[109,1555,1557],{"className":1556},[194,195,196,197],[109,1558,1560,1571,1574,1583,1586,1595,1598,1607,1610],{"className":1559},[159,197],[109,1561,1563],{"className":1562},[159,197],[109,1564,1566],{"className":1565},[159,197],[109,1567,133],{"className":1568,"style":1570},[159,1569,197],"boldsymbol","margin-right:0.0421em;",[109,1572,654],{"className":1573},[687,197],[109,1575,1577],{"className":1576},[159,197],[109,1578,1580],{"className":1579},[159,197],[109,1581,374],{"className":1582},[159,1569,197],[109,1584,654],{"className":1585},[687,197],[109,1587,1589],{"className":1588},[159,197],[109,1590,1592],{"className":1591},[159,197],[109,1593,578],{"className":1594},[159,1569,197],[109,1596,654],{"className":1597},[687,197],[109,1599,1601],{"className":1600},[159,197],[109,1602,1604],{"className":1603},[159,197],[109,1605,859],{"className":1606},[159,1569,197],[109,1608,654],{"className":1609},[687,197],[109,1611,1613],{"className":1612},[159,197],[109,1614,1616],{"className":1615},[159,197],[109,1617,862],{"className":1618,"style":1619},[159,1569,197],"margin-right:0.0185em;",[109,1621,1623,1626],{"style":1622},"top:-3em;",[109,1624],{"className":1625,"style":1553},[189],[109,1627,1628],{},[109,1629,1309],{"className":1630},[1535],[109,1632,206],{"className":1633},[205],[109,1635,1637],{"className":1636},[177],[109,1638,1641],{"className":1639,"style":1640},[181],"height:0.8882em;",[109,1642],{},[109,1644],{"className":1645,"style":1646},[691],"margin-right:0.2778em;",[109,1648],{"className":1649,"style":692},[691],[109,1651,1653],{"className":1652},[1535,1536],[109,1654,1656,1704],{"className":1655},[172,173],[109,1657,1659,1701],{"className":1658},[177],[109,1660,1663,1688],{"className":1661,"style":1662},[181],"height:1.05em;",[109,1664,1666,1670],{"style":1665},"top:-1.8557em;margin-left:0em;",[109,1667],{"className":1668,"style":1669},[189],"height:3.05em;",[109,1671,1673],{"className":1672},[194,195,196,197],[109,1674,1676,1679,1683],{"className":1675},[159,197],[109,1677,1346],{"className":1678},[159,163,197],[109,1680,1349],{"className":1681},[1682,197],"mrel",[109,1684,1353],{"className":1685,"style":1687},[159,1686,197],"mathcal","margin-right:0.0738em;",[109,1689,1691,1694],{"style":1690},"top:-3.05em;",[109,1692],{"className":1693,"style":1669},[189],[109,1695,1696],{},[109,1697,1341],{"className":1698},[1535,1699,1700],"op-symbol","large-op",[109,1702,206],{"className":1703},[205],[109,1705,1707],{"className":1706},[177],[109,1708,1711],{"className":1709,"style":1710},[181],"height:1.3217em;",[109,1712],{},[109,1714],{"className":1715,"style":692},[691],[109,1717,1719],{"className":1718},[1535,1536],[109,1720,1722,1805],{"className":1721},[172,173],[109,1723,1725,1802],{"className":1724},[177],[109,1726,1728,1792],{"className":1727,"style":1662},[181],[109,1729,1730,1733],{"style":1665},[109,1731],{"className":1732,"style":1669},[189],[109,1734,1736],{"className":1735},[194,195,196,197],[109,1737,1739,1742,1745],{"className":1738},[159,197],[109,1740,136],{"className":1741,"style":201},[159,163,197],[109,1743,1349],{"className":1744},[1682,197],[109,1746,1748,1752],{"className":1747},[159,197],[109,1749,1368],{"className":1750,"style":1751},[159,1686,197],"margin-right:0.0822em;",[109,1753,1755],{"className":1754},[168],[109,1756,1758,1783],{"className":1757},[172,173],[109,1759,1761,1780],{"className":1760},[177],[109,1762,1765],{"className":1763,"style":1764},[181],"height:0.3281em;",[109,1766,1768,1772],{"style":1767},"top:-2.357em;margin-left:-0.0822em;margin-right:0.0714em;",[109,1769],{"className":1770,"style":1771},[189],"height:2.5em;",[109,1773,1777],{"className":1774},[194,1775,1776,197],"reset-size3","size1",[109,1778,1346],{"className":1779},[159,163,197],[109,1781,206],{"className":1782},[205],[109,1784,1786],{"className":1785},[177],[109,1787,1790],{"className":1788,"style":1789},[181],"height:0.143em;",[109,1791],{},[109,1793,1794,1797],{"style":1690},[109,1795],{"className":1796,"style":1669},[189],[109,1798,1799],{},[109,1800,1341],{"className":1801},[1535,1699,1700],[109,1803,206],{"className":1804},[205],[109,1806,1808],{"className":1807},[177],[109,1809,1812],{"className":1810,"style":1811},[181],"height:1.3944em;",[109,1813],{},[109,1815,1818],{"className":1816},[1817],"mopen",[109,1819,1374],{"className":1820}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means that for every item ",[109,2591,2593,2606],{"className":2592},[112],[109,2594,2596],{"className":2595},[116],[118,2597,2598],{"xmlns":120},[122,2599,2600,2604],{},[125,2601,2602],{},[131,2603,1346],{},[138,2605,1346],{"encoding":140},[109,2607,2609],{"className":2608,"ariaHidden":146},[145],[109,2610,2612,2616],{"className":2611},[150],[109,2613],{"className":2614,"style":2615},[154],"height:0.6595em;",[109,2617,1346],{"className":2618},[159,163]," and every variant ",[109,2621,2623,2636],{"className":2622},[112],[109,2624,2626],{"className":2625},[116],[118,2627,2628],{"xmlns":120},[122,2629,2630,2634],{},[125,2631,2632],{},[131,2633,136],{},[138,2635,136],{"encoding":140},[109,2637,2639],{"className":2638,"ariaHidden":146},[145],[109,2640,2642,2645],{"className":2641},[150],[109,2643],{"className":2644,"style":247},[154],[109,2646,136],{"className":2647,"style":201},[159,163]," we pay ",[109,2650,2652,2671],{"className":2651},[112],[109,2653,2655],{"className":2654},[116],[118,2656,2657],{"xmlns":120},[122,2658,2659,2669],{},[125,2660,2661],{},[267,2662,2663,2665,2667],{},[131,2664,271],{},[131,2666,136],{},[275,2668,277],{},[138,2670,280],{"encoding":140},[109,2672,2674],{"className":2673,"ariaHidden":146},[145],[109,2675,2677,2680],{"className":2676},[150],[109,2678],{"className":2679,"style":290},[154],[109,2681,2683,2686],{"className":2682},[159],[109,2684,271],{"className":2685,"style":297},[159,163],[109,2687,2689],{"className":2688},[168],[109,2690,2692,2729],{"className":2691},[172,173],[109,2693,2695,2726],{"className":2694},[177],[109,2696,2698,2709],{"className":2697,"style":310},[181],[109,2699,2700,2703],{"style":313},[109,2701],{"className":2702,"style":190},[189],[109,2704,2706],{"className":2705},[194,195,196,197],[109,2707,136],{"className":2708,"style":201},[159,163,197],[109,2710,2711,2714],{"style":325},[109,2712],{"className":2713,"style":190},[189],[109,2715,2717],{"className":2716},[194,195,196,197],[109,2718,2720],{"className":2719},[159,197],[109,2721,2723],{"className":2722},[159,338,197],[109,2724,277],{"className":2725},[159,197],[109,2727,206],{"className":2728},[205],[109,2730,2732],{"className":2731},[177],[109,2733,2735],{"className":2734,"style":351},[181],[109,2736],{}," if the variant is chosen, and also for every\nused offer ",[109,2739,2741,2754],{"className":2740},[112],[109,2742,2744],{"className":2743},[116],[118,2745,2746],{"xmlns":120},[122,2747,2748,2752],{},[125,2749,2750],{},[131,2751,377],{},[138,2753,377],{"encoding":140},[109,2755,2757],{"className":2756,"ariaHidden":146},[145],[109,2758,2760,2763],{"className":2759},[150],[109,2761],{"className":2762,"style":247},[154],[109,2764,377],{"className":2765},[159,163]," of that variant we pay ",[109,2768,2770,2789],{"className":2769},[112],[109,2771,2773],{"className":2772},[116],[118,2774,2775],{"xmlns":120},[122,2776,2777,2787],{},[125,2778,2779],{},[267,2780,2781,2783,2785],{},[131,2782,271],{},[131,2784,377],{},[275,2786,768],{},[138,2788,771],{"encoding":140},[109,2790,2792],{"className":2791,"ariaHidden":146},[145],[109,2793,2795,2798],{"className":2794},[150],[109,2796],{"className":2797,"style":290},[154],[109,2799,2801,2804],{"className":2800},[159],[109,2802,271],{"className":2803,"style":297},[159,163],[109,2805,2807],{"className":2806},[168],[109,2808,2810,2847],{"className":2809},[172,173],[109,2811,2813,2844],{"className":2812},[177],[109,2814,2816,2827],{"className":2815,"style":310},[181],[109,2817,2818,2821],{"style":313},[109,2819],{"className":2820,"style":190},[189],[109,2822,2824],{"className":2823},[194,195,196,197],[109,2825,377],{"className":2826},[159,163,197],[109,2828,2829,2832],{"style":325},[109,2830],{"className":2831,"style":190},[189],[109,2833,2835],{"className":2834},[194,195,196,197],[109,2836,2838],{"className":2837},[159,197],[109,2839,2841],{"className":2840},[159,338,197],[109,2842,768],{"className":2843},[159,197],[109,2845,206],{"className":2846},[205],[109,2848,2850],{"className":2849},[177],[109,2851,2853],{"className":2852,"style":351},[181],[109,2854],{}," per pack bought as well as the\nfixed cost ",[109,2857,2859,2878],{"className":2858},[112],[109,2860,2862],{"className":2861},[116],[118,2863,2864],{"xmlns":120},[122,2865,2866,2876],{},[125,2867,2868],{},[267,2869,2870,2872,2874],{},[131,2871,271],{},[131,2873,377],{},[275,2875,486],{},[138,2877,489],{"encoding":140},[109,2879,2881],{"className":2880,"ariaHidden":146},[145],[109,2882,2884,2887],{"className":2883},[150],[109,2885],{"className":2886,"style":499},[154],[109,2888,2890,2893],{"className":2889},[159],[109,2891,271],{"className":2892,"style":297},[159,163],[109,2894,2896],{"className":2895},[168],[109,2897,2899,2936],{"className":2898},[172,173],[109,2900,2902,2933],{"className":2901},[177],[109,2903,2905,2916],{"className":2904,"style":518},[181],[109,2906,2907,2910],{"style":313},[109,2908],{"className":2909,"style":190},[189],[109,2911,2913],{"className":2912},[194,195,196,197],[109,2914,377],{"className":2915},[159,163,197],[109,2917,2918,2921],{"style":325},[109,2919],{"className":2920,"style":190},[189],[109,2922,2924],{"className":2923},[194,195,196,197],[109,2925,2927],{"className":2926},[159,197],[109,2928,2930],{"className":2929},[159,338,197],[109,2931,486],{"className":2932},[159,197],[109,2934,206],{"className":2935},[205],[109,2937,2939],{"className":2938},[177],[109,2940,2942],{"className":2941,"style":351},[181],[109,2943],{},". The second sum is over the\nsuppliers: the charge ",[109,2946,2948,2967],{"className":2947},[112],[109,2949,2951],{"className":2950},[116],[118,2952,2953],{"xmlns":120},[122,2954,2955,2965],{},[125,2956,2957],{},[267,2958,2959,2961,2963],{},[131,2960,271],{},[131,2962,862],{},[275,2964,974],{},[138,2966,977],{"encoding":140},[109,2968,2970],{"className":2969,"ariaHidden":146},[145],[109,2971,2973,2976],{"className":2972},[150],[109,2974],{"className":2975,"style":987},[154],[109,2977,2979,2982],{"className":2978},[159],[109,2980,271],{"className":2981,"style":297},[159,163],[109,2983,2985],{"className":2984},[168],[109,2986,2988,3025],{"className":2987},[172,173],[109,2989,2991,3022],{"className":2990},[177],[109,2992,2994,3005],{"className":2993,"style":310},[181],[109,2995,2996,2999],{"style":1008},[109,2997],{"className":2998,"style":190},[189],[109,3000,3002],{"className":3001},[194,195,196,197],[109,3003,862],{"className":3004,"style":905},[159,163,197],[109,3006,3007,3010],{"style":325},[109,3008],{"className":3009,"style":190},[189],[109,3011,3013],{"className":3012},[194,195,196,197],[109,3014,3016],{"className":3015},[159,197],[109,3017,3019],{"className":3018},[159,338,197],[109,3020,974],{"className":3021},[159,197],[109,3023,206],{"className":3024},[205],[109,3026,3028],{"className":3027},[177],[109,3029,3031],{"className":3030,"style":1044},[181],[109,3032],{}," if the supplier is used, less the\nshipping ",[109,3035,3037,3057],{"className":3036},[112],[109,3038,3040],{"className":3039},[116],[118,3041,3042],{"xmlns":120},[122,3043,3044,3054],{},[125,3045,3046],{},[267,3047,3048,3050,3052],{},[131,3049,271],{},[131,3051,862],{},[275,3053,1183],{},[138,3055,3056],{"encoding":140},"C^{\\text{ship}}_k",[109,3058,3060],{"className":3059,"ariaHidden":146},[145],[109,3061,3063,3066],{"className":3062},[150],[109,3064],{"className":3065,"style":1196},[154],[109,3067,3069,3072],{"className":3068},[159],[109,3070,271],{"className":3071,"style":297},[159,163],[109,3073,3075],{"className":3074},[168],[109,3076,3078,3115],{"className":3077},[172,173],[109,3079,3081,3112],{"className":3080},[177],[109,3082,3084,3095],{"className":3083,"style":1218},[181],[109,3085,3086,3089],{"style":1221},[109,3087],{"className":3088,"style":190},[189],[109,3090,3092],{"className":3091},[194,195,196,197],[109,3093,862],{"className":3094,"style":905},[159,163,197],[109,3096,3097,3100],{"style":1233},[109,3098],{"className":3099,"style":190},[189],[109,3101,3103],{"className":3102},[194,195,196,197],[109,3104,3106],{"className":3105},[159,197],[109,3107,3109],{"className":3108},[159,338,197],[109,3110,1183],{"className":3111},[159,197],[109,3113,206],{"className":3114},[205],[109,3116,3118],{"className":3117},[177],[109,3119,3121],{"className":3120,"style":1258},[181],[109,3122],{}," if shipping is free.",[71,3125,3127],{"id":3126},"constraints","Constraints",[11,3129,3130,3131,3167,3168,3203,3204,3239],{},"This problem has many constraints that restrict the decision variables meaningfully. For ",[109,3132,3134,3148],{"className":3133},[112],[109,3135,3137],{"className":3136},[116],[118,3138,3139],{"xmlns":120},[122,3140,3141,3145],{},[125,3142,3143],{},[131,3144,133],{"mathvariant":1317},[138,3146,3147],{"encoding":140},"\\boldsymbol{z}",[109,3149,3151],{"className":3150,"ariaHidden":146},[145],[109,3152,3154,3158],{"className":3153},[150],[109,3155],{"className":3156,"style":3157},[154],"height:0.4444em;",[109,3159,3161],{"className":3160},[159],[109,3162,3164],{"className":3163},[159],[109,3165,133],{"className":3166,"style":1570},[159,1569],", ",[109,3169,3171,3185],{"className":3170},[112],[109,3172,3174],{"className":3173},[116],[118,3175,3176],{"xmlns":120},[122,3177,3178,3182],{},[125,3179,3180],{},[131,3181,374],{"mathvariant":1317},[138,3183,3184],{"encoding":140},"\\boldsymbol{x}",[109,3186,3188],{"className":3187,"ariaHidden":146},[145],[109,3189,3191,3194],{"className":3190},[150],[109,3192],{"className":3193,"style":3157},[154],[109,3195,3197],{"className":3196},[159],[109,3198,3200],{"className":3199},[159],[109,3201,374],{"className":3202},[159,1569]," and ",[109,3205,3207,3221],{"className":3206},[112],[109,3208,3210],{"className":3209},[116],[118,3211,3212],{"xmlns":120},[122,3213,3214,3218],{},[125,3215,3216],{},[131,3217,578],{"mathvariant":1317},[138,3219,3220],{"encoding":140},"\\boldsymbol{n}",[109,3222,3224],{"className":3223,"ariaHidden":146},[145],[109,3225,3227,3230],{"className":3226},[150],[109,3228],{"className":3229,"style":3157},[154],[109,3231,3233],{"className":3232},[159],[109,3234,3236],{"className":3235},[159],[109,3237,578],{"className":3238},[159,1569],", we have the following five:",[79,3241,3242,3260],{},[82,3243,3244],{},[85,3245,3246,3249,3251,3254,3257],{},[88,3247,3248],{},"row",[88,3250],{},[88,3252,3253],{},"for every",[88,3255,3256],{},"known as",[88,3258,3259],{},"in words",[101,3261,3262,3691,4990,5256,5523],{},[85,3263,3264,3267,3630,3685,3688],{},[106,3265,3266],{},"(1)",[106,3268,3269],{},[109,3270,3272,3333],{"className":3271},[112],[109,3273,3275],{"className":3274},[116],[118,3276,3277],{"xmlns":120},[122,3278,3279,3330],{},[125,3280,3281,3292,3294,3300,3303,3309,3311,3322,3324],{},[267,3282,3283,3286,3288,3290],{},[131,3284,3285],{},"N",[131,3287,377],{},[131,3289,1309],{},[652,3291,1312],{},[275,3293,1385],{},[128,3295,3296,3298],{},[131,3297,374],{},[131,3299,377],{},[652,3301,3302],{},"≤",[128,3304,3305,3307],{},[131,3306,578],{},[131,3308,377],{},[652,3310,3302],{},[267,3312,3313,3315,3317,3320],{},[131,3314,3285],{},[131,3316,377],{},[131,3318,3319],{},"max",[652,3321,1312],{},[275,3323,1385],{},[128,3325,3326,3328],{},[131,3327,374],{},[131,3329,377],{},[138,3331,3332],{"encoding":140},"N^{\\min}_o\\, x_o \\le n_o \\le N^{\\max}_o\\, x_o",[109,3334,3336,3462,3518],{"className":3335,"ariaHidden":146},[145],[109,3337,3339,3343,3410,3413,3453,3456,3459],{"className":3338},[150],[109,3340],{"className":3341,"style":3342},[154],"height:1.0775em;vertical-align:-0.247em;",[109,3344,3346,3350],{"className":3345},[159],[109,3347,3285],{"className":3348,"style":3349},[159,163],"margin-right:0.109em;",[109,3351,3353],{"className":3352},[168],[109,3354,3356,3402],{"className":3355},[172,173],[109,3357,3359,3399],{"className":3358},[177],[109,3360,3363,3375],{"className":3361,"style":3362},[181],"height:0.8305em;",[109,3364,3366,3369],{"style":3365},"top:-2.453em;margin-left:-0.109em;margin-right:0.05em;",[109,3367],{"className":3368,"style":190},[189],[109,3370,3372],{"className":3371},[194,195,196,197],[109,3373,377],{"className":3374},[159,163,197],[109,3376,3377,3380],{"style":325},[109,3378],{"className":3379,"style":190},[189],[109,3381,3383],{"className":3382},[194,195,196,197],[109,3384,3386],{"className":3385},[159,197],[109,3387,3389,3393,3396],{"className":3388},[1535,197],[109,3390,3392],{"className":3391},[197],"m",[109,3394,1346],{"className":3395},[197],[109,3397,578],{"className":3398},[197],[109,3400,206],{"className":3401},[205],[109,3403,3405],{"className":3404},[177],[109,3406,3408],{"className":3407,"style":351},[181],[109,3409],{},[109,3411],{"className":3412,"style":692},[691],[109,3414,3416,3419],{"className":3415},[159],[109,3417,374],{"className":3418},[159,163],[109,3420,3422],{"className":3421},[168],[109,3423,3425,3445],{"className":3424},[172,173],[109,3426,3428,3442],{"className":3427},[177],[109,3429,3431],{"className":3430,"style":182},[181],[109,3432,3433,3436],{"style":410},[109,3434],{"className":3435,"style":190},[189],[109,3437,3439],{"className":3438},[194,195,196,197],[109,3440,377],{"className":3441},[159,163,197],[109,3443,206],{"className":3444},[205],[109,3446,3448],{"className":3447},[177],[109,3449,3451],{"className":3450,"style":213},[181],[109,3452],{},[109,3454],{"className":3455,"style":1646},[691],[109,3457,3302],{"className":3458},[1682],[109,3460],{"className":3461,"style":1646},[691],[109,3463,3465,3469,3509,3512,3515],{"className":3464},[150],[109,3466],{"className":3467,"style":3468},[154],"height:0.786em;vertical-align:-0.15em;",[109,3470,3472,3475],{"className":3471},[159],[109,3473,578],{"className":3474},[159,163],[109,3476,3478],{"className":3477},[168],[109,3479,3481,3501],{"className":3480},[172,173],[109,3482,3484,3498],{"className":3483},[177],[109,3485,3487],{"className":3486,"style":182},[181],[109,3488,3489,3492],{"style":410},[109,3490],{"className":3491,"style":190},[189],[109,3493,3495],{"className":3494},[194,195,196,197],[109,3496,377],{"className":3497},[159,163,197],[109,3499,206],{"className":3500},[205],[109,3502,3504],{"className":3503},[177],[109,3505,3507],{"className":3506,"style":213},[181],[109,3508],{},[109,3510],{"className":3511,"style":1646},[691],[109,3513,3302],{"className":3514},[1682],[109,3516],{"className":3517,"style":1646},[691],[109,3519,3521,3524,3587,3590],{"className":3520},[150],[109,3522],{"className":3523,"style":499},[154],[109,3525,3527,3530],{"className":3526},[159],[109,3528,3285],{"className":3529,"style":3349},[159,163],[109,3531,3533],{"className":3532},[168],[109,3534,3536,3579],{"className":3535},[172,173],[109,3537,3539,3576],{"className":3538},[177],[109,3540,3542,3553],{"className":3541,"style":518},[181],[109,3543,3544,3547],{"style":3365},[109,3545],{"className":3546,"style":190},[189],[109,3548,3550],{"className":3549},[194,195,196,197],[109,3551,377],{"className":3552},[159,163,197],[109,3554,3555,3558],{"style":325},[109,3556],{"className":3557,"style":190},[189],[109,3559,3561],{"className":3560},[194,195,196,197],[109,3562,3564],{"className":3563},[159,197],[109,3565,3567,3570,3573],{"className":3566},[1535,197],[109,3568,3392],{"className":3569},[197],[109,3571,24],{"className":3572},[197],[109,3574,374],{"className":3575},[197],[109,3577,206],{"className":3578},[205],[109,3580,3582],{"className":3581},[177],[109,3583,3585],{"className":3584,"style":351},[181],[109,3586],{},[109,3588],{"className":3589,"style":692},[691],[109,3591,3593,3596],{"className":3592},[159],[109,3594,374],{"className":3595},[159,163],[109,3597,3599],{"className":3598},[168],[109,3600,3602,3622],{"className":3601},[172,173],[109,3603,3605,3619],{"className":3604},[177],[109,3606,3608],{"className":3607,"style":182},[181],[109,3609,3610,3613],{"style":410},[109,3611],{"className":3612,"style":190},[189],[109,3614,3616],{"className":3615},[194,195,196,197],[109,3617,377],{"className":3618},[159,163,197],[109,3620,206],{"className":3621},[205],[109,3623,3625],{"className":3624},[177],[109,3626,3628],{"className":3627,"style":213},[181],[109,3629],{},[106,3631,435,3632],{},[109,3633,3635,3653],{"className":3634},[112],[109,3636,3638],{"className":3637},[116],[118,3639,3640],{"xmlns":120},[122,3641,3642,3650],{},[125,3643,3644,3646,3648],{},[131,3645,377],{},[652,3647,1349],{},[131,3649,1413],{"mathvariant":1352},[138,3651,3652],{"encoding":140},"o 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chosen variant must be covered. Each offer yields ",[109,4353,4355,4375],{"className":4354},[112],[109,4356,4358],{"className":4357},[116],[118,4359,4360],{"xmlns":120},[122,4361,4362,4372],{},[125,4363,4364],{},[267,4365,4366,4368,4370],{},[131,4367,3732],{},[131,4369,377],{},[275,4371,768],{},[138,4373,4374],{"encoding":140},"Q^{\\text{pack}}_o",[109,4376,4378],{"className":4377,"ariaHidden":146},[145],[109,4379,4381,4384],{"className":4380},[150],[109,4382],{"className":4383,"style":290},[154],[109,4385,4387,4390],{"className":4386},[159],[109,4388,3732],{"className":4389},[159,163],[109,4391,4393],{"className":4392},[168],[109,4394,4396,4433],{"className":4395},[172,173],[109,4397,4399,4430],{"className":4398},[177],[109,4400,4402,4413],{"className":4401,"style":310},[181],[109,4403,4404,4407],{"style":3912},[109,4405],{"className":4406,"style":190},[189],[109,4408,4410],{"className":4409},[194,195,196,197],[109,4411,377],{"className":4412},[159,163,197],[109,4414,4415,4418],{"style":325},[109,4416],{"className":4417,"style":190},[189],[109,4419,4421],{"className":4420},[194,195,196,197],[109,4422,4424],{"className":4423},[159,197],[109,4425,4427],{"className":4426},[159,338,197],[109,4428,768],{"className":4429},[159,197],[109,4431,206],{"className":4432},[205],[109,4434,4436],{"className":4435},[177],[109,4437,4439],{"className":4438,"style":351},[181],[109,4440],{}," units and loses ",[109,4443,4445,4465],{"className":4444},[112],[109,4446,4448],{"className":4447},[116],[118,4449,4450],{"xmlns":120},[122,4451,4452,4462],{},[125,4453,4454],{},[267,4455,4456,4458,4460],{},[131,4457,3732],{},[131,4459,377],{},[275,4461,3753],{},[138,4463,4464],{"encoding":140},"Q^{\\text{lost}}_o",[109,4466,4468],{"className":4467,"ariaHidden":146},[145],[109,4469,4471,4474],{"className":4470},[150],[109,4472],{"className":4473,"style":290},[154],[109,4475,4477,4480],{"className":4476},[159],[109,4478,3732],{"className":4479},[159,163],[109,4481,4483],{"className":4482},[168],[109,4484,4486,4523],{"className":4485},[172,173],[109,4487,4489,4520],{"className":4488},[177],[109,4490,4492,4503],{"className":4491,"style":310},[181],[109,4493,4494,4497],{"style":3912},[109,4495],{"className":4496,"style":190},[189],[109,4498,4500],{"className":4499},[194,195,196,197],[109,4501,377],{"className":4502},[159,163,197],[109,4504,4505,4508],{"style":325},[109,4506],{"className":4507,"style":190},[189],[109,4509,4511],{"className":4510},[194,195,196,197],[109,4512,4514],{"className":4513},[159,197],[109,4515,4517],{"className":4516},[159,338,197],[109,4518,3753],{"className":4519},[159,197],[109,4521,206],{"className":4522},[205],[109,4524,4526],{"className":4525},[177],[109,4527,4529],{"className":4528,"style":351},[181],[109,4530],{},". For example, if ",[109,4533,4535,4561],{"className":4534},[112],[109,4536,4538],{"className":4537},[116],[118,4539,4540],{"xmlns":120},[122,4541,4542,4558],{},[125,4543,4544,4552,4555],{},[267,4545,4546,4548,4550],{},[131,4547,3732],{},[131,4549,136],{},[275,4551,3773],{},[652,4553,4554],{},"=",[648,4556,4557],{},"4",[138,4559,4560],{"encoding":140},"Q^{\\text{need}}_v = 4",[109,4562,4564,4636],{"className":4563,"ariaHidden":146},[145],[109,4565,4567,4570,4627,4630,4633],{"className":4566},[150],[109,4568],{"className":4569,"style":290},[154],[109,4571,4573,4576],{"className":4572},[159],[109,4574,3732],{"className":4575},[159,163],[109,4577,4579],{"className":4578},[168],[109,4580,4582,4619],{"className":4581},[172,173],[109,4583,4585,4616],{"className":4584},[177],[109,4586,4588,4599],{"className":4587,"style":310},[181],[109,4589,4590,4593],{"style":3912},[109,4591],{"className":4592,"style":190},[189],[109,4594,4596],{"className":4595},[194,195,196,197],[109,4597,136],{"className":4598,"style":201},[159,163,197],[109,4600,4601,4604],{"style":325},[109,4602],{"className":4603,"style":190},[189],[109,4605,4607],{"className":4606},[194,195,196,197],[109,4608,4610],{"className":4609},[159,197],[109,4611,4613],{"className":4612},[159,338,197],[109,4614,3773],{"className":4615},[159,197],[109,4617,206],{"className":4618},[205],[109,4620,4622],{"className":4621},[177],[109,4623,4625],{"className":4624,"style":351},[181],[109,4626],{},[109,4628],{"className":4629,"style":1646},[691],[109,4631,4554],{"className":4632},[1682],[109,4634],{"className":4635,"style":1646},[691],[109,4637,4639,4643],{"className":4638},[150],[109,4640],{"className":4641,"style":4642},[154],"height:0.6444em;",[109,4644,4557],{"className":4645},[159],"  and  ",[109,4648,4650,4674],{"className":4649},[112],[109,4651,4653],{"className":4652},[116],[118,4654,4655],{"xmlns":120},[122,4656,4657,4671],{},[125,4658,4659,4667,4669],{},[267,4660,4661,4663,4665],{},[131,4662,3732],{},[131,4664,377],{},[275,4666,3753],{},[652,4668,4554],{},[648,4670,662],{},[138,4672,4673],{"encoding":140},"Q^{\\text{lost}}_o = 2",[109,4675,4677,4749],{"className":4676,"ariaHidden":146},[145],[109,4678,4680,4683,4740,4743,4746],{"className":4679},[150],[109,4681],{"className":4682,"style":290},[154],[109,4684,4686,4689],{"className":4685},[159],[109,4687,3732],{"className":4688},[159,163],[109,4690,4692],{"className":4691},[168],[109,4693,4695,4732],{"className":4694},[172,173],[109,4696,4698,4729],{"className":4697},[177],[109,4699,4701,4712],{"className":4700,"style":310},[181],[109,4702,4703,4706],{"style":3912},[109,4704],{"className":4705,"style":190},[189],[109,4707,4709],{"className":4708},[194,195,196,197],[109,4710,377],{"className":4711},[159,163,197],[109,4713,4714,4717],{"style":325},[109,4715],{"className":4716,"style":190},[189],[109,4718,4720],{"className":4719},[194,195,196,197],[109,4721,4723],{"className":4722},[159,197],[109,4724,4726],{"className":4725},[159,338,197],[109,4727,3753],{"className":4728},[159,197],[109,4730,206],{"className":4731},[205],[109,4733,4735],{"className":4734},[177],[109,4736,4738],{"className":4737,"style":351},[181],[109,4739],{},[109,4741],{"className":4742,"style":1646},[691],[109,4744,4554],{"className":4745},[1682],[109,4747],{"className":4748,"style":1646},[691],[109,4750,4752,4755],{"className":4751},[150],[109,4753],{"className":4754,"style":4642},[154],[109,4756,662],{"className":4757},[159],", the row reads ",[109,4760,4762,4794],{"className":4761},[112],[109,4763,4765],{"className":4764},[116],[118,4766,4767],{"xmlns":120},[122,4768,4769,4791],{},[125,4770,4771,4777,4779,4781,4787,4789],{},[128,4772,4773,4775],{},[131,4774,578],{},[131,4776,377],{},[652,4778,1174],{},[648,4780,662],{},[128,4782,4783,4785],{},[131,4784,374],{},[131,4786,377],{},[652,4788,3764],{},[648,4790,4557],{},[138,4792,4793],{"encoding":140},"n_o - 2x_o \\ge 4",[109,4795,4797,4853,4912],{"className":4796,"ariaHidden":146},[145],[109,4798,4800,4804,4844,4847,4850],{"className":4799},[150],[109,4801],{"className":4802,"style":4803},[154],"height:0.7333em;vertical-align:-0.15em;",[109,4805,4807,4810],{"className":4806},[159],[109,4808,578],{"className":4809},[159,163],[109,4811,4813],{"className":4812},[168],[109,4814,4816,4836],{"className":4815},[172,173],[109,4817,4819,4833],{"className":4818},[177],[109,4820,4822],{"className":4821,"style":182},[181],[109,4823,4824,4827],{"style":410},[109,4825],{"className":4826,"style":190},[189],[109,4828,4830],{"className":4829},[194,195,196,197],[109,4831,377],{"className":4832},[159,163,197],[109,4834,206],{"className":4835},[205],[109,4837,4839],{"className":4838},[177],[109,4840,4842],{"className":4841,"style":213},[181],[109,4843],{},[109,4845],{"className":4846,"style":1931},[691],[109,4848,1174],{"className":4849},[1935],[109,4851],{"className":4852,"style":1931},[691],[109,4854,4856,4860,4863,4903,4906,4909],{"className":4855},[150],[109,4857],{"className":4858,"style":4859},[154],"height:0.7944em;vertical-align:-0.15em;",[109,4861,662],{"className":4862},[159],[109,4864,4866,4869],{"className":4865},[159],[109,4867,374],{"className":4868},[159,163],[109,4870,4872],{"className":4871},[168],[109,4873,4875,4895],{"className":4874},[172,173],[109,4876,4878,4892],{"className":4877},[177],[109,4879,4881],{"className":4880,"style":182},[181],[109,4882,4883,4886],{"style":410},[109,4884],{"className":4885,"style":190},[189],[109,4887,4889],{"className":4888},[194,195,196,197],[109,4890,377],{"className":4891},[159,163,197],[109,4893,206],{"className":4894},[205],[109,4896,4898],{"className":4897},[177],[109,4899,4901],{"className":4900,"style":213},[181],[109,4902],{},[109,4904],{"className":4905,"style":1646},[691],[109,4907,3764],{"className":4908},[1682],[109,4910],{"className":4911,"style":1646},[691],[109,4913,4915,4918],{"className":4914},[150],[109,4916],{"className":4917,"style":4642},[154],[109,4919,4557],{"className":4920},[159],", so 6 packs are needed. A second offer used would lose its ",[109,4923,4925,4944],{"className":4924},[112],[109,4926,4928],{"className":4927},[116],[118,4929,4930],{"xmlns":120},[122,4931,4932,4941],{},[125,4933,4934],{},[4935,4936,4937,4939],"msup",{},[131,4938,3732],{},[275,4940,3753],{},[138,4942,4943],{"encoding":140},"Q^{\\text{lost}}",[109,4945,4947],{"className":4946,"ariaHidden":146},[145],[109,4948,4950,4954],{"className":4949},[150],[109,4951],{"className":4952,"style":4953},[154],"height:1.0435em;vertical-align:-0.1944em;",[109,4955,4957,4960],{"className":4956},[159],[109,4958,3732],{"className":4959},[159,163],[109,4961,4963],{"className":4962},[168],[109,4964,4966],{"className":4965},[172],[109,4967,4969],{"className":4968},[177],[109,4970,4972],{"className":4971,"style":310},[181],[109,4973,4974,4977],{"style":325},[109,4975],{"className":4976,"style":190},[189],[109,4978,4980],{"className":4979},[194,195,196,197],[109,4981,4983],{"className":4982},[159,197],[109,4984,4986],{"className":4985},[159,338,197],[109,4987,3753],{"className":4988},[159,197]," as well. For a variant not chosen, the right-hand side is 0, so buying nothing is allowed.",[85,4991,4992,4995,5195,5250,5253],{},[106,4993,4994],{},"(3)",[106,4996,4997],{},[109,4998,5000,5038],{"className":4999},[112],[109,5001,5003],{"className":5002},[116],[118,5004,5005],{"xmlns":120},[122,5006,5007,5035],{},[125,5008,5009,5025,5031,5033],{},[128,5010,5011,5013],{},[652,5012,1341],{},[125,5014,5015,5017,5019],{},[131,5016,136],{},[652,5018,1349],{},[128,5020,5021,5023],{},[131,5022,1368],{"mathvariant":1352},[131,5024,1346],{},[128,5026,5027,5029],{},[131,5028,133],{},[131,5030,136],{},[652,5032,4554],{},[648,5034,657],{},[138,5036,5037],{"encoding":140},"\\sum_{v \\in \\mathcal{V}_i} z_v = 1",[109,5039,5041,5186],{"className":5040,"ariaHidden":146},[145],[109,5042,5044,5048,5134,5137,5177,5180,5183],{"className":5043},[150],[109,5045],{"className":5046,"style":5047},[154],"height:1.1498em;vertical-align:-0.3998em;",[109,5049,5051,5054],{"className":5050},[1535],[109,5052,1341],{"className":5053,"style":3802},[1535,1699,3801],[109,5055,5057],{"className":5056},[168],[109,5058,5060,5126],{"className":5059},[172,173],[109,5061,5063,5123],{"className":5062},[177],[109,5064,5066],{"className":5065,"style":3815},[181],[109,5067,5068,5071],{"style":3818},[109,5069],{"className":5070,"style":190},[189],[109,5072,5074],{"className":5073},[194,195,196,197],[109,5075,5077,5080,5083],{"className":5076},[159,197],[109,5078,136],{"className":5079,"style":201},[159,163,197],[109,5081,1349],{"className":5082},[1682,197],[109,5084,5086,5089],{"className":5085},[159,197],[109,5087,1368],{"className":5088,"style":1751},[159,1686,197],[109,5090,5092],{"className":5091},[168],[109,5093,5095,5115],{"className":5094},[172,173],[109,5096,5098,5112],{"className":5097},[177],[109,5099,5101],{"className":5100,"style":1764},[181],[109,5102,5103,5106],{"style":1767},[109,5104],{"className":5105,"style":1771},[189],[109,5107,5109],{"className":5108},[194,1775,1776,197],[109,5110,1346],{"className":5111},[159,163,197],[109,5113,206],{"className":5114},[205],[109,5116,5118],{"className":5117},[177],[109,5119,5121],{"className":5120,"style":1789},[181],[109,5122],{},[109,5124,206],{"className":5125},[205],[109,5127,5129],{"className":5128},[177],[109,5130,5132],{"className":5131,"style":3883},[181],[109,5133],{},[109,5135],{"className":5136,"style":692},[691],[109,5138,5140,5143],{"className":5139},[159],[109,5141,133],{"className":5142,"style":164},[159,163],[109,5144,5146],{"className":5145},[168],[109,5147,5149,5169],{"className":5148},[172,173],[109,5150,5152,5166],{"className":5151},[177],[109,5153,5155],{"className":5154,"style":182},[181],[109,5156,5157,5160],{"style":185},[109,5158],{"className":5159,"style":190},[189],[109,5161,5163],{"className":5162},[194,195,196,197],[109,5164,136],{"className":5165,"style":201},[159,163,197],[109,5167,206],{"className":5168},[205],[109,5170,5172],{"className":5171},[177],[109,5173,5175],{"className":5174,"style":213},[181],[109,5176],{},[109,5178],{"className":5179,"style":1646},[691],[109,5181,4554],{"className":5182},[1682],[109,5184],{"className":5185,"style":1646},[691],[109,5187,5189,5192],{"className":5188},[150],[109,5190],{"className":5191,"style":4642},[154],[109,5193,657],{"className":5194},[159],[106,5196,5197,5198],{},"item ",[109,5199,5201,5219],{"className":5200},[112],[109,5202,5204],{"className":5203},[116],[118,5205,5206],{"xmlns":120},[122,5207,5208,5216],{},[125,5209,5210,5212,5214],{},[131,5211,1346],{},[652,5213,1349],{},[131,5215,1353],{"mathvariant":1352},[138,5217,5218],{"encoding":140},"i \\in \\mathcal{I}",[109,5220,5222,5241],{"className":5221,"ariaHidden":146},[145],[109,5223,5225,5229,5232,5235,5238],{"className":5224},[150],[109,5226],{"className":5227,"style":5228},[154],"height:0.6986em;vertical-align:-0.0391em;",[109,5230,1346],{"className":5231},[159,163],[109,5233],{"className":5234,"style":1646},[691],[109,5236,1349],{"className":5237},[1682],[109,5239],{"className":5240,"style":1646},[691],[109,5242,5244,5247],{"className":5243},[150],[109,5245],{"className":5246,"style":3681},[154],[109,5248,1353],{"className":5249,"style":1687},[159,1686],[106,5251,5252],{},"exactly-one row (assignment, set partitioning)",[106,5254,5255],{},"Each item has exactly one variant chosen.",[85,5257,5258,5261,5461,5517,5520],{},[106,5259,5260],{},"(4)",[106,5262,5263],{},[109,5264,5266,5305],{"className":5265},[112],[109,5267,5269],{"className":5268},[116],[118,5270,5271],{"xmlns":120},[122,5272,5273,5302],{},[125,5274,5275,5292,5298,5300],{},[128,5276,5277,5279],{},[652,5278,1341],{},[125,5280,5281,5283,5285],{},[131,5282,377],{},[652,5284,1349],{},[128,5286,5287,5289],{},[131,5288,1413],{"mathvariant":1352},[131,5290,5291],{},"e",[128,5293,5294,5296],{},[131,5295,374],{},[131,5297,377],{},[652,5299,3302],{},[648,5301,657],{},[138,5303,5304],{"encoding":140},"\\sum_{o \\in \\mathcal{O}_e} x_o \\le 1",[109,5306,5308,5452],{"className":5307,"ariaHidden":146},[145],[109,5309,5311,5314,5400,5403,5443,5446,5449],{"className":5310},[150],[109,5312],{"className":5313,"style":5047},[154],[109,5315,5317,5320],{"className":5316},[1535],[109,5318,1341],{"className":5319,"style":3802},[1535,1699,3801],[109,5321,5323],{"className":5322},[168],[109,5324,5326,5392],{"className":5325},[172,173],[109,5327,5329,5389],{"className":5328},[177],[109,5330,5332],{"className":5331,"style":3815},[181],[109,5333,5334,5337],{"style":3818},[109,5335],{"className":5336,"style":190},[189],[109,5338,5340],{"className":5339},[194,195,196,197],[109,5341,5343,5346,5349],{"className":5342},[159,197],[109,5344,377],{"className":5345},[159,163,197],[109,5347,1349],{"className":5348},[1682,197],[109,5350,5352,5355],{"className":5351},[159,197],[109,5353,1413],{"className":5354,"style":1984},[159,1686,197],[109,5356,5358],{"className":5357},[168],[109,5359,5361,5381],{"className":5360},[172,173],[109,5362,5364,5378],{"className":5363},[177],[109,5365,5367],{"className":5366,"style":1997},[181],[109,5368,5369,5372],{"style":2000},[109,5370],{"className":5371,"style":1771},[189],[109,5373,5375],{"className":5374},[194,1775,1776,197],[109,5376,5291],{"className":5377},[159,163,197],[109,5379,206],{"className":5380},[205],[109,5382,5384],{"className":5383},[177],[109,5385,5387],{"className":5386,"style":1789},[181],[109,5388],{},[109,5390,206],{"className":5391},[205],[109,5393,5395],{"className":5394},[177],[109,5396,5398],{"className":5397,"style":3883},[181],[109,5399],{},[109,5401],{"className":5402,"style":692},[691],[109,5404,5406,5409],{"className":5405},[159],[109,5407,374],{"className":5408},[159,163],[109,5410,5412],{"className":5411},[168],[109,5413,5415,5435],{"className":5414},[172,173],[109,5416,5418,5432],{"className":5417},[177],[109,5419,5421],{"className":5420,"style":182},[181],[109,5422,5423,5426],{"style":410},[109,5424],{"className":5425,"style":190},[189],[109,5427,5429],{"className":5428},[194,195,196,197],[109,5430,377],{"className":5431},[159,163,197],[109,5433,206],{"className":5434},[205],[109,5436,5438],{"className":5437},[177],[109,5439,5441],{"className":5440,"style":213},[181],[109,5442],{},[109,5444],{"className":5445,"style":1646},[691],[109,5447,3302],{"className":5448},[1682],[109,5450],{"className":5451,"style":1646},[691],[109,5453,5455,5458],{"className":5454},[150],[109,5456],{"className":5457,"style":4642},[154],[109,5459,657],{"className":5460},[159],[106,5462,5463,5464],{},"exclusive set ",[109,5465,5467,5486],{"className":5466},[112],[109,5468,5470],{"className":5469},[116],[118,5471,5472],{"xmlns":120},[122,5473,5474,5483],{},[125,5475,5476,5478,5480],{},[131,5477,5291],{},[652,5479,1349],{},[131,5481,5482],{"mathvariant":1352},"E",[138,5484,5485],{"encoding":140},"e \\in \\mathcal{E}",[109,5487,5489,5507],{"className":5488,"ariaHidden":146},[145],[109,5490,5492,5495,5498,5501,5504],{"className":5491},[150],[109,5493],{"className":5494,"style":3662},[154],[109,5496,5291],{"className":5497},[159,163],[109,5499],{"className":5500,"style":1646},[691],[109,5502,1349],{"className":5503},[1682],[109,5505],{"className":5506,"style":1646},[691],[109,5508,5510,5513],{"className":5509},[150],[109,5511],{"className":5512,"style":3681},[154],[109,5514,5482],{"className":5515,"style":5516},[159,1686],"margin-right:0.0894em;",[106,5518,5519],{},"at-most-one row (set packing)",[106,5521,5522],{},"At most one offer of an exclusive set is used. For example, a product's prices for 1–9 packs and 10–99 packs cannot be mixed.",[85,5524,5525,5528,5860,5915,5918],{},[106,5526,5527],{},"(5)",[106,5529,5530],{},[109,5531,5533,5589],{"className":5532},[112],[109,5534,5536],{"className":5535},[116],[118,5537,5538],{"xmlns":120},[122,5539,5540,5586],{},[125,5541,5542,5559,5567,5569,5575,5577],{},[128,5543,5544,5546],{},[652,5545,1341],{},[125,5547,5548,5550,5552],{},[131,5549,377],{},[652,5551,1349],{},[128,5553,5554,5556],{},[131,5555,1413],{"mathvariant":1352},[131,5557,5558],{},"r",[267,5560,5561,5563,5565],{},[131,5562,3732],{},[131,5564,377],{},[275,5566,768],{},[275,5568,1385],{},[128,5570,5571,5573],{},[131,5572,578],{},[131,5574,377],{},[652,5576,3302],{},[267,5578,5579,5581,5583],{},[131,5580,3732],{},[131,5582,5558],{},[275,5584,5585],{},"stock",[138,5587,5588],{"encoding":140},"\\sum_{o \\in \\mathcal{O}_r} Q^{\\text{pack}}_o\\, n_o \\le Q^{\\text{stock}}_r",[109,5590,5592,5797],{"className":5591,"ariaHidden":146},[145],[109,5593,5595,5599,5685,5688,5745,5748,5788,5791,5794],{"className":5594},[150],[109,5596],{"className":5597,"style":5598},[154],"height:1.2489em;vertical-align:-0.3998em;",[109,5600,5602,5605],{"className":5601},[1535],[109,5603,1341],{"className":5604,"style":3802},[1535,1699,3801],[109,5606,5608],{"className":5607},[168],[109,5609,5611,5677],{"className":5610},[172,173],[109,5612,5614,5674],{"className":5613},[177],[109,5615,5617],{"className":5616,"style":3815},[181],[109,5618,5619,5622],{"style":3818},[109,5620],{"className":5621,"style":190},[189],[109,5623,5625],{"className":5624},[194,195,196,197],[109,5626,5628,5631,5634],{"className":5627},[159,197],[109,5629,377],{"className":5630},[159,163,197],[109,5632,1349],{"className":5633},[1682,197],[109,5635,5637,5640],{"className":5636},[159,197],[109,5638,1413],{"className":5639,"style":1984},[159,1686,197],[109,5641,5643],{"className":5642},[168],[109,5644,5646,5666],{"className":5645},[172,173],[109,5647,5649,5663],{"className":5648},[177],[109,5650,5652],{"className":5651,"style":1997},[181],[109,5653,5654,5657],{"style":2000},[109,5655],{"className":5656,"style":1771},[189],[109,5658,5660],{"className":5659},[194,1775,1776,197],[109,5661,5558],{"className":5662,"style":1984},[159,163,197],[109,5664,206],{"className":5665},[205],[109,5667,5669],{"className":5668},[177],[109,5670,5672],{"className":5671,"style":1789},[181],[109,5673],{},[109,5675,206],{"className":5676},[205],[109,5678,5680],{"className":5679},[177],[109,5681,5683],{"className":5682,"style":3883},[181],[109,5684],{},[109,5686],{"className":5687,"style":692},[691],[109,5689,5691,5694],{"className":5690},[159],[109,5692,3732],{"className":5693},[159,163],[109,5695,5697],{"className":5696},[168],[109,5698,5700,5737],{"className":5699},[172,173],[109,5701,5703,5734],{"className":5702},[177],[109,5704,5706,5717],{"className":5705,"style":310},[181],[109,5707,5708,5711],{"style":3912},[109,5709],{"className":5710,"style":190},[189],[109,5712,5714],{"className":5713},[194,195,196,197],[109,5715,377],{"className":5716},[159,163,197],[109,5718,5719,5722],{"style":325},[109,5720],{"className":5721,"style":190},[189],[109,5723,5725],{"className":5724},[194,195,196,197],[109,5726,5728],{"className":5727},[159,197],[109,5729,5731],{"className":5730},[159,338,197],[109,5732,768],{"className":5733},[159,197],[109,5735,206],{"className":5736},[205],[109,5738,5740],{"className":5739},[177],[109,5741,5743],{"className":5742,"style":351},[181],[109,5744],{},[109,5746],{"className":5747,"style":692},[691],[109,5749,5751,5754],{"className":5750},[159],[109,5752,578],{"className":5753},[159,163],[109,5755,5757],{"className":5756},[168],[109,5758,5760,5780],{"className":5759},[172,173],[109,5761,5763,5777],{"className":5762},[177],[109,5764,5766],{"className":5765,"style":182},[181],[109,5767,5768,5771],{"style":410},[109,5769],{"className":5770,"style":190},[189],[109,5772,5774],{"className":5773},[194,195,196,197],[109,5775,377],{"className":5776},[159,163,197],[109,5778,206],{"className":5779},[205],[109,5781,5783],{"className":5782},[177],[109,5784,5786],{"className":5785,"style":213},[181],[109,5787],{},[109,5789],{"className":5790,"style":1646},[691],[109,5792,3302],{"className":5793},[1682],[109,5795],{"className":5796,"style":1646},[691],[109,5798,5800,5803],{"className":5799},[150],[109,5801],{"className":5802,"style":290},[154],[109,5804,5806,5809],{"className":5805},[159],[109,5807,3732],{"className":5808},[159,163],[109,5810,5812],{"className":5811},[168],[109,5813,5815,5852],{"className":5814},[172,173],[109,5816,5818,5849],{"className":5817},[177],[109,5819,5821,5832],{"className":5820,"style":310},[181],[109,5822,5823,5826],{"style":3912},[109,5824],{"className":5825,"style":190},[189],[109,5827,5829],{"className":5828},[194,195,196,197],[109,5830,5558],{"className":5831,"style":1984},[159,163,197],[109,5833,5834,5837],{"style":325},[109,5835],{"className":5836,"style":190},[189],[109,5838,5840],{"className":5839},[194,195,196,197],[109,5841,5843],{"className":5842},[159,197],[109,5844,5846],{"className":5845},[159,338,197],[109,5847,5585],{"className":5848},[159,197],[109,5850,206],{"className":5851},[205],[109,5853,5855],{"className":5854},[177],[109,5856,5858],{"className":5857,"style":351},[181],[109,5859],{},[106,5861,5862,5863],{},"stock set ",[109,5864,5866,5885],{"className":5865},[112],[109,5867,5869],{"className":5868},[116],[118,5870,5871],{"xmlns":120},[122,5872,5873,5882],{},[125,5874,5875,5877,5879],{},[131,5876,5558],{},[652,5878,1349],{},[131,5880,5881],{"mathvariant":1352},"R",[138,5883,5884],{"encoding":140},"r \\in \\mathcal{R}",[109,5886,5888,5906],{"className":5887,"ariaHidden":146},[145],[109,5889,5891,5894,5897,5900,5903],{"className":5890},[150],[109,5892],{"className":5893,"style":3662},[154],[109,5895,5558],{"className":5896,"style":1984},[159,163],[109,5898],{"className":5899,"style":1646},[691],[109,5901,1349],{"className":5902},[1682],[109,5904],{"className":5905,"style":1646},[691],[109,5907,5909,5912],{"className":5908},[150],[109,5910],{"className":5911,"style":3681},[154],[109,5913,5881],{"className":5914},[159,1686],[106,5916,5917],{},"capacity row (integer knapsack)",[106,5919,5920],{},"Do not order more units than the supplier has in stock.",[11,5922,5923],{},"For the supplier variables, we have two more:",[79,5925,5926,5940],{},[82,5927,5928],{},[85,5929,5930,5932,5934,5936,5938],{},[88,5931,3248],{},[88,5933],{},[88,5935,3253],{},[88,5937,3256],{},[88,5939,3259],{},[101,5941,5942,6790],{},[85,5943,5944,5947,6257,6311,6403],{},[106,5945,5946],{},"(6)",[106,5948,5949],{},[109,5950,5952,6008],{"className":5951},[112],[109,5953,5955],{"className":5954},[116],[118,5956,5957],{"xmlns":120},[122,5958,5959,6005],{},[125,5960,5961,5977,5983,5985,5989,5995,5997,5999],{},[128,5962,5963,5965],{},[652,5964,1341],{},[125,5966,5967,5969,5971],{},[131,5968,377],{},[652,5970,1349],{},[128,5972,5973,5975],{},[131,5974,1413],{"mathvariant":1352},[131,5976,862],{},[128,5978,5979,5981],{},[131,5980,374],{},[131,5982,377],{},[652,5984,3302],{},[652,5986,5988],{"stretchy":5987},"false","∣",[128,5990,5991,5993],{},[131,5992,1413],{"mathvariant":1352},[131,5994,862],{},[652,5996,5988],{"stretchy":5987},[275,5998,1385],{},[128,6000,6001,6003],{},[131,6002,859],{},[131,6004,862],{},[138,6006,6007],{"encoding":140},"\\sum_{o \\in \\mathcal{O}_k} x_o \\le \\lvert \\mathcal{O}_k \\rvert\\, u_k",[109,6009,6011,6160],{"className":6010,"ariaHidden":146},[145],[109,6012,6014,6018,6108,6111,6151,6154,6157],{"className":6013},[150],[109,6015],{"className":6016,"style":6017},[154],"height:1.1556em;vertical-align:-0.4056em;",[109,6019,6021,6024],{"className":6020},[1535],[109,6022,1341],{"className":6023,"style":3802},[1535,1699,3801],[109,6025,6027],{"className":6026},[168],[109,6028,6030,6099],{"className":6029},[172,173],[109,6031,6033,6096],{"className":6032},[177],[109,6034,6036],{"className":6035,"style":3815},[181],[109,6037,6038,6041],{"style":3818},[109,6039],{"className":6040,"style":190},[189],[109,6042,6044],{"className":6043},[194,195,196,197],[109,6045,6047,6050,6053],{"className":6046},[159,197],[109,6048,377],{"className":6049},[159,163,197],[109,6051,1349],{"className":6052},[1682,197],[109,6054,6056,6059],{"className":6055},[159,197],[109,6057,1413],{"className":6058,"style":1984},[159,1686,197],[109,6060,6062],{"className":6061},[168],[109,6063,6065,6087],{"className":6064},[172,173],[109,6066,6068,6084],{"className":6067},[177],[109,6069,6072],{"className":6070,"style":6071},[181],"height:0.3448em;",[109,6073,6075,6078],{"style":6074},"top:-2.3488em;margin-left:-0.0278em;margin-right:0.0714em;",[109,6076],{"className":6077,"style":1771},[189],[109,6079,6081],{"className":6080},[194,1775,1776,197],[109,6082,862],{"className":6083,"style":905},[159,163,197],[109,6085,206],{"className":6086},[205],[109,6088,6090],{"className":6089},[177],[109,6091,6094],{"className":6092,"style":6093},[181],"height:0.1512em;",[109,6095],{},[109,6097,206],{"className":6098},[205],[109,6100,6102],{"className":6101},[177],[109,6103,6106],{"className":6104,"style":6105},[181],"height:0.4056em;",[109,6107],{},[109,6109],{"className":6110,"style":692},[691],[109,6112,6114,6117],{"className":6113},[159],[109,6115,374],{"className":6116},[159,163],[109,6118,6120],{"className":6119},[168],[109,6121,6123,6143],{"className":6122},[172,173],[109,6124,6126,6140],{"className":6125},[177],[109,6127,6129],{"className":6128,"style":182},[181],[109,6130,6131,6134],{"style":410},[109,6132],{"className":6133,"style":190},[189],[109,6135,6137],{"className":6136},[194,195,196,197],[109,6138,377],{"className":6139},[159,163,197],[109,6141,206],{"className":6142},[205],[109,6144,6146],{"className":6145},[177],[109,6147,6149],{"className":6148,"style":213},[181],[109,6150],{},[109,6152],{"className":6153,"style":1646},[691],[109,6155,3302],{"className":6156},[1682],[109,6158],{"className":6159,"style":1646},[691],[109,6161,6163,6167,6170,6211,6214,6217],{"className":6162},[150],[109,6164],{"className":6165,"style":6166},[154],"height:1em;vertical-align:-0.25em;",[109,6168,5988],{"className":6169},[1817],[109,6171,6173,6176],{"className":6172},[159],[109,6174,1413],{"className":6175,"style":1984},[159,1686],[109,6177,6179],{"className":6178},[168],[109,6180,6182,6203],{"className":6181},[172,173],[109,6183,6185,6200],{"className":6184},[177],[109,6186,6188],{"className":6187,"style":893},[181],[109,6189,6191,6194],{"style":6190},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[109,6192],{"className":6193,"style":190},[189],[109,6195,6197],{"className":6196},[194,195,196,197],[109,6198,862],{"className":6199,"style":905},[159,163,197],[109,6201,206],{"className":6202},[205],[109,6204,6206],{"className":6205},[177],[109,6207,6209],{"className":6208,"style":213},[181],[109,6210],{},[109,6212,5988],{"className":6213},[2268],[109,6215],{"className":6216,"style":692},[691],[109,6218,6220,6223],{"className":6219},[159],[109,6221,859],{"className":6222},[159,163],[109,6224,6226],{"className":6225},[168],[109,6227,6229,6249],{"className":6228},[172,173],[109,6230,6232,6246],{"className":6231},[177],[109,6233,6235],{"className":6234,"style":893},[181],[109,6236,6237,6240],{"style":410},[109,6238],{"className":6239,"style":190},[189],[109,6241,6243],{"className":6242},[194,195,196,197],[109,6244,862],{"className":6245,"style":905},[159,163,197],[109,6247,206],{"className":6248},[205],[109,6250,6252],{"className":6251},[177],[109,6253,6255],{"className":6254,"style":213},[181],[109,6256],{},[106,6258,921,6259],{},[109,6260,6262,6280],{"className":6261},[112],[109,6263,6265],{"className":6264},[116],[118,6266,6267],{"xmlns":120},[122,6268,6269,6277],{},[125,6270,6271,6273,6275],{},[131,6272,862],{},[652,6274,1349],{},[131,6276,1476],{"mathvariant":1352},[138,6278,6279],{"encoding":140},"k \\in \\mathcal{K}",[109,6281,6283,6302],{"className":6282,"ariaHidden":146},[145],[109,6284,6286,6290,6293,6296,6299],{"className":6285},[150],[109,6287],{"className":6288,"style":6289},[154],"height:0.7335em;vertical-align:-0.0391em;",[109,6291,862],{"className":6292,"style":905},[159,163],[109,6294],{"className":6295,"style":1646},[691],[109,6297,1349],{"className":6298},[1682],[109,6300],{"className":6301,"style":1646},[691],[109,6303,6305,6308],{"className":6304},[150],[109,6306],{"className":6307,"style":3681},[154],[109,6309,1476],{"className":6310,"style":2333},[159,1686],[106,6312,6313,6314,6402],{},"linking row for the fixed charge ",[109,6315,6317,6336],{"className":6316},[112],[109,6318,6320],{"className":6319},[116],[118,6321,6322],{"xmlns":120},[122,6323,6324,6334],{},[125,6325,6326],{},[267,6327,6328,6330,6332],{},[131,6329,271],{},[131,6331,862],{},[275,6333,974],{},[138,6335,977],{"encoding":140},[109,6337,6339],{"className":6338,"ariaHidden":146},[145],[109,6340,6342,6345],{"className":6341},[150],[109,6343],{"className":6344,"style":987},[154],[109,6346,6348,6351],{"className":6347},[159],[109,6349,271],{"className":6350,"style":297},[159,163],[109,6352,6354],{"className":6353},[168],[109,6355,6357,6394],{"className":6356},[172,173],[109,6358,6360,6391],{"className":6359},[177],[109,6361,6363,6374],{"className":6362,"style":310},[181],[109,6364,6365,6368],{"style":1008},[109,6366],{"className":6367,"style":190},[189],[109,6369,6371],{"className":6370},[194,195,196,197],[109,6372,862],{"className":6373,"style":905},[159,163,197],[109,6375,6376,6379],{"style":325},[109,6377],{"className":6378,"style":190},[189],[109,6380,6382],{"className":6381},[194,195,196,197],[109,6383,6385],{"className":6384},[159,197],[109,6386,6388],{"className":6387},[159,338,197],[109,6389,974],{"className":6390},[159,197],[109,6392,206],{"className":6393},[205],[109,6395,6397],{"className":6396},[177],[109,6398,6400],{"className":6399,"style":1044},[181],[109,6401],{},", in summed big-M form",[106,6404,6405,6406,6434,6435,6504,6505,6593,6594,6697,6698,1518],{},"If any offer of supplier ",[109,6407,6409,6422],{"className":6408},[112],[109,6410,6412],{"className":6411},[116],[118,6413,6414],{"xmlns":120},[122,6415,6416,6420],{},[125,6417,6418],{},[131,6419,862],{},[138,6421,862],{"encoding":140},[109,6423,6425],{"className":6424,"ariaHidden":146},[145],[109,6426,6428,6431],{"className":6427},[150],[109,6429],{"className":6430,"style":947},[154],[109,6432,862],{"className":6433,"style":905},[159,163]," is used, 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must be 1, and the charge 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is paid. Here ",[109,6595,6597,6624],{"className":6596},[112],[109,6598,6600],{"className":6599},[116],[118,6601,6602],{"xmlns":120},[122,6603,6604,6621],{},[125,6605,6606,6609,6611,6613,6619],{},[131,6607,6608],{},"M",[652,6610,4554],{},[652,6612,5988],{"stretchy":5987},[128,6614,6615,6617],{},[131,6616,1413],{"mathvariant":1352},[131,6618,862],{},[652,6620,5988],{"stretchy":5987},[138,6622,6623],{"encoding":140},"M = \\lvert \\mathcal{O}_k \\rvert",[109,6625,6627,6645],{"className":6626,"ariaHidden":146},[145],[109,6628,6630,6633,6636,6639,6642],{"className":6629},[150],[109,6631],{"className":6632,"style":3681},[154],[109,6634,6608],{"className":6635,"style":3349},[159,163],[109,6637],{"className":6638,"style":1646},[691],[109,6640,4554],{"className":6641},[1682],[109,6643],{"className":6644,"style":1646},[691],[109,6646,6648,6651,6654,6694],{"className":6647},[150],[109,6649],{"className":6650,"style":6166},[154],[109,6652,5988],{"className":6653},[1817],[109,6655,6657,6660],{"className":6656},[159],[109,6658,1413],{"className":6659,"style":1984},[159,1686],[109,6661,6663],{"className":6662},[168],[109,6664,6666,6686],{"className":6665},[172,173],[109,6667,6669,6683],{"className":6668},[177],[109,6670,6672],{"className":6671,"style":893},[181],[109,6673,6674,6677],{"style":6190},[109,6675],{"className":6676,"style":190},[189],[109,6678,6680],{"className":6679},[194,195,196,197],[109,6681,862],{"className":6682,"style":905},[159,163,197],[109,6684,206],{"className":6685},[205],[109,6687,6689],{"className":6688},[177],[109,6690,6692],{"className":6691,"style":213},[181],[109,6693],{},[109,6695,5988],{"className":6696},[2268],", the number of its offers, the most the sum can be, so the row is harmless when ",[109,6699,6701,6723],{"className":6700},[112],[109,6702,6704],{"className":6703},[116],[118,6705,6706],{"xmlns":120},[122,6707,6708,6720],{},[125,6709,6710,6716,6718],{},[128,6711,6712,6714],{},[131,6713,859],{},[131,6715,862],{},[652,6717,4554],{},[648,6719,657],{},[138,6721,6722],{"encoding":140},"u_k = 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1",[109,7398,7400,7455],{"className":7399,"ariaHidden":146},[145],[109,7401,7403,7406,7446,7449,7452],{"className":7402},[150],[109,7404],{"className":7405,"style":1082},[154],[109,7407,7409,7412],{"className":7408},[159],[109,7410,1067],{"className":7411,"style":1089},[159,163],[109,7413,7415],{"className":7414},[168],[109,7416,7418,7438],{"className":7417},[172,173],[109,7419,7421,7435],{"className":7420},[177],[109,7422,7424],{"className":7423,"style":893},[181],[109,7425,7426,7429],{"style":1104},[109,7427],{"className":7428,"style":190},[189],[109,7430,7432],{"className":7431},[194,195,196,197],[109,7433,862],{"className":7434,"style":905},[159,163,197],[109,7436,206],{"className":7437},[205],[109,7439,7441],{"className":7440},[177],[109,7442,7444],{"className":7443,"style":213},[181],[109,7445],{},[109,7447],{"className":7448,"style":1646},[691],[109,7450,4554],{"className":7451},[1682],[109,7453],{"className":7454,"style":1646},[691],[109,7456,7458,7461],{"className":7457},[150],[109,7459],{"className":7460,"style":4642},[154],[109,7462,657],{"className":7463},[159],", the spend is at least ",[109,7466,7468,7488],{"className":7467},[112],[109,7469,7471],{"className":7470},[116],[118,7472,7473],{"xmlns":120},[122,7474,7475,7485],{},[125,7476,7477],{},[267,7478,7479,7481,7483],{},[131,7480,6831],{},[131,7482,862],{},[275,7484,6870],{},[138,7486,7487],{"encoding":140},"S^{\\text{free}}_k",[109,7489,7491],{"className":7490,"ariaHidden":146},[145],[109,7492,7494,7497],{"className":7493},[150],[109,7495],{"className":7496,"style":987},[154],[109,7498,7500,7503],{"className":7499},[159],[109,7501,6831],{"className":7502,"style":6990},[159,163],[109,7504,7506],{"className":7505},[168],[109,7507,7509,7546],{"className":7508},[172,173],[109,7510,7512,7543],{"className":7511},[177],[109,7513,7515,7526],{"className":7514,"style":310},[181],[109,7516,7517,7520],{"style":7236},[109,7518],{"className":7519,"style":190},[189],[109,7521,7523],{"className":7522},[194,195,196,197],[109,7524,862],{"className":7525,"style":905},[159,163,197],[109,7527,7528,7531],{"style":325},[109,7529],{"className":7530,"style":190},[189],[109,7532,7534],{"className":7533},[194,195,196,197],[109,7535,7537],{"className":7536},[159,197],[109,7538,7540],{"className":7539},[159,338,197],[109,7541,6870],{"className":7542},[159,197],[109,7544,206],{"className":7545},[205],[109,7547,7549],{"className":7548},[177],[109,7550,7552],{"className":7551,"style":1044},[181],[109,7553],{},"\"",[106,7556,7557,7558,7649,7650,7678,7679,7767,7768,7860],{},"Shipping can be free, ",[109,7559,7561,7582],{"className":7560},[112],[109,7562,7564],{"className":7563},[116],[118,7565,7566],{"xmlns":120},[122,7567,7568,7580],{},[125,7569,7570,7576,7578],{},[128,7571,7572,7574],{},[131,7573,1067],{},[131,7575,862],{},[652,7577,4554],{},[648,7579,657],{},[138,7581,7396],{"encoding":140},[109,7583,7585,7640],{"className":7584,"ariaHidden":146},[145],[109,7586,7588,7591,7631,7634,7637],{"className":7587},[150],[109,7589],{"className":7590,"style":1082},[154],[109,7592,7594,7597],{"className":7593},[159],[109,7595,1067],{"className":7596,"style":1089},[159,163],[109,7598,7600],{"className":7599},[168],[109,7601,7603,7623],{"className":7602},[172,173],[109,7604,7606,7620],{"className":7605},[177],[109,7607,7609],{"className":7608,"style":893},[181],[109,7610,7611,7614],{"style":1104},[109,7612],{"className":7613,"style":190},[189],[109,7615,7617],{"className":7616},[194,195,196,197],[109,7618,862],{"className":7619,"style":905},[159,163,197],[109,7621,206],{"className":7622},[205],[109,7624,7626],{"className":7625},[177],[109,7627,7629],{"className":7628,"style":213},[181],[109,7630],{},[109,7632],{"className":7633,"style":1646},[691],[109,7635,4554],{"className":7636},[1682],[109,7638],{"className":7639,"style":1646},[691],[109,7641,7643,7646],{"className":7642},[150],[109,7644],{"className":7645,"style":4642},[154],[109,7647,657],{"className":7648},[159],", only if the spend with supplier ",[109,7651,7653,7666],{"className":7652},[112],[109,7654,7656],{"className":7655},[116],[118,7657,7658],{"xmlns":120},[122,7659,7660,7664],{},[125,7661,7662],{},[131,7663,862],{},[138,7665,862],{"encoding":140},[109,7667,7669],{"className":7668,"ariaHidden":146},[145],[109,7670,7672,7675],{"className":7671},[150],[109,7673],{"className":7674,"style":947},[154],[109,7676,862],{"className":7677,"style":905},[159,163]," reaches ",[109,7680,7682,7701],{"className":7681},[112],[109,7683,7685],{"className":7684},[116],[118,7686,7687],{"xmlns":120},[122,7688,7689,7699],{},[125,7690,7691],{},[267,7692,7693,7695,7697],{},[131,7694,6831],{},[131,7696,862],{},[275,7698,6870],{},[138,7700,7487],{"encoding":140},[109,7702,7704],{"className":7703,"ariaHidden":146},[145],[109,7705,7707,7710],{"className":7706},[150],[109,7708],{"className":7709,"style":987},[154],[109,7711,7713,7716],{"className":7712},[159],[109,7714,6831],{"className":7715,"style":6990},[159,163],[109,7717,7719],{"className":7718},[168],[109,7720,7722,7759],{"className":7721},[172,173],[109,7723,7725,7756],{"className":7724},[177],[109,7726,7728,7739],{"className":7727,"style":310},[181],[109,7729,7730,7733],{"style":7236},[109,7731],{"className":7732,"style":190},[189],[109,7734,7736],{"className":7735},[194,195,196,197],[109,7737,862],{"className":7738,"style":905},[159,163,197],[109,7740,7741,7744],{"style":325},[109,7742],{"className":7743,"style":190},[189],[109,7745,7747],{"className":7746},[194,195,196,197],[109,7748,7750],{"className":7749},[159,197],[109,7751,7753],{"className":7752},[159,338,197],[109,7754,6870],{"className":7755},[159,197],[109,7757,206],{"className":7758},[205],[109,7760,7762],{"className":7761},[177],[109,7763,7765],{"className":7764,"style":1044},[181],[109,7766],{},". With ",[109,7769,7771,7793],{"className":7770},[112],[109,7772,7774],{"className":7773},[116],[118,7775,7776],{"xmlns":120},[122,7777,7778,7790],{},[125,7779,7780,7786,7788],{},[128,7781,7782,7784],{},[131,7783,1067],{},[131,7785,862],{},[652,7787,4554],{},[648,7789,650],{},[138,7791,7792],{"encoding":140},"f_k = 0",[109,7794,7796,7851],{"className":7795,"ariaHidden":146},[145],[109,7797,7799,7802,7842,7845,7848],{"className":7798},[150],[109,7800],{"className":7801,"style":1082},[154],[109,7803,7805,7808],{"className":7804},[159],[109,7806,1067],{"className":7807,"style":1089},[159,163],[109,7809,7811],{"className":7810},[168],[109,7812,7814,7834],{"className":7813},[172,173],[109,7815,7817,7831],{"className":7816},[177],[109,7818,7820],{"className":7819,"style":893},[181],[109,7821,7822,7825],{"style":1104},[109,7823],{"className":7824,"style":190},[189],[109,7826,7828],{"className":7827},[194,195,196,197],[109,7829,862],{"className":7830,"style":905},[159,163,197],[109,7832,206],{"className":7833},[205],[109,7835,7837],{"className":7836},[177],[109,7838,7840],{"className":7839,"style":213},[181],[109,7841],{},[109,7843],{"className":7844,"style":1646},[691],[109,7846,4554],{"className":7847},[1682],[109,7849],{"className":7850,"style":1646},[691],[109,7852,7854,7857],{"className":7853},[150],[109,7855],{"className":7856,"style":4642},[154],[109,7858,650],{"className":7859},[159]," the row holds trivially.",[66,7862,7864],{"id":7863},"cp-sat","CP-SAT",[11,7866,7867,7868,7871,7872,7874],{},"Here you can load the MILP from its zipped ",[61,7869,7870],{},".mps"," file and solve it with CP-SAT. Note that CP-SAT's performance can be quite sensitive to the number of ",[61,7873,63],{}," as well as the time limit.",[7876,7877,7882],"pre",{"className":7878,"code":7879,"language":7880,"meta":7881,"style":7881},"language-python shiki shiki-themes github-dark","from ortools.linear_solver.python import model_builder as mb\nimport gzip\n\n# define the model and load the instance data (21538 variables, 35578 constraint rows)\nmodel = mb.Model()\nmodel.import_from_mps_string(gzip.open(f\"{DATA}\u002F{INSTANCE}.mps.gz\", \"rt\").read())\n\nsolver = mb.Solver(\"CP-SAT\")\nsolver.set_time_limit_in_seconds(60)\nWORKERS = 8 # number of CP-SAT \"workers\" (roughly equal to number of CPUs)\nsolver.set_solver_specific_parameters(f\"num_workers:{WORKERS}\")\nstatus = solver.solve(model)\n\nprint(f\"status    {status.name}\")\nprint(f\"cost      {solver.objective_value:,.0f}\")\nprint(f\"time      {solver.wall_time:.1f} s on {WORKERS} workers\")\n","python","",[61,7883,7884,7908,7916,7923,7930,7941,7973,7978,7995,8006,8020,8038,8049,8054,8080,8106],{"__ignoreMap":7881},[109,7885,7888,7892,7896,7899,7902,7905],{"class":7886,"line":7887},"line",1,[109,7889,7891],{"class":7890},"snl16","from",[109,7893,7895],{"class":7894},"s95oV"," ortools.linear_solver.python ",[109,7897,7898],{"class":7890},"import",[109,7900,7901],{"class":7894}," model_builder ",[109,7903,7904],{"class":7890},"as",[109,7906,7907],{"class":7894}," mb\n",[109,7909,7911,7913],{"class":7886,"line":7910},2,[109,7912,7898],{"class":7890},[109,7914,7915],{"class":7894}," gzip\n",[109,7917,7919],{"class":7886,"line":7918},3,[109,7920,7922],{"emptyLinePlaceholder":7921},true,"\n",[109,7924,7926],{"class":7886,"line":7925},4,[109,7927,7929],{"class":7928},"sAwPA","# define the model and load the instance data (21538 variables, 35578 constraint rows)\n",[109,7931,7933,7936,7938],{"class":7886,"line":7932},5,[109,7934,7935],{"class":7894},"model ",[109,7937,4554],{"class":7890},[109,7939,7940],{"class":7894}," mb.Model()\n",[109,7942,7944,7947,7949,7952,7956,7959,7962,7965,7967,7970],{"class":7886,"line":7943},6,[109,7945,7946],{"class":7894},"model.import_from_mps_string(gzip.open(",[109,7948,1067],{"class":7890},[109,7950,7554],{"class":7951},"sU2Wk",[109,7953,7955],{"class":7954},"sDLfK","{DATA}",[109,7957,7958],{"class":7951},"\u002F",[109,7960,7961],{"class":7954},"{INSTANCE}",[109,7963,7964],{"class":7951},".mps.gz\"",[109,7966,3167],{"class":7894},[109,7968,7969],{"class":7951},"\"rt\"",[109,7971,7972],{"class":7894},").read())\n",[109,7974,7976],{"class":7886,"line":7975},7,[109,7977,7922],{"emptyLinePlaceholder":7921},[109,7979,7981,7984,7986,7989,7992],{"class":7886,"line":7980},8,[109,7982,7983],{"class":7894},"solver ",[109,7985,4554],{"class":7890},[109,7987,7988],{"class":7894}," mb.Solver(",[109,7990,7991],{"class":7951},"\"CP-SAT\"",[109,7993,7994],{"class":7894},")\n",[109,7996,7998,8001,8004],{"class":7886,"line":7997},9,[109,7999,8000],{"class":7894},"solver.set_time_limit_in_seconds(",[109,8002,8003],{"class":7954},"60",[109,8005,7994],{"class":7894},[109,8007,8009,8011,8014,8017],{"class":7886,"line":8008},10,[109,8010,63],{"class":7954},[109,8012,8013],{"class":7890}," =",[109,8015,8016],{"class":7954}," 8",[109,8018,8019],{"class":7928}," # number of CP-SAT \"workers\" (roughly equal to number of CPUs)\n",[109,8021,8023,8026,8028,8031,8034,8036],{"class":7886,"line":8022},11,[109,8024,8025],{"class":7894},"solver.set_solver_specific_parameters(",[109,8027,1067],{"class":7890},[109,8029,8030],{"class":7951},"\"num_workers:",[109,8032,8033],{"class":7954},"{WORKERS}",[109,8035,7554],{"class":7951},[109,8037,7994],{"class":7894},[109,8039,8041,8044,8046],{"class":7886,"line":8040},12,[109,8042,8043],{"class":7894},"status ",[109,8045,4554],{"class":7890},[109,8047,8048],{"class":7894}," solver.solve(model)\n",[109,8050,8052],{"class":7886,"line":8051},13,[109,8053,7922],{"emptyLinePlaceholder":7921},[109,8055,8057,8060,8062,8064,8067,8070,8073,8076,8078],{"class":7886,"line":8056},14,[109,8058,8059],{"class":7954},"print",[109,8061,1374],{"class":7894},[109,8063,1067],{"class":7890},[109,8065,8066],{"class":7951},"\"status    ",[109,8068,8069],{"class":7954},"{",[109,8071,8072],{"class":7894},"status.name",[109,8074,8075],{"class":7954},"}",[109,8077,7554],{"class":7951},[109,8079,7994],{"class":7894},[109,8081,8083,8085,8087,8089,8092,8094,8097,8100,8102,8104],{"class":7886,"line":8082},15,[109,8084,8059],{"class":7954},[109,8086,1374],{"class":7894},[109,8088,1067],{"class":7890},[109,8090,8091],{"class":7951},"\"cost      ",[109,8093,8069],{"class":7954},[109,8095,8096],{"class":7894},"solver.objective_value",[109,8098,8099],{"class":7890},":,.0f",[109,8101,8075],{"class":7954},[109,8103,7554],{"class":7951},[109,8105,7994],{"class":7894},[109,8107,8109,8111,8113,8115,8118,8120,8123,8126,8128,8131,8133,8136],{"class":7886,"line":8108},16,[109,8110,8059],{"class":7954},[109,8112,1374],{"class":7894},[109,8114,1067],{"class":7890},[109,8116,8117],{"class":7951},"\"time      ",[109,8119,8069],{"class":7954},[109,8121,8122],{"class":7894},"solver.wall_time",[109,8124,8125],{"class":7890},":.1f",[109,8127,8075],{"class":7954},[109,8129,8130],{"class":7951}," s on ",[109,8132,8033],{"class":7954},[109,8134,8135],{"class":7951}," workers\"",[109,8137,7994],{"class":7894},[8139,8140],"cell-output",{":lines":8141,"label":8142},"[\"status    FEASIBLE\",\"cost      162,129,469,300\",\"time      60.1 s on 8 workers\"]","Output",[66,8144,8146],{"id":8145},"quicopt","Quicopt",[11,8148,8149],{},"Below we rebuild the model with Pyomo (to illustrate how it's done), and then send it to Quicopt's API service for solving.",[7876,8151,8153],{"className":7878,"code":8152,"language":7880,"meta":7881,"style":7881},"import pyomo.environ as pyo\nfrom quicopt import Client\nfrom quicopt.structured import active, block, semiinteger\n\n# define a Pyomo model\nmodel = pyo.ConcreteModel()\n",[61,8154,8155,8167,8179,8191,8195,8200],{"__ignoreMap":7881},[109,8156,8157,8159,8162,8164],{"class":7886,"line":7887},[109,8158,7898],{"class":7890},[109,8160,8161],{"class":7894}," pyomo.environ ",[109,8163,7904],{"class":7890},[109,8165,8166],{"class":7894}," pyo\n",[109,8168,8169,8171,8174,8176],{"class":7886,"line":7910},[109,8170,7891],{"class":7890},[109,8172,8173],{"class":7894}," quicopt ",[109,8175,7898],{"class":7890},[109,8177,8178],{"class":7894}," Client\n",[109,8180,8181,8183,8186,8188],{"class":7886,"line":7918},[109,8182,7891],{"class":7890},[109,8184,8185],{"class":7894}," quicopt.structured ",[109,8187,7898],{"class":7890},[109,8189,8190],{"class":7894}," active, block, semiinteger\n",[109,8192,8193],{"class":7886,"line":7925},[109,8194,7922],{"emptyLinePlaceholder":7921},[109,8196,8197],{"class":7886,"line":7932},[109,8198,8199],{"class":7928},"# define a Pyomo model\n",[109,8201,8202,8204,8206],{"class":7886,"line":7943},[109,8203,7935],{"class":7894},[109,8205,4554],{"class":7890},[109,8207,8208],{"class":7894}," pyo.ConcreteModel()\n",[11,8210,8211,8212,3203,8267,8301,8302,8305,8306,8340],{},"Next we introduce the decision variables ",[109,8213,8215,8233],{"className":8214},[112],[109,8216,8218],{"className":8217},[116],[118,8219,8220],{"xmlns":120},[122,8221,8222,8230],{},[125,8223,8224,8226,8228],{},[131,8225,133],{"mathvariant":1317},[652,8227,654],{"separator":146},[131,8229,578],{"mathvariant":1317},[138,8231,8232],{"encoding":140},"\\boldsymbol{z}, \\boldsymbol{n}",[109,8234,8236],{"className":8235,"ariaHidden":146},[145],[109,8237,8239,8243,8252,8255,8258],{"className":8238},[150],[109,8240],{"className":8241,"style":8242},[154],"height:0.6389em;vertical-align:-0.1944em;",[109,8244,8246],{"className":8245},[159],[109,8247,8249],{"className":8248},[159],[109,8250,133],{"className":8251,"style":1570},[159,1569],[109,8253,654],{"className":8254},[687],[109,8256],{"className":8257,"style":692},[691],[109,8259,8261],{"className":8260},[159],[109,8262,8264],{"className":8263},[159],[109,8265,578],{"className":8266},[159,1569],[109,8268,8270,8283],{"className":8269},[112],[109,8271,8273],{"className":8272},[116],[118,8274,8275],{"xmlns":120},[122,8276,8277,8281],{},[125,8278,8279],{},[131,8280,374],{"mathvariant":1317},[138,8282,3184],{"encoding":140},[109,8284,8286],{"className":8285,"ariaHidden":146},[145],[109,8287,8289,8292],{"className":8288},[150],[109,8290],{"className":8291,"style":3157},[154],[109,8293,8295],{"className":8294},[159],[109,8296,8298],{"className":8297},[159],[109,8299,374],{"className":8300},[159,1569]," that we already encountered above. Note that ",[56,8303,8304],{},"constraint type (1)"," is already covered in the Pyomo declaration of ",[109,8307,8309,8322],{"className":8308},[112],[109,8310,8312],{"className":8311},[116],[118,8313,8314],{"xmlns":120},[122,8315,8316,8320],{},[125,8317,8318],{},[131,8319,578],{"mathvariant":1317},[138,8321,3220],{"encoding":140},[109,8323,8325],{"className":8324,"ariaHidden":146},[145],[109,8326,8328,8331],{"className":8327},[150],[109,8329],{"className":8330,"style":3157},[154],[109,8332,8334],{"className":8333},[159],[109,8335,8337],{"className":8336},[159],[109,8338,578],{"className":8339},[159,1569],":",[7876,8342,8344],{"className":7878,"code":8343,"language":7880,"meta":7881,"style":7881},"# introduce the binary variables z\nmodel.z = pyo.Var(V, within=pyo.Binary)\n\n# integer numbers of packs ordered, constraint type (1) via bounds and semiinteger\nmodel.n = pyo.Var(O, within=pyo.NonNegativeIntegers, bounds=lambda model, o: (N_min[o], N_max[o]))\nsemiinteger(model.n)\n\n# shortcut aliases for n and z\nz, n = model.z, model.n\n\n# the binary variables x become indicators of the pack counts n\nx = {o: active(n[o]) for o in O}\n\n# tell the solver how the variables are related\nfor i in I:\n  block(model, f\"item_{i}\", [n[o] for v in V_i[i] for o in O_v[v]] + [z[v] for v in V_i[i]])\n",[61,8345,8346,8351,8370,8374,8379,8405,8410,8414,8419,8429,8433,8438,8460,8464,8469,8481],{"__ignoreMap":7881},[109,8347,8348],{"class":7886,"line":7887},[109,8349,8350],{"class":7928},"# introduce the binary variables z\n",[109,8352,8353,8356,8358,8361,8365,8367],{"class":7886,"line":7910},[109,8354,8355],{"class":7894},"model.z ",[109,8357,4554],{"class":7890},[109,8359,8360],{"class":7894}," pyo.Var(V, ",[109,8362,8364],{"class":8363},"s9osk","within",[109,8366,4554],{"class":7890},[109,8368,8369],{"class":7894},"pyo.Binary)\n",[109,8371,8372],{"class":7886,"line":7918},[109,8373,7922],{"emptyLinePlaceholder":7921},[109,8375,8376],{"class":7886,"line":7925},[109,8377,8378],{"class":7928},"# integer numbers of packs ordered, constraint type (1) via bounds and semiinteger\n",[109,8380,8381,8384,8386,8389,8391,8393,8396,8399,8402],{"class":7886,"line":7932},[109,8382,8383],{"class":7894},"model.n ",[109,8385,4554],{"class":7890},[109,8387,8388],{"class":7894}," pyo.Var(O, ",[109,8390,8364],{"class":8363},[109,8392,4554],{"class":7890},[109,8394,8395],{"class":7894},"pyo.NonNegativeIntegers, ",[109,8397,8398],{"class":8363},"bounds",[109,8400,8401],{"class":7890},"=lambda",[109,8403,8404],{"class":7894}," model, o: (N_min[o], N_max[o]))\n",[109,8406,8407],{"class":7886,"line":7943},[109,8408,8409],{"class":7894},"semiinteger(model.n)\n",[109,8411,8412],{"class":7886,"line":7975},[109,8413,7922],{"emptyLinePlaceholder":7921},[109,8415,8416],{"class":7886,"line":7980},[109,8417,8418],{"class":7928},"# shortcut aliases for n and z\n",[109,8420,8421,8424,8426],{"class":7886,"line":7997},[109,8422,8423],{"class":7894},"z, n ",[109,8425,4554],{"class":7890},[109,8427,8428],{"class":7894}," model.z, model.n\n",[109,8430,8431],{"class":7886,"line":8008},[109,8432,7922],{"emptyLinePlaceholder":7921},[109,8434,8435],{"class":7886,"line":8022},[109,8436,8437],{"class":7928},"# the binary variables x become indicators of the pack counts n\n",[109,8439,8440,8443,8445,8448,8451,8454,8457],{"class":7886,"line":8040},[109,8441,8442],{"class":7894},"x ",[109,8444,4554],{"class":7890},[109,8446,8447],{"class":7894}," {o: active(n[o]) ",[109,8449,8450],{"class":7890},"for",[109,8452,8453],{"class":7894}," o ",[109,8455,8456],{"class":7890},"in",[109,8458,8459],{"class":7894}," O}\n",[109,8461,8462],{"class":7886,"line":8051},[109,8463,7922],{"emptyLinePlaceholder":7921},[109,8465,8466],{"class":7886,"line":8056},[109,8467,8468],{"class":7928},"# tell the solver how the variables are related\n",[109,8470,8471,8473,8476,8478],{"class":7886,"line":8082},[109,8472,8450],{"class":7890},[109,8474,8475],{"class":7894}," i ",[109,8477,8456],{"class":7890},[109,8479,8480],{"class":7894}," I:\n",[109,8482,8483,8486,8488,8491,8493,8495,8497,8499,8502,8504,8507,8509,8512,8514,8516,8518,8521,8523,8526,8528,8530,8532],{"class":7886,"line":8108},[109,8484,8485],{"class":7894},"  block(model, ",[109,8487,1067],{"class":7890},[109,8489,8490],{"class":7951},"\"item_",[109,8492,8069],{"class":7954},[109,8494,1346],{"class":7894},[109,8496,8075],{"class":7954},[109,8498,7554],{"class":7951},[109,8500,8501],{"class":7894},", [n[o] ",[109,8503,8450],{"class":7890},[109,8505,8506],{"class":7894}," v ",[109,8508,8456],{"class":7890},[109,8510,8511],{"class":7894}," V_i[i] ",[109,8513,8450],{"class":7890},[109,8515,8453],{"class":7894},[109,8517,8456],{"class":7890},[109,8519,8520],{"class":7894}," O_v[v]] ",[109,8522,1396],{"class":7890},[109,8524,8525],{"class":7894}," [z[v] ",[109,8527,8450],{"class":7890},[109,8529,8506],{"class":7894},[109,8531,8456],{"class":7890},[109,8533,8534],{"class":7894}," V_i[i]])\n",[11,8536,8537],{},"Next, we define the remaining constraints, followed by the objective function:",[7876,8539,8541],{"className":7878,"code":8540,"language":7880,"meta":7881,"style":7881},"# constraint type (2)\nmodel.covered = pyo.Constraint(V, rule=lambda model, v: sum(Q_pack[o] * n[o] - Q_lost[o] * x[o] for o in O_v[v]) >= Q_need[v] * z[v])\n\n# constraint type (3)\nmodel.one_variant = pyo.Constraint(I, rule=lambda model, i: sum(z[v] for v in V_i[i]) == 1)\n\n# constraint type (4)\nmodel.exclusive = pyo.Constraint(E, rule=lambda model, e: sum(x[o] for o in O_e[e]) \u003C= 1)\n\n# constraint type (5)\nmodel.stock = pyo.Constraint(R, rule=lambda model, r: sum(Q_pack[o] * n[o] for o in O_r[r]) \u003C= Q_stock[r])\n\n# the objective function, including the final two constraint types\npyo_obj_expr = (sum(C_choice[v] * z[v] for v in V)\n              + sum(C_pack[o] * n[o] + C_use[o] * x[o] for o in O)\n              + sum(pyo.Expr_if(IF=sum(x[o] for o in O_k[k]) >= 1,\n                                THEN=C_fee[k],\n                                ELSE=0) for k in K)  # constraint (6)\n              + sum(pyo.Expr_if(IF=sum(S_pack[o] * n[o] + S_use[o] * x[o] for o in O_k[k]) >= S_free[k],\n                                THEN=-C_ship[k],\n                                ELSE=0) for k in K)) # constraint (7)\n\n# give the objective to Pyomo\nmodel.cost = pyo.Objective(sense=pyo.minimize, expr=pyo_obj_expr)\n",[61,8542,8543,8548,8609,8613,8618,8657,8661,8666,8704,8708,8713,8752,8756,8761,8790,8823,8857,8868,8894,8938,8949,8972,8977,8983],{"__ignoreMap":7881},[109,8544,8545],{"class":7886,"line":7887},[109,8546,8547],{"class":7928},"# constraint type (2)\n",[109,8549,8550,8553,8555,8558,8561,8563,8566,8569,8572,8575,8578,8581,8584,8586,8589,8591,8593,8595,8598,8601,8604,8606],{"class":7886,"line":7910},[109,8551,8552],{"class":7894},"model.covered ",[109,8554,4554],{"class":7890},[109,8556,8557],{"class":7894}," pyo.Constraint(V, ",[109,8559,8560],{"class":8363},"rule",[109,8562,8401],{"class":7890},[109,8564,8565],{"class":7894}," model, v: ",[109,8567,8568],{"class":7954},"sum",[109,8570,8571],{"class":7894},"(Q_pack[o] ",[109,8573,8574],{"class":7890},"*",[109,8576,8577],{"class":7894}," n[o] ",[109,8579,8580],{"class":7890},"-",[109,8582,8583],{"class":7894}," Q_lost[o] ",[109,8585,8574],{"class":7890},[109,8587,8588],{"class":7894}," x[o] ",[109,8590,8450],{"class":7890},[109,8592,8453],{"class":7894},[109,8594,8456],{"class":7890},[109,8596,8597],{"class":7894}," O_v[v]) ",[109,8599,8600],{"class":7890},">=",[109,8602,8603],{"class":7894}," Q_need[v] ",[109,8605,8574],{"class":7890},[109,8607,8608],{"class":7894}," z[v])\n",[109,8610,8611],{"class":7886,"line":7918},[109,8612,7922],{"emptyLinePlaceholder":7921},[109,8614,8615],{"class":7886,"line":7925},[109,8616,8617],{"class":7928},"# constraint type (3)\n",[109,8619,8620,8623,8625,8628,8630,8632,8635,8637,8640,8642,8644,8646,8649,8652,8655],{"class":7886,"line":7932},[109,8621,8622],{"class":7894},"model.one_variant ",[109,8624,4554],{"class":7890},[109,8626,8627],{"class":7894}," pyo.Constraint(I, ",[109,8629,8560],{"class":8363},[109,8631,8401],{"class":7890},[109,8633,8634],{"class":7894}," model, i: ",[109,8636,8568],{"class":7954},[109,8638,8639],{"class":7894},"(z[v] ",[109,8641,8450],{"class":7890},[109,8643,8506],{"class":7894},[109,8645,8456],{"class":7890},[109,8647,8648],{"class":7894}," V_i[i]) ",[109,8650,8651],{"class":7890},"==",[109,8653,8654],{"class":7954}," 1",[109,8656,7994],{"class":7894},[109,8658,8659],{"class":7886,"line":7943},[109,8660,7922],{"emptyLinePlaceholder":7921},[109,8662,8663],{"class":7886,"line":7975},[109,8664,8665],{"class":7928},"# constraint type (4)\n",[109,8667,8668,8671,8673,8676,8678,8680,8683,8685,8688,8690,8692,8694,8697,8700,8702],{"class":7886,"line":7980},[109,8669,8670],{"class":7894},"model.exclusive ",[109,8672,4554],{"class":7890},[109,8674,8675],{"class":7894}," pyo.Constraint(E, ",[109,8677,8560],{"class":8363},[109,8679,8401],{"class":7890},[109,8681,8682],{"class":7894}," model, e: ",[109,8684,8568],{"class":7954},[109,8686,8687],{"class":7894},"(x[o] ",[109,8689,8450],{"class":7890},[109,8691,8453],{"class":7894},[109,8693,8456],{"class":7890},[109,8695,8696],{"class":7894}," O_e[e]) ",[109,8698,8699],{"class":7890},"\u003C=",[109,8701,8654],{"class":7954},[109,8703,7994],{"class":7894},[109,8705,8706],{"class":7886,"line":7997},[109,8707,7922],{"emptyLinePlaceholder":7921},[109,8709,8710],{"class":7886,"line":8008},[109,8711,8712],{"class":7928},"# constraint type (5)\n",[109,8714,8715,8718,8720,8723,8725,8727,8730,8732,8734,8736,8738,8740,8742,8744,8747,8749],{"class":7886,"line":8022},[109,8716,8717],{"class":7894},"model.stock ",[109,8719,4554],{"class":7890},[109,8721,8722],{"class":7894}," pyo.Constraint(R, ",[109,8724,8560],{"class":8363},[109,8726,8401],{"class":7890},[109,8728,8729],{"class":7894}," model, r: ",[109,8731,8568],{"class":7954},[109,8733,8571],{"class":7894},[109,8735,8574],{"class":7890},[109,8737,8577],{"class":7894},[109,8739,8450],{"class":7890},[109,8741,8453],{"class":7894},[109,8743,8456],{"class":7890},[109,8745,8746],{"class":7894}," O_r[r]) ",[109,8748,8699],{"class":7890},[109,8750,8751],{"class":7894}," Q_stock[r])\n",[109,8753,8754],{"class":7886,"line":8040},[109,8755,7922],{"emptyLinePlaceholder":7921},[109,8757,8758],{"class":7886,"line":8051},[109,8759,8760],{"class":7928},"# the objective function, including the final two constraint types\n",[109,8762,8763,8766,8768,8771,8773,8776,8778,8781,8783,8785,8787],{"class":7886,"line":8056},[109,8764,8765],{"class":7894},"pyo_obj_expr ",[109,8767,4554],{"class":7890},[109,8769,8770],{"class":7894}," (",[109,8772,8568],{"class":7954},[109,8774,8775],{"class":7894},"(C_choice[v] ",[109,8777,8574],{"class":7890},[109,8779,8780],{"class":7894}," z[v] ",[109,8782,8450],{"class":7890},[109,8784,8506],{"class":7894},[109,8786,8456],{"class":7890},[109,8788,8789],{"class":7894}," V)\n",[109,8791,8792,8795,8798,8801,8803,8805,8807,8810,8812,8814,8816,8818,8820],{"class":7886,"line":8082},[109,8793,8794],{"class":7890},"              +",[109,8796,8797],{"class":7954}," sum",[109,8799,8800],{"class":7894},"(C_pack[o] ",[109,8802,8574],{"class":7890},[109,8804,8577],{"class":7894},[109,8806,1396],{"class":7890},[109,8808,8809],{"class":7894}," C_use[o] ",[109,8811,8574],{"class":7890},[109,8813,8588],{"class":7894},[109,8815,8450],{"class":7890},[109,8817,8453],{"class":7894},[109,8819,8456],{"class":7890},[109,8821,8822],{"class":7894}," O)\n",[109,8824,8825,8827,8829,8832,8835,8837,8839,8841,8843,8845,8847,8850,8852,8854],{"class":7886,"line":8108},[109,8826,8794],{"class":7890},[109,8828,8797],{"class":7954},[109,8830,8831],{"class":7894},"(pyo.Expr_if(",[109,8833,8834],{"class":8363},"IF",[109,8836,4554],{"class":7890},[109,8838,8568],{"class":7954},[109,8840,8687],{"class":7894},[109,8842,8450],{"class":7890},[109,8844,8453],{"class":7894},[109,8846,8456],{"class":7890},[109,8848,8849],{"class":7894}," O_k[k]) ",[109,8851,8600],{"class":7890},[109,8853,8654],{"class":7954},[109,8855,8856],{"class":7894},",\n",[109,8858,8860,8863,8865],{"class":7886,"line":8859},17,[109,8861,8862],{"class":8363},"                                THEN",[109,8864,4554],{"class":7890},[109,8866,8867],{"class":7894},"C_fee[k],\n",[109,8869,8871,8874,8876,8878,8881,8883,8886,8888,8891],{"class":7886,"line":8870},18,[109,8872,8873],{"class":8363},"                                ELSE",[109,8875,4554],{"class":7890},[109,8877,650],{"class":7954},[109,8879,8880],{"class":7894},") ",[109,8882,8450],{"class":7890},[109,8884,8885],{"class":7894}," k ",[109,8887,8456],{"class":7890},[109,8889,8890],{"class":7894}," K)  ",[109,8892,8893],{"class":7928},"# constraint (6)\n",[109,8895,8897,8899,8901,8903,8905,8907,8909,8912,8914,8916,8918,8921,8923,8925,8927,8929,8931,8933,8935],{"class":7886,"line":8896},19,[109,8898,8794],{"class":7890},[109,8900,8797],{"class":7954},[109,8902,8831],{"class":7894},[109,8904,8834],{"class":8363},[109,8906,4554],{"class":7890},[109,8908,8568],{"class":7954},[109,8910,8911],{"class":7894},"(S_pack[o] ",[109,8913,8574],{"class":7890},[109,8915,8577],{"class":7894},[109,8917,1396],{"class":7890},[109,8919,8920],{"class":7894}," S_use[o] ",[109,8922,8574],{"class":7890},[109,8924,8588],{"class":7894},[109,8926,8450],{"class":7890},[109,8928,8453],{"class":7894},[109,8930,8456],{"class":7890},[109,8932,8849],{"class":7894},[109,8934,8600],{"class":7890},[109,8936,8937],{"class":7894}," S_free[k],\n",[109,8939,8941,8943,8946],{"class":7886,"line":8940},20,[109,8942,8862],{"class":8363},[109,8944,8945],{"class":7890},"=-",[109,8947,8948],{"class":7894},"C_ship[k],\n",[109,8950,8952,8954,8956,8958,8960,8962,8964,8966,8969],{"class":7886,"line":8951},21,[109,8953,8873],{"class":8363},[109,8955,4554],{"class":7890},[109,8957,650],{"class":7954},[109,8959,8880],{"class":7894},[109,8961,8450],{"class":7890},[109,8963,8885],{"class":7894},[109,8965,8456],{"class":7890},[109,8967,8968],{"class":7894}," K)) ",[109,8970,8971],{"class":7928},"# constraint (7)\n",[109,8973,8975],{"class":7886,"line":8974},22,[109,8976,7922],{"emptyLinePlaceholder":7921},[109,8978,8980],{"class":7886,"line":8979},23,[109,8981,8982],{"class":7928},"# give the objective to Pyomo\n",[109,8984,8986,8989,8991,8994,8997,8999,9002,9005,9007],{"class":7886,"line":8985},24,[109,8987,8988],{"class":7894},"model.cost ",[109,8990,4554],{"class":7890},[109,8992,8993],{"class":7894}," pyo.Objective(",[109,8995,8996],{"class":8363},"sense",[109,8998,4554],{"class":7890},[109,9000,9001],{"class":7894},"pyo.minimize, ",[109,9003,9004],{"class":8363},"expr",[109,9006,4554],{"class":7890},[109,9008,9009],{"class":7894},"pyo_obj_expr)\n",[11,9011,9012],{},"And then we send it off:",[7876,9014,9016],{"className":7878,"code":9015,"language":7880,"meta":7881,"style":7881},"# solve with our API service\nresult = Client().solve(model, time_limit=2.0, gzip=True)\nprint(result.display)\n",[61,9017,9018,9023,9053],{"__ignoreMap":7881},[109,9019,9020],{"class":7886,"line":7887},[109,9021,9022],{"class":7928},"# solve with our API service\n",[109,9024,9025,9028,9030,9033,9036,9038,9041,9043,9046,9048,9051],{"class":7886,"line":7910},[109,9026,9027],{"class":7894},"result ",[109,9029,4554],{"class":7890},[109,9031,9032],{"class":7894}," Client().solve(model, ",[109,9034,9035],{"class":8363},"time_limit",[109,9037,4554],{"class":7890},[109,9039,9040],{"class":7954},"2.0",[109,9042,3167],{"class":7894},[109,9044,9045],{"class":8363},"gzip",[109,9047,4554],{"class":7890},[109,9049,9050],{"class":7954},"True",[109,9052,7994],{"class":7894},[109,9054,9055,9057],{"class":7886,"line":7918},[109,9056,8059],{"class":7954},[109,9058,9059],{"class":7894},"(result.display)\n",[8139,9061],{":lines":9062,"label":8142},"[\"▄▀▀▀▄        ▄                     █\",\"█   █ █   █ ▄▄  ▄▀▀▀▄ ▄▀▀▀▄ █▀▀▀▄ ▀█▀▀\",\"█ ▀▄▀ █   █  █  █   ▄ █   █ █▄▄▄▀  █  ▄\",\" ▀▀ ▀  ▀▀▀▀ ▀▀▀  ▀▀▀   ▀▀▀  █       ▀▀\",\"\",\"├── shots\",\"│   └── 1 · primal   162044219300   1.98s  ◀ best\",\"├── status:     heuristic\",\"├── feasible:   true\",\"├── objective:  1.620442193e11\",\"├── x:          x1=1, x10=0, x100=0, x1000=0, x10000=0, x10001=0, …  (10933 variables)\",\"├── time_limit: 2 s\",\"└── solve_time: 1.9852 s\",\"\",\"    Intel Xeon Gold 6338N @ 2.20GHz · 16 cores, 1 core per shot\"]",[11,9064,9065],{},"Let's inspect the solution quality:",[7876,9067,9069],{"className":7878,"code":9068,"language":7880,"meta":7881,"style":7881},"print(f\"CP-SAT's total cost                              {solver.objective_value:,.0f}\")\nprint(f\"Total cost from Quicopt's API service            {result.objective:,.0f}\")\n",[61,9070,9071,9094],{"__ignoreMap":7881},[109,9072,9073,9075,9077,9079,9082,9084,9086,9088,9090,9092],{"class":7886,"line":7887},[109,9074,8059],{"class":7954},[109,9076,1374],{"class":7894},[109,9078,1067],{"class":7890},[109,9080,9081],{"class":7951},"\"CP-SAT's total cost                              ",[109,9083,8069],{"class":7954},[109,9085,8096],{"class":7894},[109,9087,8099],{"class":7890},[109,9089,8075],{"class":7954},[109,9091,7554],{"class":7951},[109,9093,7994],{"class":7894},[109,9095,9096,9098,9100,9102,9105,9107,9110,9112,9114,9116],{"class":7886,"line":7910},[109,9097,8059],{"class":7954},[109,9099,1374],{"class":7894},[109,9101,1067],{"class":7890},[109,9103,9104],{"class":7951},"\"Total cost from Quicopt's API service            ",[109,9106,8069],{"class":7954},[109,9108,9109],{"class":7894},"result.objective",[109,9111,8099],{"class":7890},[109,9113,8075],{"class":7954},[109,9115,7554],{"class":7951},[109,9117,7994],{"class":7894},[8139,9119],{":lines":9120,"label":8142},"[\"CP-SAT's total cost                              162,129,469,300\",\"Total cost from Quicopt's API service            162,044,219,300\"]",[11,9122,9123,9124,9127,9128,9131,9132,9136],{},"Quicopt found a ",[56,9125,9126],{},"better solution"," in less than ",[56,9129,9130],{},"two seconds!"," CP-SAT invested 60s on eight CPUs (which actually sometimes suffices to find the same solution as Quicopt). The ",[24,9133,9135],{"href":40,"rel":9134},[28],"notebook"," on our GitHub has additional proofs that the solution of Quicopt's solver is correct in the original MILP formulation.",[9138,9139,9140],"style",{},"html pre.shiki code .snl16, html code.shiki .snl16{--shiki-default:#F97583}html pre.shiki code .s95oV, html code.shiki .s95oV{--shiki-default:#E1E4E8}html pre.shiki code .sAwPA, html code.shiki .sAwPA{--shiki-default:#6A737D}html pre.shiki code .sU2Wk, html code.shiki .sU2Wk{--shiki-default:#9ECBFF}html pre.shiki code .sDLfK, html code.shiki .sDLfK{--shiki-default:#79B8FF}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html pre.shiki code .s9osk, html code.shiki .s9osk{--shiki-default:#FFAB70}",{"title":7881,"searchDepth":7910,"depth":7910,"links":9142},[9143,9148,9149],{"id":68,"depth":7910,"text":69,"children":9144},[9145,9146,9147],{"id":73,"depth":7918,"text":74},{"id":1264,"depth":7918,"text":1265},{"id":3126,"depth":7918,"text":3127},{"id":7863,"depth":7910,"text":7864},{"id":8145,"depth":7910,"text":8146},"Let us take a deep dive into the so-called \"internet-shopping optimization problem\", also known as \"supplier selection and order allocation\". This is a very common problem that occurs basically whenever a large number of items needs to be ordered from a large number of suppliers. As it involves integer quantities, it is usually formulated as a mixed-integer linear problem (MILP). Unfortunately, MILPs are known to be NP-hard - which more or less says that they can be very hard to solve.","md",null,{},false,"\u002Fuse-cases\u002Finternet-shopping",{"title":6,"description":9150},"use-cases\u002Finternet-shopping","pxlEQH91AlkYS9N6bryjhfDpHrX6mY8eiGSltGSn_N4",{"id":9160,"title":9161,"body":9162,"description":17795,"extension":9151,"faq":9152,"meta":17796,"navigation":7921,"noindex":9154,"path":17797,"seo":17798,"stem":17799,"__hash__":17800},"content\u002Fuse-cases\u002Finternet-shopping.de.md","Das Internet-Shopping-Optimierungsproblem",{"type":8,"value":9163,"toc":17786},[9164,9185,9201,9211,9215,9219,9222,10300,10304,10323,11552,12087,12091,12198,14818,14821,16732,16734,16743,16933,16936,16938,16941,16991,17121,17279,17282,17672,17675,17713,17715,17718,17766,17768,17784],[11,9165,9166,9167,9170,9171,9174,9175,9180,9181,9184],{},"Wir tauchen tief in das sogenannte „Internet-Shopping-Optimierungsproblem“ ein, auch bekannt als „Lieferantenauswahl und Bestellaufteilung“. Es tritt praktisch immer dann auf, wenn eine große Zahl von Artikeln bei einer großen Zahl von Lieferanten bestellt werden muss. Da es um ",[15,9168,9169],{},"ganzzahlige"," Mengen geht, wird es üblicherweise als ",[15,9172,9173],{},"gemischt-ganzzahliges lineares Problem"," (MILP) formuliert. Leider gelten MILPs als ",[24,9176,9179],{"href":9177,"rel":9178},"https:\u002F\u002Fde.wikipedia.org\u002Fwiki\u002FNP-Schwere",[28],"NP-schwer"," – das heißt mehr oder weniger, dass sie ",[15,9182,9183],{},"sehr"," schwer zu lösen sein können.",[11,9186,9187,9188,9192,9193,9196,9197,9200],{},"Sie können alle Ergebnisse selbst nachvollziehen, indem Sie ",[24,9189,9191],{"href":40,"rel":9190},[28],"dieses Notebook"," ausführen. Beachten Sie dabei: Der Open-Source-Solver CP-SAT läuft ",[15,9194,9195],{},"auf Ihrem Rechner",", Quicopt dagegen auf ",[15,9198,9199],{},"unserem API-Server",". Die Hardware ist also nicht identisch, wie sie es für einen streng sauberen Vergleich sein müsste.",[11,9202,9203,9204,9207,9208,9210],{},"Selbst wenn Sie CP-SAT auf schnelleren CPUs ausführen als denen unseres Testservers, werden Sie trotzdem feststellen, dass ",[56,9205,9206],{},"Quicopt deutlich schneller ist!"," Außerdem nutzt CP-SAT 8 Kerne, Quicopt dagegen nur einen! (Die Zahl der Kerne von CP-SAT lässt sich über ",[61,9209,63],{}," einstellen, siehe unten. Bedenken Sie schließlich, dass CP-SAT nicht deterministisch arbeitet: Die Laufzeiten schwanken also recht stark, ebenso die Qualität der Lösung.)",[66,9212,9214],{"id":9213},"das-problem-als-milp","Das Problem als MILP",[71,9216,9218],{"id":9217},"variablen","Variablen",[11,9220,9221],{},"Die Variablen unseres Optimierungsproblems stehen in der folgenden Tabelle:",[79,9223,9224,9240],{},[82,9225,9226],{},[85,9227,9228,9231,9234,9237],{},[88,9229,9230],{},"Symbol",[88,9232,9233],{},"Werte",[88,9235,9236],{},"Bedeutung",[88,9238,9239],{},"Kosten, die sie 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Angebote)",[106,10009,10010],{},[109,10011,10013,10032],{"className":10012},[112],[109,10014,10016],{"className":10015},[116],[118,10017,10018],{"xmlns":120},[122,10019,10020,10030],{},[125,10021,10022],{},[267,10023,10024,10026,10028],{},[131,10025,271],{},[131,10027,862],{},[275,10029,974],{},[138,10031,977],{"encoding":140},[109,10033,10035],{"className":10034,"ariaHidden":146},[145],[109,10036,10038,10041],{"className":10037},[150],[109,10039],{"className":10040,"style":987},[154],[109,10042,10044,10047],{"className":10043},[159],[109,10045,271],{"className":10046,"style":297},[159,163],[109,10048,10050],{"className":10049},[168],[109,10051,10053,10090],{"className":10052},[172,173],[109,10054,10056,10087],{"className":10055},[177],[109,10057,10059,10070],{"className":10058,"style":310},[181],[109,10060,10061,10064],{"style":1008},[109,10062],{"className":10063,"style":190},[189],[109,10065,10067],{"className":10066},[194,195,196,197],[109,10068,862],{"className":10069,"style":905},[159,163,197],[109,10071,10072,10075],{"style":325},[109,10073],{"className":10074,"style":190},[189],[109,10076,10078],{"className":10077},[194,195,196,197],[109,10079,10081],{"className":10080},[159,197],[109,10082,10084],{"className":10083},[159,338,197],[109,10085,974],{"className":10086},[159,197],[109,10088,206],{"className":10089},[205],[109,10091,10093],{"className":10092},[177],[109,10094,10096],{"className":10095,"style":1044},[181],[109,10097],{},[85,10099,10100,10171,10173,10204],{},[106,10101,10102],{},[109,10103,10105,10122],{"className":10104},[112],[109,10106,10108],{"className":10107},[116],[118,10109,10110],{"xmlns":120},[122,10111,10112,10120],{},[125,10113,10114],{},[128,10115,10116,10118],{},[131,10117,1067],{},[131,10119,862],{},[138,10121,1072],{"encoding":140},[109,10123,10125],{"className":10124,"ariaHidden":146},[145],[109,10126,10128,10131],{"className":10127},[150],[109,10129],{"className":10130,"style":1082},[154],[109,10132,10134,10137],{"className":10133},[159],[109,10135,1067],{"className":10136,"style":1089},[159,163],[109,10138,10140],{"className":10139},[168],[109,10141,10143,10163],{"className":10142},[172,173],[109,10144,10146,10160],{"className":10145},[177],[109,10147,10149],{"className":10148,"style":893},[181],[109,10150,10151,10154],{"style":1104},[109,10152],{"className":10153,"style":190},[189],[109,10155,10157],{"className":10156},[194,195,196,197],[109,10158,862],{"className":10159,"style":905},[159,163,197],[109,10161,206],{"className":10162},[205],[109,10164,10166],{"className":10165},[177],[109,10167,10169],{"className":10168,"style":213},[181],[109,10170],{},[106,10172,9317],{},[106,10174,9978,10175,10203],{},[109,10176,10178,10191],{"className":10177},[112],[109,10179,10181],{"className":10180},[116],[118,10182,10183],{"xmlns":120},[122,10184,10185,10189],{},[125,10186,10187],{},[131,10188,862],{},[138,10190,862],{"encoding":140},[109,10192,10194],{"className":10193,"ariaHidden":146},[145],[109,10195,10197,10200],{"className":10196},[150],[109,10198],{"className":10199,"style":947},[154],[109,10201,862],{"className":10202,"style":905},[159,163]," liefert versandkostenfrei: nein\u002Fja",[106,10205,10206,10299],{},[109,10207,10209,10230],{"className":10208},[112],[109,10210,10212],{"className":10211},[116],[118,10213,10214],{"xmlns":120},[122,10215,10216,10228],{},[125,10217,10218,10220],{},[652,10219,1174],{},[267,10221,10222,10224,10226],{},[131,10223,271],{},[131,10225,862],{},[275,10227,1183],{},[138,10229,1186],{"encoding":140},[109,10231,10233],{"className":10232,"ariaHidden":146},[145],[109,10234,10236,10239,10242],{"className":10235},[150],[109,10237],{"className":10238,"style":1196},[154],[109,10240,1174],{"className":10241},[159],[109,10243,10245,10248],{"className":10244},[159],[109,10246,271],{"className":10247,"style":297},[159,163],[109,10249,10251],{"className":10250},[168],[109,10252,10254,10291],{"className":10253},[172,173],[109,10255,10257,10288],{"className":10256},[177],[109,10258,10260,10271],{"className":10259,"style":1218},[181],[109,10261,10262,10265],{"style":1221},[109,10263],{"className":10264,"style":190},[189],[109,10266,10268],{"className":10267},[194,195,196,197],[109,10269,862],{"className":10270,"style":905},[159,163,197],[109,10272,10273,10276],{"style":1233},[109,10274],{"className":10275,"style":190},[189],[109,10277,10279],{"className":10278},[194,195,196,197],[109,10280,10282],{"className":10281},[159,197],[109,10283,10285],{"className":10284},[159,338,197],[109,10286,1183],{"className":10287},[159,197],[109,10289,206],{"className":10290},[205],[109,10292,10294],{"className":10293},[177],[109,10295,10297],{"className":10296,"style":1258},[181],[109,10298],{}," (eine Ersparnis)",[71,10301,10303],{"id":10302},"zielfunktion","Zielfunktion",[11,10305,10306,10307,10310,10311,10314,10315,10318,10319,10322],{},"Das Problem hat vier „Ebenen“: die zu kaufenden ",[56,10308,10309],{},"Artikel",", die ",[56,10312,10313],{},"Varianten"," jedes Artikels, die ",[56,10316,10317],{},"Angebote"," für jede Variante und schließlich die ",[56,10320,10321],{},"Lieferanten",", die diese Angebote machen. Die Kostenfunktion besteht aus zwei 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bedeutet: Für jeden Artikel ",[109,11556,11558,11571],{"className":11557},[112],[109,11559,11561],{"className":11560},[116],[118,11562,11563],{"xmlns":120},[122,11564,11565,11569],{},[125,11566,11567],{},[131,11568,1346],{},[138,11570,1346],{"encoding":140},[109,11572,11574],{"className":11573,"ariaHidden":146},[145],[109,11575,11577,11580],{"className":11576},[150],[109,11578],{"className":11579,"style":2615},[154],[109,11581,1346],{"className":11582},[159,163]," und jede Variante ",[109,11585,11587,11600],{"className":11586},[112],[109,11588,11590],{"className":11589},[116],[118,11591,11592],{"xmlns":120},[122,11593,11594,11598],{},[125,11595,11596],{},[131,11597,136],{},[138,11599,136],{"encoding":140},[109,11601,11603],{"className":11602,"ariaHidden":146},[145],[109,11604,11606,11609],{"className":11605},[150],[109,11607],{"className":11608,"style":247},[154],[109,11610,136],{"className":11611,"style":201},[159,163]," zahlen wir 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wenn die Variante gewählt wird, und außerdem für jedes verwendete Angebot ",[109,11703,11705,11718],{"className":11704},[112],[109,11706,11708],{"className":11707},[116],[118,11709,11710],{"xmlns":120},[122,11711,11712,11716],{},[125,11713,11714],{},[131,11715,377],{},[138,11717,377],{"encoding":140},[109,11719,11721],{"className":11720,"ariaHidden":146},[145],[109,11722,11724,11727],{"className":11723},[150],[109,11725],{"className":11726,"style":247},[154],[109,11728,377],{"className":11729},[159,163]," dieser Variante 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pro gekaufter Packung sowie die Fixkosten ",[109,11821,11823,11842],{"className":11822},[112],[109,11824,11826],{"className":11825},[116],[118,11827,11828],{"xmlns":120},[122,11829,11830,11840],{},[125,11831,11832],{},[267,11833,11834,11836,11838],{},[131,11835,271],{},[131,11837,377],{},[275,11839,486],{},[138,11841,489],{"encoding":140},[109,11843,11845],{"className":11844,"ariaHidden":146},[145],[109,11846,11848,11851],{"className":11847},[150],[109,11849],{"className":11850,"style":499},[154],[109,11852,11854,11857],{"className":11853},[159],[109,11855,271],{"className":11856,"style":297},[159,163],[109,11858,11860],{"className":11859},[168],[109,11861,11863,11900],{"className":11862},[172,173],[109,11864,11866,11897],{"className":11865},[177],[109,11867,11869,11880],{"className":11868,"style":518},[181],[109,11870,11871,11874],{"style":313},[109,11872],{"className":11873,"style":190},[189],[109,11875,11877],{"className":11876},[194,195,196,197],[109,11878,377],{"className":11879},[159,163,197],[109,11881,11882,11885],{"style":325},[109,11883],{"className":11884,"style":190},[189],[109,11886,11888],{"className":11887},[194,195,196,197],[109,11889,11891],{"className":11890},[159,197],[109,11892,11894],{"className":11893},[159,338,197],[109,11895,486],{"className":11896},[159,197],[109,11898,206],{"className":11899},[205],[109,11901,11903],{"className":11902},[177],[109,11904,11906],{"className":11905,"style":351},[181],[109,11907],{},". Die zweite Summe läuft über die Lieferanten: die Gebühr ",[109,11910,11912,11931],{"className":11911},[112],[109,11913,11915],{"className":11914},[116],[118,11916,11917],{"xmlns":120},[122,11918,11919,11929],{},[125,11920,11921],{},[267,11922,11923,11925,11927],{},[131,11924,271],{},[131,11926,862],{},[275,11928,974],{},[138,11930,977],{"encoding":140},[109,11932,11934],{"className":11933,"ariaHidden":146},[145],[109,11935,11937,11940],{"className":11936},[150],[109,11938],{"className":11939,"style":987},[154],[109,11941,11943,11946],{"className":11942},[159],[109,11944,271],{"className":11945,"style":297},[159,163],[109,11947,11949],{"className":11948},[168],[109,11950,11952,11989],{"className":11951},[172,173],[109,11953,11955,11986],{"className":11954},[177],[109,11956,11958,11969],{"className":11957,"style":310},[181],[109,11959,11960,11963],{"style":1008},[109,11961],{"className":11962,"style":190},[189],[109,11964,11966],{"className":11965},[194,195,196,197],[109,11967,862],{"className":11968,"style":905},[159,163,197],[109,11970,11971,11974],{"style":325},[109,11972],{"className":11973,"style":190},[189],[109,11975,11977],{"className":11976},[194,195,196,197],[109,11978,11980],{"className":11979},[159,197],[109,11981,11983],{"className":11982},[159,338,197],[109,11984,974],{"className":11985},[159,197],[109,11987,206],{"className":11988},[205],[109,11990,11992],{"className":11991},[177],[109,11993,11995],{"className":11994,"style":1044},[181],[109,11996],{},", wenn der Lieferant verwendet wird, abzüglich der Versandkosten ",[109,11999,12001,12020],{"className":12000},[112],[109,12002,12004],{"className":12003},[116],[118,12005,12006],{"xmlns":120},[122,12007,12008,12018],{},[125,12009,12010],{},[267,12011,12012,12014,12016],{},[131,12013,271],{},[131,12015,862],{},[275,12017,1183],{},[138,12019,3056],{"encoding":140},[109,12021,12023],{"className":12022,"ariaHidden":146},[145],[109,12024,12026,12029],{"className":12025},[150],[109,12027],{"className":12028,"style":1196},[154],[109,12030,12032,12035],{"className":12031},[159],[109,12033,271],{"className":12034,"style":297},[159,163],[109,12036,12038],{"className":12037},[168],[109,12039,12041,12078],{"className":12040},[172,173],[109,12042,12044,12075],{"className":12043},[177],[109,12045,12047,12058],{"className":12046,"style":1218},[181],[109,12048,12049,12052],{"style":1221},[109,12050],{"className":12051,"style":190},[189],[109,12053,12055],{"className":12054},[194,195,196,197],[109,12056,862],{"className":12057,"style":905},[159,163,197],[109,12059,12060,12063],{"style":1233},[109,12061],{"className":12062,"style":190},[189],[109,12064,12066],{"className":12065},[194,195,196,197],[109,12067,12069],{"className":12068},[159,197],[109,12070,12072],{"className":12071},[159,338,197],[109,12073,1183],{"className":12074},[159,197],[109,12076,206],{"className":12077},[205],[109,12079,12081],{"className":12080},[177],[109,12082,12084],{"className":12083,"style":1258},[181],[109,12085],{},", wenn der Versand kostenfrei ist.",[71,12088,12090],{"id":12089},"nebenbedingungen","Nebenbedingungen",[11,12092,12093,12094,3167,12128,12162,12163,12197],{},"Dieses Problem hat viele Nebenbedingungen, die die Entscheidungsvariablen sinnvoll einschränken. Für ",[109,12095,12097,12110],{"className":12096},[112],[109,12098,12100],{"className":12099},[116],[118,12101,12102],{"xmlns":120},[122,12103,12104,12108],{},[125,12105,12106],{},[131,12107,133],{"mathvariant":1317},[138,12109,3147],{"encoding":140},[109,12111,12113],{"className":12112,"ariaHidden":146},[145],[109,12114,12116,12119],{"className":12115},[150],[109,12117],{"className":12118,"style":3157},[154],[109,12120,12122],{"className":12121},[159],[109,12123,12125],{"className":12124},[159],[109,12126,133],{"className":12127,"style":1570},[159,1569],[109,12129,12131,12144],{"className":12130},[112],[109,12132,12134],{"className":12133},[116],[118,12135,12136],{"xmlns":120},[122,12137,12138,12142],{},[125,12139,12140],{},[131,12141,374],{"mathvariant":1317},[138,12143,3184],{"encoding":140},[109,12145,12147],{"className":12146,"ariaHidden":146},[145],[109,12148,12150,12153],{"className":12149},[150],[109,12151],{"className":12152,"style":3157},[154],[109,12154,12156],{"className":12155},[159],[109,12157,12159],{"className":12158},[159],[109,12160,374],{"className":12161},[159,1569]," und ",[109,12164,12166,12179],{"className":12165},[112],[109,12167,12169],{"className":12168},[116],[118,12170,12171],{"xmlns":120},[122,12172,12173,12177],{},[125,12174,12175],{},[131,12176,578],{"mathvariant":1317},[138,12178,3220],{"encoding":140},[109,12180,12182],{"className":12181,"ariaHidden":146},[145],[109,12183,12185,12188],{"className":12184},[150],[109,12186],{"className":12187,"style":3157},[154],[109,12189,12191],{"className":12190},[159],[109,12192,12194],{"className":12193},[159],[109,12195,578],{"className":12196},[159,1569]," gelten die folgenden fünf:",[79,12199,12200,12218],{},[82,12201,12202],{},[85,12203,12204,12207,12209,12212,12215],{},[88,12205,12206],{},"Zeile",[88,12208],{},[88,12210,12211],{},"für jede\u002Fn",[88,12213,12214],{},"bekannt als",[88,12216,12217],{},"in Worten",[101,12219,12220,12635,13905,14166,14427],{},[85,12221,12222,12224,12577,12629,12632],{},[106,12223,3266],{},[106,12225,12226],{},[109,12227,12229,12286],{"className":12228},[112],[109,12230,12232],{"className":12231},[116],[118,12233,12234],{"xmlns":120},[122,12235,12236,12284],{},[125,12237,12238,12248,12250,12256,12258,12264,12266,12276,12278],{},[267,12239,12240,12242,12244,12246],{},[131,12241,3285],{},[131,12243,377],{},[131,12245,1309],{},[652,12247,1312],{},[275,12249,1385],{},[128,12251,12252,12254],{},[131,12253,374],{},[131,12255,377],{},[652,12257,3302],{},[128,12259,12260,12262],{},[131,12261,578],{},[131,12263,377],{},[652,12265,3302],{},[267,12267,12268,12270,12272,12274],{},[131,12269,3285],{},[131,12271,377],{},[131,12273,3319],{},[652,12275,1312],{},[275,12277,1385],{},[128,12279,12280,12282],{},[131,12281,374],{},[131,12283,377],{},[138,12285,3332],{"encoding":140},[109,12287,12289,12410,12465],{"className":12288,"ariaHidden":146},[145],[109,12290,12292,12295,12358,12361,12401,12404,12407],{"className":12291},[150],[109,12293],{"className":12294,"style":3342},[154],[109,12296,12298,12301],{"className":12297},[159],[109,12299,3285],{"className":12300,"style":3349},[159,163],[109,12302,12304],{"className":12303},[168],[109,12305,12307,12350],{"className":12306},[172,173],[109,12308,12310,12347],{"className":12309},[177],[109,12311,12313,12324],{"className":12312,"style":3362},[181],[109,12314,12315,12318],{"style":3365},[109,12316],{"className":12317,"style":190},[189],[109,12319,12321],{"className":12320},[194,195,196,197],[109,12322,377],{"className":12323},[159,163,197],[109,12325,12326,12329],{"style":325},[109,12327],{"className":12328,"style":190},[189],[109,12330,12332],{"className":12331},[194,195,196,197],[109,12333,12335],{"className":12334},[159,197],[109,12336,12338,12341,12344],{"className":12337},[1535,197],[109,12339,3392],{"className":12340},[197],[109,12342,1346],{"className":12343},[197],[109,12345,578],{"className":12346},[197],[109,12348,206],{"className":12349},[205],[109,12351,12353],{"className":12352},[177],[109,12354,12356],{"className":12355,"style":351},[181],[109,12357],{},[109,12359],{"className":12360,"style":692},[691],[109,12362,12364,12367],{"className":12363},[159],[109,12365,374],{"className":12366},[159,163],[109,12368,12370],{"className":12369},[168],[109,12371,12373,12393],{"className":12372},[172,173],[109,12374,12376,12390],{"className":12375},[177],[109,12377,12379],{"className":12378,"style":182},[181],[109,12380,12381,12384],{"style":410},[109,12382],{"className":12383,"style":190},[189],[109,12385,12387],{"className":12386},[194,195,196,197],[109,12388,377],{"className":12389},[159,163,197],[109,12391,206],{"className":12392},[205],[109,12394,12396],{"className":12395},[177],[109,12397,12399],{"className":12398,"style":213},[181],[109,12400],{},[109,12402],{"className":12403,"style":1646},[691],[109,12405,3302],{"className":12406},[1682],[109,12408],{"className":12409,"style":1646},[691],[109,12411,12413,12416,12456,12459,12462],{"className":12412},[150],[109,12414],{"className":12415,"style":3468},[154],[109,12417,12419,12422],{"className":12418},[159],[109,12420,578],{"className":12421},[159,163],[109,12423,12425],{"className":12424},[168],[109,12426,12428,12448],{"className":12427},[172,173],[109,12429,12431,12445],{"className":12430},[177],[109,12432,12434],{"className":12433,"style":182},[181],[109,12435,12436,12439],{"style":410},[109,12437],{"className":12438,"style":190},[189],[109,12440,12442],{"className":12441},[194,195,196,197],[109,12443,377],{"className":12444},[159,163,197],[109,12446,206],{"className":12447},[205],[109,12449,12451],{"className":12450},[177],[109,12452,12454],{"className":12453,"style":213},[181],[109,12455],{},[109,12457],{"className":12458,"style":1646},[691],[109,12460,3302],{"className":12461},[1682],[109,12463],{"className":12464,"style":1646},[691],[109,12466,12468,12471,12534,12537],{"className":12467},[150],[109,12469],{"className":12470,"style":499},[154],[109,12472,12474,12477],{"className":12473},[159],[109,12475,3285],{"className":12476,"style":3349},[159,163],[109,12478,12480],{"className":12479},[168],[109,12481,12483,12526],{"className":12482},[172,173],[109,12484,12486,12523],{"className":12485},[177],[109,12487,12489,12500],{"className":12488,"style":518},[181],[109,12490,12491,12494],{"style":3365},[109,12492],{"className":12493,"style":190},[189],[109,12495,12497],{"className":12496},[194,195,196,197],[109,12498,377],{"className":12499},[159,163,197],[109,12501,12502,12505],{"style":325},[109,12503],{"className":12504,"style":190},[189],[109,12506,12508],{"className":12507},[194,195,196,197],[109,12509,12511],{"className":12510},[159,197],[109,12512,12514,12517,12520],{"className":12513},[1535,197],[109,12515,3392],{"className":12516},[197],[109,12518,24],{"className":12519},[197],[109,12521,374],{"className":12522},[197],[109,12524,206],{"className":12525},[205],[109,12527,12529],{"className":12528},[177],[109,12530,12532],{"className":12531,"style":351},[181],[109,12533],{},[109,12535],{"className":12536,"style":692},[691],[109,12538,12540,12543],{"className":12539},[159],[109,12541,374],{"className":12542},[159,163],[109,12544,12546],{"className":12545},[168],[109,12547,12549,12569],{"className":12548},[172,173],[109,12550,12552,12566],{"className":12551},[177],[109,12553,12555],{"className":12554,"style":182},[181],[109,12556,12557,12560],{"style":410},[109,12558],{"className":12559,"style":190},[189],[109,12561,12563],{"className":12562},[194,195,196,197],[109,12564,377],{"className":12565},[159,163,197],[109,12567,206],{"className":12568},[205],[109,12570,12572],{"className":12571},[177],[109,12573,12575],{"className":12574,"style":213},[181],[109,12576],{},[106,12578,9517,12579],{},[109,12580,12582,12599],{"className":12581},[112],[109,12583,12585],{"className":12584},[116],[118,12586,12587],{"xmlns":120},[122,12588,12589,12597],{},[125,12590,12591,12593,12595],{},[131,12592,377],{},[652,12594,1349],{},[131,12596,1413],{"mathvariant":1352},[138,12598,3652],{"encoding":140},[109,12600,12602,12620],{"className":12601,"ariaHidden":146},[145],[109,12603,12605,12608,12611,12614,12617],{"className":12604},[150],[109,12606],{"className":12607,"style":3662},[154],[109,12609,377],{"className":12610},[159,163],[109,12612],{"className":12613,"style":1646},[691],[109,12615,1349],{"className":12616},[1682],[109,12618],{"className":12619,"style":1646},[691],[109,12621,12623,12626],{"className":12622},[150],[109,12624],{"className":12625,"style":3681},[154],[109,12627,1413],{"className":12628,"style":1984},[159,1686],[106,12630,12631],{},"semiganzzahlige Variablen, Big-M-Zeilen",[106,12633,12634],{},"Packungen stammen nur aus verwendeten Angeboten, und zwar zwischen der kleinsten und der größten Menge, die das Angebot zulässt.",[85,12636,12637,12639,13154,13206,13278],{},[106,12638,3695],{},[106,12640,12641],{},[109,12642,12644,12723],{"className":12643},[112],[109,12645,12647],{"className":12646},[116],[118,12648,12649],{"xmlns":120},[122,12650,12651,12721],{},[125,12652,12653,12669,12671,12679,12685,12687,12695,12701,12703,12705,12713,12715],{},[128,12654,12655,12657],{},[652,12656,1341],{},[125,12658,12659,12661,12663],{},[131,12660,377],{},[652,12662,1349],{},[128,12664,12665,12667],{},[131,12666,1413],{"mathvariant":1352},[131,12668,136],{},[652,12670,1374],{"fence":146,"stretchy":146,"minsize":1418,"maxsize":1418},[267,12672,12673,12675,12677],{},[131,12674,3732],{},[131,12676,377],{},[275,12678,768],{},[128,12680,12681,12683],{},[131,12682,578],{},[131,12684,377],{},[652,12686,1174],{},[267,12688,12689,12691,12693],{},[131,12690,3732],{},[131,12692,377],{},[275,12694,3753],{},[128,12696,12697,12699],{},[131,12698,374],{},[131,12700,377],{},[652,12702,1455],{"fence":146,"stretchy":146,"minsize":1418,"maxsize":1418},[652,12704,3764],{},[267,12706,12707,12709,12711],{},[131,12708,3732],{},[131,12710,136],{},[275,12712,3773],{},[275,12714,1385],{},[128,12716,12717,12719],{},[131,12718,133],{},[131,12720,136],{},[138,12722,3784],{"encoding":140},[109,12724,12726,12930,13048],{"className":12725,"ariaHidden":146},[145],[109,12727,12729,12732,12818,12824,12881,12921,12924,12927],{"className":12728},[150],[109,12730],{"className":12731,"style":3794},[154],[109,12733,12735,12738],{"className":12734},[1535],[109,12736,1341],{"className":12737,"style":3802},[1535,1699,3801],[109,12739,12741],{"className":12740},[168],[109,12742,12744,12810],{"className":12743},[172,173],[109,12745,12747,12807],{"className":12746},[177],[109,12748,12750],{"className":12749,"style":3815},[181],[109,12751,12752,12755],{"style":3818},[109,12753],{"className":12754,"style":190},[189],[109,12756,12758],{"className":12757},[194,195,196,197],[109,12759,12761,12764,12767],{"className":12760},[159,197],[109,12762,377],{"className":12763},[159,163,197],[109,12765,1349],{"className":12766},[1682,197],[109,12768,12770,12773],{"className":12769},[159,197],[109,12771,1413],{"className":12772,"style":1984},[159,1686,197],[109,12774,12776],{"className":12775},[168],[109,12777,12779,12799],{"className":12778},[172,173],[109,12780,12782,12796],{"className":12781},[177],[109,12783,12785],{"className":12784,"style":1997},[181],[109,12786,12787,12790],{"style":2000},[109,12788],{"className":12789,"style":1771},[189],[109,12791,12793],{"className":12792},[194,1775,1776,197],[109,12794,136],{"className":12795,"style":201},[159,163,197],[109,12797,206],{"className":12798},[205],[109,12800,12802],{"className":12801},[177],[109,12803,12805],{"className":12804,"style":1789},[181],[109,12806],{},[109,12808,206],{"className":12809},[205],[109,12811,12813],{"className":12812},[177],[109,12814,12816],{"className":12815,"style":3883},[181],[109,12817],{},[109,12819,12821],{"className":12820},[1817],[109,12822,1374],{"className":12823},[1821,1776],[109,12825,12827,12830],{"className":12826},[159],[109,12828,3732],{"className":12829},[159,163],[109,12831,12833],{"className":12832},[168],[109,12834,12836,12873],{"className":12835},[172,173],[109,12837,12839,12870],{"className":12838},[177],[109,12840,12842,12853],{"className":12841,"style":310},[181],[109,12843,12844,12847],{"style":3912},[109,12845],{"className":12846,"style":190},[189],[109,12848,12850],{"className":12849},[194,195,196,197],[109,12851,377],{"className":12852},[159,163,197],[109,12854,12855,12858],{"style":325},[109,12856],{"className":12857,"style":190},[189],[109,12859,12861],{"className":12860},[194,195,196,197],[109,12862,12864],{"className":12863},[159,197],[109,12865,12867],{"className":12866},[159,338,197],[109,12868,768],{"className":12869},[159,197],[109,12871,206],{"className":12872},[205],[109,12874,12876],{"className":12875},[177],[109,12877,12879],{"className":12878,"style":351},[181],[109,12880],{},[109,12882,12884,12887],{"className":12883},[159],[109,12885,578],{"className":12886},[159,163],[109,12888,12890],{"className":12889},[168],[109,12891,12893,12913],{"className":12892},[172,173],[109,12894,12896,12910],{"className":12895},[177],[109,12897,12899],{"className":12898,"style":182},[181],[109,12900,12901,12904],{"style":410},[109,12902],{"className":12903,"style":190},[189],[109,12905,12907],{"className":12906},[194,195,196,197],[109,12908,377],{"className":12909},[159,163,197],[109,12911,206],{"className":12912},[205],[109,12914,12916],{"className":12915},[177],[109,12917,12919],{"className":12918,"style":213},[181],[109,12920],{},[109,12922],{"className":12923,"style":1931},[691],[109,12925,1174],{"className":12926},[1935],[109,12928],{"className":12929,"style":1931},[691],[109,12931,12933,12936,12993,13033,13039,13042,13045],{"className":12932},[150],[109,12934],{"className":12935,"style":4005},[154],[109,12937,12939,12942],{"className":12938},[159],[109,12940,3732],{"className":12941},[1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(Überdeckung), eingeschaltet durch ",[109,13210,13212,13229],{"className":13211},[112],[109,13213,13215],{"className":13214},[116],[118,13216,13217],{"xmlns":120},[122,13218,13219,13227],{},[125,13220,13221],{},[128,13222,13223,13225],{},[131,13224,133],{},[131,13226,136],{},[138,13228,141],{"encoding":140},[109,13230,13232],{"className":13231,"ariaHidden":146},[145],[109,13233,13235,13238],{"className":13234},[150],[109,13236],{"className":13237,"style":155},[154],[109,13239,13241,13244],{"className":13240},[159],[109,13242,133],{"className":13243,"style":164},[159,163],[109,13245,13247],{"className":13246},[168],[109,13248,13250,13270],{"className":13249},[172,173],[109,13251,13253,13267],{"className":13252},[177],[109,13254,13256],{"className":13255,"style":182},[181],[109,13257,13258,13261],{"style":185},[109,13259],{"className":13260,"style":190},[189],[109,13262,13264],{"className":13263},[194,195,196,197],[109,13265,136],{"className":13266,"style":201},[159,163,197],[109,13268,206],{"className":13269},[205],[109,13271,13273],{"className":13272},[177],[109,13274,13276],{"className":13275,"style":213},[181],[109,13277],{},[106,13279,13280,13281,13369,13370,13458,13459,12162,13569,13679,13680,13839,13840,13904],{},"Die gewählte Variante muss gedeckt werden. Jedes Angebot liefert ",[109,13282,13284,13303],{"className":13283},[112],[109,13285,13287],{"className":13286},[116],[118,13288,13289],{"xmlns":120},[122,13290,13291,13301],{},[125,13292,13293],{},[267,13294,13295,13297,13299],{},[131,13296,3732],{},[131,13298,377],{},[275,13300,768],{},[138,13302,4374],{"encoding":140},[109,13304,13306],{"className":13305,"ariaHidden":146},[145],[109,13307,13309,13312],{"className":13308},[150],[109,13310],{"className":13311,"style":290},[154],[109,13313,13315,13318],{"className":13314},[159],[109,13316,3732],{"className":13317},[159,163],[109,13319,13321],{"className":13320},[168],[109,13322,13324,13361],{"className":13323},[172,173],[109,13325,13327,13358],{"className":13326},[177],[109,13328,13330,13341],{"className":13329,"style":310},[181],[109,13331,13332,13335],{"style":3912},[109,13333],{"className":13334,"style":190},[189],[109,13336,13338],{"className":13337},[194,195,196,197],[109,13339,377],{"className":13340},[159,163,197],[109,13342,13343,13346],{"style":325},[109,13344],{"className":13345,"style":190},[189],[109,13347,13349],{"className":13348},[194,195,196,197],[109,13350,13352],{"className":13351},[159,197],[109,13353,13355],{"className":13354},[159,338,197],[109,13356,768],{"className":13357},[159,197],[109,13359,206],{"className":13360},[205],[109,13362,13364],{"className":13363},[177],[109,13365,13367],{"className":13366,"style":351},[181],[109,13368],{}," Einheiten und verliert ",[109,13371,13373,13392],{"className":13372},[112],[109,13374,13376],{"className":13375},[116],[118,13377,13378],{"xmlns":120},[122,13379,13380,13390],{},[125,13381,13382],{},[267,13383,13384,13386,13388],{},[131,13385,3732],{},[131,13387,377],{},[275,13389,3753],{},[138,13391,4464],{"encoding":140},[109,13393,13395],{"className":13394,"ariaHidden":146},[145],[109,13396,13398,13401],{"className":13397},[150],[109,13399],{"className":13400,"style":290},[154],[109,13402,13404,13407],{"className":13403},[159],[109,13405,3732],{"className":13406},[159,163],[109,13408,13410],{"className":13409},[168],[109,13411,13413,13450],{"className":13412},[172,173],[109,13414,13416,13447],{"className":13415},[177],[109,13417,13419,13430],{"className":13418,"style":310},[181],[109,13420,13421,13424],{"style":3912},[109,13422],{"className":13423,"style":190},[189],[109,13425,13427],{"className":13426},[194,195,196,197],[109,13428,377],{"className":13429},[159,163,197],[109,13431,13432,13435],{"style":325},[109,13433],{"className":13434,"style":190},[189],[109,13436,13438],{"className":13437},[194,195,196,197],[109,13439,13441],{"className":13440},[159,197],[109,13442,13444],{"className":13443},[159,338,197],[109,13445,3753],{"className":13446},[159,197],[109,13448,206],{"className":13449},[205],[109,13451,13453],{"className":13452},[177],[109,13454,13456],{"className":13455,"style":351},[181],[109,13457],{},". Gilt zum Beispiel ",[109,13460,13462,13485],{"className":13461},[112],[109,13463,13465],{"className":13464},[116],[118,13466,13467],{"xmlns":120},[122,13468,13469,13483],{},[125,13470,13471,13479,13481],{},[267,13472,13473,13475,13477],{},[131,13474,3732],{},[131,13476,136],{},[275,13478,3773],{},[652,13480,4554],{},[648,13482,4557],{},[138,13484,4560],{"encoding":140},[109,13486,13488,13560],{"className":13487,"ariaHidden":146},[145],[109,13489,13491,13494,13551,13554,13557],{"className":13490},[150],[109,13492],{"className":13493,"style":290},[154],[109,13495,13497,13500],{"className":13496},[159],[109,13498,3732],{"className":13499},[159,163],[109,13501,13503],{"className":13502},[168],[109,13504,13506,13543],{"className":13505},[172,173],[109,13507,13509,13540],{"className":13508},[177],[109,13510,13512,13523],{"className":13511,"style":310},[181],[109,13513,13514,13517],{"style":3912},[109,13515],{"className":13516,"style":190},[189],[109,13518,13520],{"className":13519},[194,195,196,197],[109,13521,136],{"className":13522,"style":201},[159,163,197],[109,13524,13525,13528],{"style":325},[109,13526],{"className":13527,"style":190},[189],[109,13529,13531],{"className":13530},[194,195,196,197],[109,13532,13534],{"className":13533},[159,197],[109,13535,13537],{"className":13536},[159,338,197],[109,13538,3773],{"className":13539},[159,197],[109,13541,206],{"className":13542},[205],[109,13544,13546],{"className":13545},[177],[109,13547,13549],{"className":13548,"style":351},[181],[109,13550],{},[109,13552],{"className":13553,"style":1646},[691],[109,13555,4554],{"className":13556},[1682],[109,13558],{"className":13559,"style":1646},[691],[109,13561,13563,13566],{"className":13562},[150],[109,13564],{"className":13565,"style":4642},[154],[109,13567,4557],{"className":13568},[159],[109,13570,13572,13595],{"className":13571},[112],[109,13573,13575],{"className":13574},[116],[118,13576,13577],{"xmlns":120},[122,13578,13579,13593],{},[125,13580,13581,13589,13591],{},[267,13582,13583,13585,13587],{},[131,13584,3732],{},[131,13586,377],{},[275,13588,3753],{},[652,13590,4554],{},[648,13592,662],{},[138,13594,4673],{"encoding":140},[109,13596,13598,13670],{"className":13597,"ariaHidden":146},[145],[109,13599,13601,13604,13661,13664,13667],{"className":13600},[150],[109,13602],{"className":13603,"style":290},[154],[109,13605,13607,13610],{"className":13606},[159],[109,13608,3732],{"className":13609},[159,163],[109,13611,13613],{"className":13612},[168],[109,13614,13616,13653],{"className":13615},[172,173],[109,13617,13619,13650],{"className":13618},[177],[109,13620,13622,13633],{"className":13621,"style":310},[181],[109,13623,13624,13627],{"style":3912},[109,13625],{"className":13626,"style":190},[189],[109,13628,13630],{"className":13629},[194,195,196,197],[109,13631,377],{"className":13632},[159,163,197],[109,13634,13635,13638],{"style":325},[109,13636],{"className":13637,"style":190},[189],[109,13639,13641],{"className":13640},[194,195,196,197],[109,13642,13644],{"className":13643},[159,197],[109,13645,13647],{"className":13646},[159,338,197],[109,13648,3753],{"className":13649},[159,197],[109,13651,206],{"className":13652},[205],[109,13654,13656],{"className":13655},[177],[109,13657,13659],{"className":13658,"style":351},[181],[109,13660],{},[109,13662],{"className":13663,"style":1646},[691],[109,13665,4554],{"className":13666},[1682],[109,13668],{"className":13669,"style":1646},[691],[109,13671,13673,13676],{"className":13672},[150],[109,13674],{"className":13675,"style":4642},[154],[109,13677,662],{"className":13678},[159],", lautet die Zeile 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es werden also 6 Packungen benötigt. Ein zweites verwendetes Angebot würde ebenfalls sein ",[109,13841,13843,13860],{"className":13842},[112],[109,13844,13846],{"className":13845},[116],[118,13847,13848],{"xmlns":120},[122,13849,13850,13858],{},[125,13851,13852],{},[4935,13853,13854,13856],{},[131,13855,3732],{},[275,13857,3753],{},[138,13859,4943],{"encoding":140},[109,13861,13863],{"className":13862,"ariaHidden":146},[145],[109,13864,13866,13869],{"className":13865},[150],[109,13867],{"className":13868,"style":4953},[154],[109,13870,13872,13875],{"className":13871},[159],[109,13873,3732],{"className":13874},[159,163],[109,13876,13878],{"className":13877},[168],[109,13879,13881],{"className":13880},[172],[109,13882,13884],{"className":13883},[177],[109,13885,13887],{"className":13886,"style":310},[181],[109,13888,13889,13892],{"style":325},[109,13890],{"className":13891,"style":190},[189],[109,13893,13895],{"className":13894},[194,195,196,197],[109,13896,13898],{"className":13897},[159,197],[109,13899,13901],{"className":13900},[159,338,197],[109,13902,3753],{"className":13903},[159,197]," verlieren. Bei einer nicht gewählten Variante ist die rechte Seite 0, sodass es erlaubt ist, nichts zu kaufen.",[85,13906,13907,13909,14107,14160,14163],{},[106,13908,4994],{},[106,13910,13911],{},[109,13912,13914,13951],{"className":13913},[112],[109,13915,13917],{"className":13916},[116],[118,13918,13919],{"xmlns":120},[122,13920,13921,13949],{},[125,13922,13923,13939,13945,13947],{},[128,13924,13925,13927],{},[652,13926,1341],{},[125,13928,13929,13931,13933],{},[131,13930,136],{},[652,13932,1349],{},[128,13934,13935,13937],{},[131,13936,1368],{"mathvariant":1352},[131,13938,1346],{},[128,13940,13941,13943],{},[131,13942,133],{},[131,13944,136],{},[652,13946,4554],{},[648,13948,657],{},[138,13950,5037],{"encoding":140},[109,13952,13954,14098],{"className":13953,"ariaHidden":146},[145],[109,13955,13957,13960,14046,14049,14089,14092,14095],{"className":13956},[150],[109,13958],{"className":13959,"style":5047},[154],[109,13961,13963,13966],{"className":13962},[1535],[109,13964,1341],{"className":13965,"style":3802},[1535,1699,3801],[109,13967,13969],{"className":13968},[168],[109,13970,13972,14038],{"className":13971},[172,173],[109,13973,13975,14035],{"className":13974},[177],[109,13976,13978],{"className":13977,"style":3815},[181],[109,13979,13980,13983],{"style":3818},[109,13981],{"className":13982,"style":190},[189],[109,13984,13986],{"className":13985},[194,195,196,197],[109,13987,13989,13992,13995],{"className":13988},[159,197],[109,13990,136],{"className":13991,"style":201},[159,163,197],[109,13993,1349],{"className":13994},[1682,197],[109,13996,13998,14001],{"className":13997},[159,197],[109,13999,1368],{"className":14000,"style":1751},[159,1686,197],[109,14002,14004],{"className":14003},[168],[109,14005,14007,14027],{"className":14006},[172,173],[109,14008,14010,14024],{"className":14009},[177],[109,14011,14013],{"className":14012,"style":1764},[181],[109,14014,14015,14018],{"style":1767},[109,14016],{"className":14017,"style":1771},[189],[109,14019,14021],{"className":14020},[194,1775,1776,197],[109,14022,1346],{"className":14023},[159,163,197],[109,14025,206],{"className":14026},[205],[109,14028,14030],{"className":14029},[177],[109,14031,14033],{"className":14032,"style":1789},[181],[109,14034],{},[109,14036,206],{"className":14037},[205],[109,14039,14041],{"className":14040},[177],[109,14042,14044],{"className":14043,"style":3883},[181],[109,14045],{},[109,14047],{"className":14048,"style":692},[691],[109,14050,14052,14055],{"className":14051},[159],[109,14053,133],{"className":14054,"style":164},[159,163],[109,14056,14058],{"className":14057},[168],[109,14059,14061,14081],{"className":14060},[172,173],[109,14062,14064,14078],{"className":14063},[177],[109,14065,14067],{"className":14066,"style":182},[181],[109,14068,14069,14072],{"style":185},[109,14070],{"className":14071,"style":190},[189],[109,14073,14075],{"className":14074},[194,195,196,197],[109,14076,136],{"className":14077,"style":201},[159,163,197],[109,14079,206],{"className":14080},[205],[109,14082,14084],{"className":14083},[177],[109,14085,14087],{"className":14086,"style":213},[181],[109,14088],{},[109,14090],{"className":14091,"style":1646},[691],[109,14093,4554],{"className":14094},[1682],[109,14096],{"className":14097,"style":1646},[691],[109,14099,14101,14104],{"className":14100},[150],[109,14102],{"className":14103,"style":4642},[154],[109,14105,657],{"className":14106},[159],[106,14108,14109,14110],{},"Artikel 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gewählt.",[85,14167,14168,14170,14368,14421,14424],{},[106,14169,5260],{},[106,14171,14172],{},[109,14173,14175,14212],{"className":14174},[112],[109,14176,14178],{"className":14177},[116],[118,14179,14180],{"xmlns":120},[122,14181,14182,14210],{},[125,14183,14184,14200,14206,14208],{},[128,14185,14186,14188],{},[652,14187,1341],{},[125,14189,14190,14192,14194],{},[131,14191,377],{},[652,14193,1349],{},[128,14195,14196,14198],{},[131,14197,1413],{"mathvariant":1352},[131,14199,5291],{},[128,14201,14202,14204],{},[131,14203,374],{},[131,14205,377],{},[652,14207,3302],{},[648,14209,657],{},[138,14211,5304],{"encoding":140},[109,14213,14215,14359],{"className":14214,"ariaHidden":146},[145],[109,14216,14218,14221,14307,14310,14350,14353,14356],{"className":14217},[150],[109,14219],{"className":14220,"style":5047},[154],[109,14222,14224,14227],{"className":14223},[1535],[109,14225,1341],{"className":14226,"style":3802},[1535,1699,3801],[109,14228,14230],{"className":14229},[168],[109,14231,14233,14299],{"className":14232},[172,173],[109,14234,14236,14296],{"className":14235},[177],[109,14237,14239],{"className":14238,"style":3815},[181],[109,14240,14241,14244],{"style":3818},[109,14242],{"className":14243,"style":190},[189],[109,14245,14247],{"className":14246},[194,195,196,197],[109,14248,14250,14253,14256],{"className":14249},[159,197],[109,14251,377],{"className":14252},[159,163,197],[109,14254,1349],{"className":14255},[1682,197],[109,14257,14259,14262],{"className":14258},[159,197],[109,14260,1413],{"className":14261,"style":1984},[159,1686,197],[109,14263,14265],{"className":14264},[168],[109,14266,14268,14288],{"className":14267},[172,173],[109,14269,14271,14285],{"className":14270},[177],[109,14272,14274],{"className":14273,"style":1997},[181],[109,14275,14276,14279],{"style":2000},[109,14277],{"className":14278,"style":1771},[189],[109,14280,14282],{"className":14281},[194,1775,1776,197],[109,14283,5291],{"className":14284},[159,163,197],[109,14286,206],{"className":14287},[205],[109,14289,14291],{"className":14290},[177],[109,14292,14294],{"className":14293,"style":1789},[181],[109,14295],{},[109,14297,206],{"className":14298},[205],[109,14300,14302],{"className":14301},[177],[109,14303,14305],{"className":14304,"style":3883},[181],[109,14306],{},[109,14308],{"className":14309,"style":692},[691],[109,14311,14313,14316],{"className":14312},[159],[109,14314,374],{"className":14315},[159,163],[109,14317,14319],{"className":14318},[168],[109,14320,14322,14342],{"className":14321},[172,173],[109,14323,14325,14339],{"className":14324},[177],[109,14326,14328],{"className":14327,"style":182},[181],[109,14329,14330,14333],{"style":410},[109,14331],{"className":14332,"style":190},[189],[109,14334,14336],{"className":14335},[194,195,196,197],[109,14337,377],{"className":14338},[159,163,197],[109,14340,206],{"className":14341},[205],[109,14343,14345],{"className":14344},[177],[109,14346,14348],{"className":14347,"style":213},[181],[109,14349],{},[109,14351],{"className":14352,"style":1646},[691],[109,14354,3302],{"className":14355},[1682],[109,14357],{"className":14358,"style":1646},[691],[109,14360,14362,14365],{"className":14361},[150],[109,14363],{"className":14364,"style":4642},[154],[109,14366,657],{"className":14367},[159],[106,14369,14370,14371],{},"exklusive Menge ",[109,14372,14374,14391],{"className":14373},[112],[109,14375,14377],{"className":14376},[116],[118,14378,14379],{"xmlns":120},[122,14380,14381,14389],{},[125,14382,14383,14385,14387],{},[131,14384,5291],{},[652,14386,1349],{},[131,14388,5482],{"mathvariant":1352},[138,14390,5485],{"encoding":140},[109,14392,14394,14412],{"className":14393,"ariaHidden":146},[145],[109,14395,14397,14400,14403,14406,14409],{"className":14396},[150],[109,14398],{"className":14399,"style":3662},[154],[109,14401,5291],{"className":14402},[159,163],[109,14404],{"className":14405,"style":1646},[691],[109,14407,1349],{"className":14408},[1682],[109,14410],{"className":14411,"style":1646},[691],[109,14413,14415,14418],{"className":14414},[150],[109,14416],{"className":14417,"style":3681},[154],[109,14419,5482],{"className":14420,"style":5516},[159,1686],[106,14422,14423],{},"Höchstens-eins-Zeile (Mengenpackung)",[106,14425,14426],{},"Von einer exklusiven Menge wird höchstens ein Angebot verwendet. Beispielsweise lassen sich die Preise eines Produkts für 1–9 Packungen und für 10–99 Packungen nicht mischen.",[85,14428,14429,14431,14759,14812,14815],{},[106,14430,5527],{},[106,14432,14433],{},[109,14434,14436,14489],{"className":14435},[112],[109,14437,14439],{"className":14438},[116],[118,14440,14441],{"xmlns":120},[122,14442,14443,14487],{},[125,14444,14445,14461,14469,14471,14477,14479],{},[128,14446,14447,14449],{},[652,14448,1341],{},[125,14450,14451,14453,14455],{},[131,14452,377],{},[652,14454,1349],{},[128,14456,14457,14459],{},[131,14458,1413],{"mathvariant":1352},[131,14460,5558],{},[267,14462,14463,14465,14467],{},[131,14464,3732],{},[131,14466,377],{},[275,14468,768],{},[275,14470,1385],{},[128,14472,14473,14475],{},[131,14474,578],{},[131,14476,377],{},[652,14478,3302],{},[267,14480,14481,14483,14485],{},[131,14482,3732],{},[131,14484,5558],{},[275,14486,5585],{},[138,14488,5588],{"encoding":140},[109,14490,14492,14696],{"className":14491,"ariaHidden":146},[145],[109,14493,14495,14498,14584,14587,14644,14647,14687,14690,14693],{"className":14494},[150],[109,14496],{"className":14497,"style":5598},[154],[109,14499,14501,14504],{"className":14500},[1535],[109,14502,1341],{"className":14503,"style":3802},[1535,1699,3801],[109,14505,14507],{"className":14506},[168],[109,14508,14510,14576],{"className":14509},[172,173],[109,14511,14513,14573],{"className":14512},[177],[109,14514,14516],{"className":14515,"style":3815},[181],[109,14517,14518,14521],{"style":3818},[109,14519],{"className":14520,"style":190},[189],[109,14522,14524],{"className":14523},[194,195,196,197],[109,14525,14527,14530,14533],{"className":14526},[159,197],[109,14528,377],{"className":14529},[159,163,197],[109,14531,1349],{"className":14532},[1682,197],[109,14534,14536,14539],{"className":14535},[159,197],[109,14537,1413],{"className":14538,"style":1984},[159,1686,197],[109,14540,14542],{"className":14541},[168],[109,14543,14545,14565],{"className":14544},[172,173],[109,14546,14548,14562],{"className":14547},[177],[109,14549,14551],{"className":14550,"style":1997},[181],[109,14552,14553,14556],{"style":2000},[109,14554],{"className":14555,"style":1771},[189],[109,14557,14559],{"className":14558},[194,1775,1776,197],[109,14560,5558],{"className":14561,"style":1984},[159,163,197],[109,14563,206],{"className":14564},[205],[109,14566,14568],{"className":14567},[177],[109,14569,14571],{"className":14570,"style":1789},[181],[109,14572],{},[109,14574,206],{"className":14575},[205],[109,14577,14579],{"className":14578},[177],[109,14580,14582],{"className":14581,"style":3883},[181],[109,14583],{},[109,14585],{"className":14586,"style":692},[691],[109,14588,14590,14593],{"className":14589},[159],[109,14591,3732],{"className":14592},[159,163],[109,14594,14596],{"className":14595},[168],[109,14597,14599,14636],{"className":14598},[172,173],[109,14600,14602,14633],{"className":14601},[177],[109,14603,14605,14616],{"className":14604,"style":310},[181],[109,14606,14607,14610],{"style":3912},[109,14608],{"className":14609,"style":190},[189],[109,14611,14613],{"className":14612},[194,195,196,197],[109,14614,377],{"className":14615},[159,163,197],[109,14617,14618,14621],{"style":325},[109,14619],{"className":14620,"style":190},[189],[109,14622,14624],{"className":14623},[194,195,196,197],[109,14625,14627],{"className":14626},[159,197],[109,14628,14630],{"className":14629},[159,338,197],[109,14631,768],{"className":14632},[159,197],[109,14634,206],{"className":14635},[205],[109,14637,14639],{"className":14638},[177],[109,14640,14642],{"className":14641,"style":351},[181],[109,14643],{},[109,14645],{"className":14646,"style":692},[691],[109,14648,14650,14653],{"className":14649},[159],[109,14651,578],{"className":14652},[159,163],[109,14654,14656],{"className":14655},[168],[109,14657,14659,14679],{"className":14658},[172,173],[109,14660,14662,14676],{"className":14661},[177],[109,14663,14665],{"className":14664,"style":182},[181],[109,14666,14667,14670],{"style":410},[109,14668],{"className":14669,"style":190},[189],[109,14671,14673],{"className":14672},[194,195,196,197],[109,14674,377],{"className":14675},[159,163,197],[109,14677,206],{"className":14678},[205],[109,14680,14682],{"className":14681},[177],[109,14683,14685],{"className":14684,"style":213},[181],[109,14686],{},[109,14688],{"className":14689,"style":1646},[691],[109,14691,3302],{"className":14692},[1682],[109,14694],{"className":14695,"style":1646},[691],[109,14697,14699,14702],{"className":14698},[150],[109,14700],{"className":14701,"style":290},[154],[109,14703,14705,14708],{"className":14704},[159],[109,14706,3732],{"className":14707},[159,163],[109,14709,14711],{"className":14710},[168],[109,14712,14714,14751],{"className":14713},[172,173],[109,14715,14717,14748],{"className":14716},[177],[109,14718,14720,14731],{"className":14719,"style":310},[181],[109,14721,14722,14725],{"style":3912},[109,14723],{"className":14724,"style":190},[189],[109,14726,14728],{"className":14727},[194,195,196,197],[109,14729,5558],{"className":14730,"style":1984},[159,163,197],[109,14732,14733,14736],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Lager hat.",[11,14819,14820],{},"Für die Lieferantenvariablen kommen zwei weitere hinzu:",[79,14822,14823,14837],{},[82,14824,14825],{},[85,14826,14827,14829,14831,14833,14835],{},[88,14828,12206],{},[88,14830],{},[88,14832,12211],{},[88,14834,12214],{},[88,14836,12217],{},[101,14838,14839,15672],{},[85,14840,14841,14843,15143,15195,15287],{},[106,14842,5946],{},[106,14844,14845],{},[109,14846,14848,14901],{"className":14847},[112],[109,14849,14851],{"className":14850},[116],[118,14852,14853],{"xmlns":120},[122,14854,14855,14899],{},[125,14856,14857,14873,14879,14881,14883,14889,14891,14893],{},[128,14858,14859,14861],{},[652,14860,1341],{},[125,14862,14863,14865,14867],{},[131,14864,377],{},[652,14866,1349],{},[128,14868,14869,14871],{},[131,14870,1413],{"mathvariant":1352},[131,14872,862],{},[128,14874,14875,14877],{},[131,14876,374],{},[131,14878,377],{},[652,14880,3302],{},[652,14882,5988],{"stretchy":5987},[128,14884,14885,14887],{},[131,14886,1413],{"mathvariant":1352},[131,14888,862],{},[652,14890,5988],{"stretchy":5987},[275,14892,1385],{},[128,14894,14895,14897],{},[131,14896,859],{},[131,14898,862],{},[138,14900,6007],{"encoding":140},[109,14902,14904,15048],{"className":14903,"ariaHidden":146},[145],[109,14905,14907,14910,14996,14999,15039,15042,15045],{"className":14906},[150],[109,14908],{"className":14909,"style":6017},[154],[109,14911,14913,14916],{"className":14912},[1535],[109,14914,1341],{"className":14915,"style":3802},[1535,1699,3801],[109,14917,14919],{"className":14918},[168],[109,14920,14922,14988],{"className":14921},[172,173],[109,14923,14925,14985],{"className":14924},[177],[109,14926,14928],{"className":14927,"style":3815},[181],[109,14929,14930,14933],{"style":3818},[109,14931],{"className":14932,"style":190},[189],[109,14934,14936],{"className":14935},[194,195,196,197],[109,14937,14939,14942,14945],{"className":14938},[159,197],[109,14940,377],{"className":14941},[159,163,197],[109,14943,1349],{"className":14944},[1682,197],[109,14946,14948,14951],{"className":14947},[159,197],[109,14949,1413],{"className":14950,"style":1984},[159,1686,197],[109,14952,14954],{"className":14953},[168],[109,14955,14957,14977],{"className":14956},[172,173],[109,14958,14960,14974],{"className":14959},[177],[109,14961,14963],{"className":14962,"style":6071},[181],[109,14964,14965,14968],{"style":6074},[109,14966],{"className":14967,"style":1771},[189],[109,14969,14971],{"className":14970},[194,1775,1776,197],[109,14972,862],{"className":14973,"style":905},[159,163,197],[109,14975,206],{"className":14976},[205],[109,14978,14980],{"className":14979},[177],[109,14981,14983],{"className":14982,"style":6093},[181],[109,14984],{},[109,14986,206],{"className":14987},[205],[109,14989,14991],{"className":14990},[177],[109,14992,14994],{"className":14993,"style":6105},[181],[109,14995],{},[109,14997],{"className":14998,"style":692},[691],[109,15000,15002,15005],{"className":15001},[159],[109,15003,374],{"className":15004},[159,163],[109,15006,15008],{"className":15007},[168],[109,15009,15011,15031],{"className":15010},[172,173],[109,15012,15014,15028],{"className":15013},[177],[109,15015,15017],{"className":15016,"style":182},[181],[109,15018,15019,15022],{"style":410},[109,15020],{"className":15021,"style":190},[189],[109,15023,15025],{"className":15024},[194,195,196,197],[109,15026,377],{"className":15027},[159,163,197],[109,15029,206],{"className":15030},[205],[109,15032,15034],{"className":15033},[177],[109,15035,15037],{"className":15036,"style":213},[181],[109,15038],{},[109,15040],{"className":15041,"style":1646},[691],[109,15043,3302],{"className":15044},[1682],[109,15046],{"className":15047,"style":1646},[691],[109,15049,15051,15054,15057,15097,15100,15103],{"className":15050},[150],[109,15052],{"className":15053,"style":6166},[154],[109,15055,5988],{"className":15056},[1817],[109,15058,15060,15063],{"className":15059},[159],[109,15061,1413],{"className":15062,"style":1984},[159,1686],[109,15064,15066],{"className":15065},[168],[109,15067,15069,15089],{"className":15068},[172,173],[109,15070,15072,15086],{"className":15071},[177],[109,15073,15075],{"className":15074,"style":893},[181],[109,15076,15077,15080],{"style":6190},[109,15078],{"className":15079,"style":190},[189],[109,15081,15083],{"className":15082},[194,195,196,197],[109,15084,862],{"className":15085,"style":905},[159,163,197],[109,15087,206],{"className":15088},[205],[109,15090,15092],{"className":15091},[177],[109,15093,15095],{"className":15094,"style":213},[181],[109,15096],{},[109,15098,5988],{"className":15099},[2268],[109,15101],{"className":15102,"style":692},[691],[109,15104,15106,15109],{"className":15105},[159],[109,15107,859],{"className":15108},[159,163],[109,15110,15112],{"className":15111},[168],[109,15113,15115,15135],{"className":15114},[172,173],[109,15116,15118,15132],{"className":15117},[177],[109,15119,15121],{"className":15120,"style":893},[181],[109,15122,15123,15126],{"style":410},[109,15124],{"className":15125,"style":190},[189],[109,15127,15129],{"className":15128},[194,195,196,197],[109,15130,862],{"className":15131,"style":905},[159,163,197],[109,15133,206],{"className":15134},[205],[109,15136,15138],{"className":15137},[177],[109,15139,15141],{"className":15140,"style":213},[181],[109,15142],{},[106,15144,9978,15145],{},[109,15146,15148,15165],{"className":15147},[112],[109,15149,15151],{"className":15150},[116],[118,15152,15153],{"xmlns":120},[122,15154,15155,15163],{},[125,15156,15157,15159,15161],{},[131,15158,862],{},[652,15160,1349],{},[131,15162,1476],{"mathvariant":1352},[138,15164,6279],{"encoding":140},[109,15166,15168,15186],{"className":15167,"ariaHidden":146},[145],[109,15169,15171,15174,15177,15180,15183],{"className":15170},[150],[109,15172],{"className":15173,"style":6289},[154],[109,15175,862],{"className":15176,"style":905},[159,163],[109,15178],{"className":15179,"style":1646},[691],[109,15181,1349],{"className":15182},[1682],[109,15184],{"className":15185,"style":1646},[691],[109,15187,15189,15192],{"className":15188},[150],[109,15190],{"className":15191,"style":3681},[154],[109,15193,1476],{"className":15194,"style":2333},[159,1686],[106,15196,15197,15198,15286],{},"Kopplungszeile für die Fixgebühr ",[109,15199,15201,15220],{"className":15200},[112],[109,15202,15204],{"className":15203},[116],[118,15205,15206],{"xmlns":120},[122,15207,15208,15218],{},[125,15209,15210],{},[267,15211,15212,15214,15216],{},[131,15213,271],{},[131,15215,862],{},[275,15217,974],{},[138,15219,977],{"encoding":140},[109,15221,15223],{"className":15222,"ariaHidden":146},[145],[109,15224,15226,15229],{"className":15225},[150],[109,15227],{"className":15228,"style":987},[154],[109,15230,15232,15235],{"className":15231},[159],[109,15233,271],{"className":15234,"style":297},[159,163],[109,15236,15238],{"className":15237},[168],[109,15239,15241,15278],{"className":15240},[172,173],[109,15242,15244,15275],{"className":15243},[177],[109,15245,15247,15258],{"className":15246,"style":310},[181],[109,15248,15249,15252],{"style":1008},[109,15250],{"className":15251,"style":190},[189],[109,15253,15255],{"className":15254},[194,195,196,197],[109,15256,862],{"className":15257,"style":905},[159,163,197],[109,15259,15260,15263],{"style":325},[109,15261],{"className":15262,"style":190},[189],[109,15264,15266],{"className":15265},[194,195,196,197],[109,15267,15269],{"className":15268},[159,197],[109,15270,15272],{"className":15271},[159,338,197],[109,15273,974],{"className":15274},[159,197],[109,15276,206],{"className":15277},[205],[109,15279,15281],{"className":15280},[177],[109,15282,15284],{"className":15283,"style":1044},[181],[109,15285],{},", in summierter Big-M-Form",[106,15288,15289,15290,15318,15319,15388,15389,15477,15478,15579,15580,15671],{},"Wird irgendein Angebot von Lieferant ",[109,15291,15293,15306],{"className":15292},[112],[109,15294,15296],{"className":15295},[116],[118,15297,15298],{"xmlns":120},[122,15299,15300,15304],{},[125,15301,15302],{},[131,15303,862],{},[138,15305,862],{"encoding":140},[109,15307,15309],{"className":15308,"ariaHidden":146},[145],[109,15310,15312,15315],{"className":15311},[150],[109,15313],{"className":15314,"style":947},[154],[109,15316,862],{"className":15317,"style":905},[159,163]," verwendet, muss ",[109,15320,15322,15339],{"className":15321},[112],[109,15323,15325],{"className":15324},[116],[118,15326,15327],{"xmlns":120},[122,15328,15329,15337],{},[125,15330,15331],{},[128,15332,15333,15335],{},[131,15334,859],{},[131,15336,862],{},[138,15338,865],{"encoding":140},[109,15340,15342],{"className":15341,"ariaHidden":146},[145],[109,15343,15345,15348],{"className":15344},[150],[109,15346],{"className":15347,"style":155},[154],[109,15349,15351,15354],{"className":15350},[159],[109,15352,859],{"className":15353},[159,163],[109,15355,15357],{"className":15356},[168],[109,15358,15360,15380],{"className":15359},[172,173],[109,15361,15363,15377],{"className":15362},[177],[109,15364,15366],{"className":15365,"style":893},[181],[109,15367,15368,15371],{"style":410},[109,15369],{"className":15370,"style":190},[189],[109,15372,15374],{"className":15373},[194,195,196,197],[109,15375,862],{"className":15376,"style":905},[159,163,197],[109,15378,206],{"className":15379},[205],[109,15381,15383],{"className":15382},[177],[109,15384,15386],{"className":15385,"style":213},[181],[109,15387],{}," gleich 1 sein, und die Gebühr ",[109,15390,15392,15411],{"className":15391},[112],[109,15393,15395],{"className":15394},[116],[118,15396,15397],{"xmlns":120},[122,15398,15399,15409],{},[125,15400,15401],{},[267,15402,15403,15405,15407],{},[131,15404,271],{},[131,15406,862],{},[275,15408,974],{},[138,15410,977],{"encoding":140},[109,15412,15414],{"className":15413,"ariaHidden":146},[145],[109,15415,15417,15420],{"className":15416},[150],[109,15418],{"className":15419,"style":987},[154],[109,15421,15423,15426],{"className":15422},[159],[109,15424,271],{"className":15425,"style":297},[159,163],[109,15427,15429],{"className":15428},[168],[109,15430,15432,15469],{"className":15431},[172,173],[109,15433,15435,15466],{"className":15434},[177],[109,15436,15438,15449],{"className":15437,"style":310},[181],[109,15439,15440,15443],{"style":1008},[109,15441],{"className":15442,"style":190},[189],[109,15444,15446],{"className":15445},[194,195,196,197],[109,15447,862],{"className":15448,"style":905},[159,163,197],[109,15450,15451,15454],{"style":325},[109,15452],{"className":15453,"style":190},[189],[109,15455,15457],{"className":15456},[194,195,196,197],[109,15458,15460],{"className":15459},[159,197],[109,15461,15463],{"className":15462},[159,338,197],[109,15464,974],{"className":15465},[159,197],[109,15467,206],{"className":15468},[205],[109,15470,15472],{"className":15471},[177],[109,15473,15475],{"className":15474,"style":1044},[181],[109,15476],{}," fällt an. Hier ist ",[109,15479,15481,15506],{"className":15480},[112],[109,15482,15484],{"className":15483},[116],[118,15485,15486],{"xmlns":120},[122,15487,15488,15504],{},[125,15489,15490,15492,15494,15496,15502],{},[131,15491,6608],{},[652,15493,4554],{},[652,15495,5988],{"stretchy":5987},[128,15497,15498,15500],{},[131,15499,1413],{"mathvariant":1352},[131,15501,862],{},[652,15503,5988],{"stretchy":5987},[138,15505,6623],{"encoding":140},[109,15507,15509,15527],{"className":15508,"ariaHidden":146},[145],[109,15510,15512,15515,15518,15521,15524],{"className":15511},[150],[109,15513],{"className":15514,"style":3681},[154],[109,15516,6608],{"className":15517,"style":3349},[159,163],[109,15519],{"className":15520,"style":1646},[691],[109,15522,4554],{"className":15523},[1682],[109,15525],{"className":15526,"style":1646},[691],[109,15528,15530,15533,15536,15576],{"className":15529},[150],[109,15531],{"className":15532,"style":6166},[154],[109,15534,5988],{"className":15535},[1817],[109,15537,15539,15542],{"className":15538},[159],[109,15540,1413],{"className":15541,"style":1984},[159,1686],[109,15543,15545],{"className":15544},[168],[109,15546,15548,15568],{"className":15547},[172,173],[109,15549,15551,15565],{"className":15550},[177],[109,15552,15554],{"className":15553,"style":893},[181],[109,15555,15556,15559],{"style":6190},[109,15557],{"className":15558,"style":190},[189],[109,15560,15562],{"className":15561},[194,195,196,197],[109,15563,862],{"className":15564,"style":905},[159,163,197],[109,15566,206],{"className":15567},[205],[109,15569,15571],{"className":15570},[177],[109,15572,15574],{"className":15573,"style":213},[181],[109,15575],{},[109,15577,5988],{"className":15578},[2268],", die Anzahl seiner Angebote und damit der größtmögliche Wert der Summe, sodass die Zeile für ",[109,15581,15583,15604],{"className":15582},[112],[109,15584,15586],{"className":15585},[116],[118,15587,15588],{"xmlns":120},[122,15589,15590,15602],{},[125,15591,15592,15598,15600],{},[128,15593,15594,15596],{},[131,15595,859],{},[131,15597,862],{},[652,15599,4554],{},[648,15601,657],{},[138,15603,6722],{"encoding":140},[109,15605,15607,15662],{"className":15606,"ariaHidden":146},[145],[109,15608,15610,15613,15653,15656,15659],{"className":15609},[150],[109,15611],{"className":15612,"style":155},[154],[109,15614,15616,15619],{"className":15615},[159],[109,15617,859],{"className":15618},[159,163],[109,15620,15622],{"className":15621},[168],[109,15623,15625,15645],{"className":15624},[172,173],[109,15626,15628,15642],{"className":15627},[177],[109,15629,15631],{"className":15630,"style":893},[181],[109,15632,15633,15636],{"style":410},[109,15634],{"className":15635,"style":190},[189],[109,15637,15639],{"className":15638},[194,195,196,197],[109,15640,862],{"className":15641,"style":905},[159,163,197],[109,15643,206],{"className":15644},[205],[109,15646,15648],{"className":15647},[177],[109,15649,15651],{"className":15650,"style":213},[181],[109,15652],{},[109,15654],{"className":15655,"style":1646},[691],[109,15657,4554],{"className":15658},[1682],[109,15660],{"className":15661,"style":1646},[691],[109,15663,15665,15668],{"className":15664},[150],[109,15666],{"className":15667,"style":4642},[154],[109,15669,657],{"className":15670},[159]," unschädlich ist.",[85,15673,15674,15676,16191,16243,16427],{},[106,15675,6794],{},[106,15677,15678],{},[109,15679,15681,15760],{"className":15680},[112],[109,15682,15684],{"className":15683},[116],[118,15685,15686],{"xmlns":120},[122,15687,15688,15758],{},[125,15689,15690,15706,15708,15716,15722,15724,15732,15738,15740,15742,15750,15752],{},[128,15691,15692,15694],{},[652,15693,1341],{},[125,15695,15696,15698,15700],{},[131,15697,377],{},[652,15699,1349],{},[128,15701,15702,15704],{},[131,15703,1413],{"mathvariant":1352},[131,15705,862],{},[652,15707,1374],{"fence":146,"stretchy":146,"minsize":1418,"maxsize":1418},[267,15709,15710,15712,15714],{},[131,15711,6831],{},[131,15713,377],{},[275,15715,768],{},[128,15717,15718,15720],{},[131,15719,578],{},[131,15721,377],{},[652,15723,1396],{},[267,15725,15726,15728,15730],{},[131,15727,6831],{},[131,15729,377],{},[275,15731,486],{},[128,15733,15734,15736],{},[131,15735,374],{},[131,15737,377],{},[652,15739,1455],{"fence":146,"stretchy":146,"minsize":1418,"maxsize":1418},[652,15741,3764],{},[267,15743,15744,15746,15748],{},[131,15745,6831],{},[131,15747,862],{},[275,15749,6870],{},[275,15751,1385],{},[128,15753,15754,15756],{},[131,15755,1067],{},[131,15757,862],{},[138,15759,6881],{"encoding":140},[109,15761,15763,15967,16085],{"className":15762,"ariaHidden":146},[145],[109,15764,15766,15769,15855,15861,15918,15958,15961,15964],{"className":15765},[150],[109,15767],{"className":15768,"style":6891},[154],[109,15770,15772,15775],{"className":15771},[1535],[109,15773,1341],{"className":15774,"style":3802},[1535,1699,3801],[109,15776,15778],{"className":15777},[168],[109,15779,15781,15847],{"className":15780},[172,173],[109,15782,15784,15844],{"className":15783},[177],[109,15785,15787],{"className":15786,"style":3815},[181],[109,15788,15789,15792],{"style":3818},[109,15790],{"className":15791,"style":190},[189],[109,15793,15795],{"className":15794},[194,195,196,197],[109,15796,15798,15801,15804],{"className":15797},[159,197],[109,15799,377],{"className":15800},[159,163,197],[109,15802,1349],{"className":15803},[1682,197],[109,15805,15807,15810],{"className":15806},[159,197],[109,15808,1413],{"className":15809,"style":1984},[159,1686,197],[109,15811,15813],{"className":15812},[168],[109,15814,15816,15836],{"className":15815},[172,173],[109,15817,15819,15833],{"className":15818},[177],[109,15820,15822],{"className":15821,"style":6071},[181],[109,15823,15824,15827],{"style":6074},[109,15825],{"className":15826,"style":1771},[189],[109,15828,15830],{"className":15829},[194,1775,1776,197],[109,15831,862],{"className":15832,"style":905},[159,163,197],[109,15834,206],{"className":15835},[205],[109,15837,15839],{"className":15838},[177],[109,15840,15842],{"className":15841,"style":6093},[181],[109,15843],{},[109,15845,206],{"className":15846},[205],[109,15848,15850],{"className":15849},[177],[109,15851,15853],{"className":15852,"style":6105},[181],[109,15854],{},[109,15856,15858],{"className":15857},[1817],[109,15859,1374],{"className":15860},[1821,1776],[109,15862,15864,15867],{"className":15863},[159],[109,15865,6831],{"className":15866,"style":6990},[159,163],[109,15868,15870],{"className":15869},[168],[109,15871,15873,15910],{"className":15872},[172,173],[109,15874,15876,15907],{"className":15875},[177],[109,15877,15879,15890],{"className":15878,"style":310},[181],[109,15880,15881,15884],{"style":7005},[109,15882],{"className":15883,"style":190},[189],[109,15885,15887],{"className":15886},[194,195,196,197],[109,15888,377],{"className":15889},[159,163,197],[109,15891,15892,15895],{"style":325},[109,15893],{"className":15894,"style":190},[189],[109,15896,15898],{"className":15897},[194,195,196,197],[109,15899,15901],{"className":15900},[159,197],[109,15902,15904],{"className":15903},[159,338,197],[109,15905,768],{"className":15906},[159,197],[109,15908,206],{"className":15909},[205],[109,15911,15913],{"className":15912},[177],[109,15914,15916],{"className":15915,"style":351},[181],[109,15917],{},[109,15919,15921,15924],{"className":15920},[159],[109,15922,578],{"className":15923},[159,163],[109,15925,15927],{"className":15926},[168],[109,15928,15930,15950],{"className":15929},[172,173],[109,15931,15933,15947],{"className":15932},[177],[109,15934,15936],{"className":15935,"style":182},[181],[109,15937,15938,15941],{"style":410},[109,15939],{"className":15940,"style":190},[189],[109,15942,15944],{"className":15943},[194,195,196,197],[109,15945,377],{"className":15946},[159,163,197],[109,15948,206],{"className":15949},[205],[109,15951,15953],{"className":15952},[177],[109,15954,15956],{"className":15955,"style":213},[181],[109,15957],{},[109,15959],{"className":15960,"style":1931},[691],[109,15962,1396],{"className":15963},[1935],[109,15965],{"className":15966,"style":1931},[691],[109,15968,15970,15973,16030,16070,16076,16079,16082],{"className":15969},[150],[109,15971],{"className":15972,"style":4005},[154],[109,15974,15976,15979],{"className":15975},[159],[109,15977,6831],{"className":15978,"style":6990},[159,163],[109,15980,15982],{"className":15981},[168],[109,15983,15985,16022],{"className":15984},[172,173],[109,15986,15988,16019],{"className":15987},[177],[109,15989,15991,16002],{"className":15990,"style":518},[181],[109,15992,15993,15996],{"style":7005},[109,15994],{"className":15995,"style":190},[189],[109,15997,15999],{"className":15998},[194,195,196,197],[109,16000,377],{"className":16001},[159,163,197],[109,16003,16004,16007],{"style":325},[109,16005],{"className":16006,"style":190},[189],[109,16008,16010],{"className":16009},[194,195,196,197],[109,16011,16013],{"className":16012},[159,197],[109,16014,16016],{"className":16015},[159,338,197],[109,16017,486],{"className":16018},[159,197],[109,16020,206],{"className":16021},[205],[109,16023,16025],{"className":16024},[177],[109,16026,16028],{"className":16027,"style":351},[181],[109,16029],{},[109,16031,16033,16036],{"className":16032},[159],[109,16034,374],{"className":16035},[159,163],[109,16037,16039],{"className":16038},[168],[109,16040,16042,16062],{"className":16041},[172,173],[109,16043,16045,16059],{"className":16044},[177],[109,16046,16048],{"className":16047,"style":182},[181],[109,16049,16050,16053],{"style":410},[109,16051],{"className":16052,"style":190},[189],[109,16054,16056],{"className":16055},[194,195,196,197],[109,16057,377],{"className":16058},[159,163,197],[109,16060,206],{"className":16061},[205],[109,16063,16065],{"className":16064},[177],[109,16066,16068],{"className":16067,"style":213},[181],[109,16069],{},[109,16071,16073],{"className":16072},[2268],[109,16074,1455],{"className":16075},[1821,1776],[109,16077],{"className":16078,"style":1646},[691],[109,16080,3764],{"className":16081},[1682],[109,16083],{"className":16084,"style":1646},[691],[109,16086,16088,16091,16148,16151],{"className":16087},[150],[109,16089],{"className":16090,"style":987},[154],[109,16092,16094,16097],{"className":16093},[159],[109,16095,6831],{"className":16096,"style":6990},[159,163],[109,16098,16100],{"className":16099},[168],[109,16101,16103,16140],{"className":16102},[172,173],[109,16104,16106,16137],{"className":16105},[177],[109,16107,16109,16120],{"className":16108,"style":310},[181],[109,16110,16111,16114],{"style":7236},[109,16112],{"className":16113,"style":190},[189],[109,16115,16117],{"className":16116},[194,195,196,197],[109,16118,862],{"className":16119,"style":905},[159,163,197],[109,16121,16122,16125],{"style":325},[109,16123],{"className":16124,"style":190},[189],[109,16126,16128],{"className":16127},[194,195,196,197],[109,16129,16131],{"className":16130},[159,197],[109,16132,16134],{"className":16133},[159,338,197],[109,16135,6870],{"className":16136},[159,197],[109,16138,206],{"className":16139},[205],[109,16141,16143],{"className":16142},[177],[109,16144,16146],{"className":16145,"style":1044},[181],[109,16147],{},[109,16149],{"className":16150,"style":692},[691],[109,16152,16154,16157],{"className":16153},[159],[109,16155,1067],{"className":16156,"style":1089},[159,163],[109,16158,16160],{"className":16159},[168],[109,16161,16163,16183],{"className":16162},[172,173],[109,16164,16166,16180],{"className":16165},[177],[109,16167,16169],{"className":16168,"style":893},[181],[109,16170,16171,16174],{"style":1104},[109,16172],{"className":16173,"style":190},[189],[109,16175,16177],{"className":16176},[194,195,196,197],[109,16178,862],{"className":16179,"style":905},[159,163,197],[109,16181,206],{"className":16182},[205],[109,16184,16186],{"className":16185},[177],[109,16187,16189],{"className":16188,"style":213},[181],[109,16190],{},[106,16192,9978,16193],{},[109,16194,16196,16213],{"className":16195},[112],[109,16197,16199],{"className":16198},[116],[118,16200,16201],{"xmlns":120},[122,16202,16203,16211],{},[125,16204,16205,16207,16209],{},[131,16206,862],{},[652,16208,1349],{},[131,16210,1476],{"mathvariant":1352},[138,16212,6279],{"encoding":140},[109,16214,16216,16234],{"className":16215,"ariaHidden":146},[145],[109,16217,16219,16222,16225,16228,16231],{"className":16218},[150],[109,16220],{"className":16221,"style":6289},[154],[109,16223,862],{"className":16224,"style":905},[159,163],[109,16226],{"className":16227,"style":1646},[691],[109,16229,1349],{"className":16230},[1682],[109,16232],{"className":16233,"style":1646},[691],[109,16235,16237,16240],{"className":16236},[150],[109,16238],{"className":16239,"style":3681},[154],[109,16241,1476],{"className":16242,"style":2333},[159,1686],[106,16244,16245,16246,16337,16338,16426],{},"Schwellenzeile, die Big-M-Form von „wenn ",[109,16247,16249,16270],{"className":16248},[112],[109,16250,16252],{"className":16251},[116],[118,16253,16254],{"xmlns":120},[122,16255,16256,16268],{},[125,16257,16258,16264,16266],{},[128,16259,16260,16262],{},[131,16261,1067],{},[131,16263,862],{},[652,16265,4554],{},[648,16267,657],{},[138,16269,7396],{"encoding":140},[109,16271,16273,16328],{"className":16272,"ariaHidden":146},[145],[109,16274,16276,16279,16319,16322,16325],{"className":16275},[150],[109,16277],{"className":16278,"style":1082},[154],[109,16280,16282,16285],{"className":16281},[159],[109,16283,1067],{"className":16284,"style":1089},[159,163],[109,16286,16288],{"className":16287},[168],[109,16289,16291,16311],{"className":16290},[172,173],[109,16292,16294,16308],{"className":16293},[177],[109,16295,16297],{"className":16296,"style":893},[181],[109,16298,16299,16302],{"style":1104},[109,16300],{"className":16301,"style":190},[189],[109,16303,16305],{"className":16304},[194,195,196,197],[109,16306,862],{"className":16307,"style":905},[159,163,197],[109,16309,206],{"className":16310},[205],[109,16312,16314],{"className":16313},[177],[109,16315,16317],{"className":16316,"style":213},[181],[109,16318],{},[109,16320],{"className":16321,"style":1646},[691],[109,16323,4554],{"className":16324},[1682],[109,16326],{"className":16327,"style":1646},[691],[109,16329,16331,16334],{"className":16330},[150],[109,16332],{"className":16333,"style":4642},[154],[109,16335,657],{"className":16336},[159],", betragen die Ausgaben mindestens ",[109,16339,16341,16360],{"className":16340},[112],[109,16342,16344],{"className":16343},[116],[118,16345,16346],{"xmlns":120},[122,16347,16348,16358],{},[125,16349,16350],{},[267,16351,16352,16354,16356],{},[131,16353,6831],{},[131,16355,862],{},[275,16357,6870],{},[138,16359,7487],{"encoding":140},[109,16361,16363],{"className":16362,"ariaHidden":146},[145],[109,16364,16366,16369],{"className":16365},[150],[109,16367],{"className":16368,"style":987},[154],[109,16370,16372,16375],{"className":16371},[159],[109,16373,6831],{"className":16374,"style":6990},[159,163],[109,16376,16378],{"className":16377},[168],[109,16379,16381,16418],{"className":16380},[172,173],[109,16382,16384,16415],{"className":16383},[177],[109,16385,16387,16398],{"className":16386,"style":310},[181],[109,16388,16389,16392],{"style":7236},[109,16390],{"className":16391,"style":190},[189],[109,16393,16395],{"className":16394},[194,195,196,197],[109,16396,862],{"className":16397,"style":905},[159,163,197],[109,16399,16400,16403],{"style":325},[109,16401],{"className":16402,"style":190},[189],[109,16404,16406],{"className":16405},[194,195,196,197],[109,16407,16409],{"className":16408},[159,197],[109,16410,16412],{"className":16411},[159,338,197],[109,16413,6870],{"className":16414},[159,197],[109,16416,206],{"className":16417},[205],[109,16419,16421],{"className":16420},[177],[109,16422,16424],{"className":16423,"style":1044},[181],[109,16425],{},"“",[106,16428,16429,16430,16521,16522,16550,16551,16639,16640,16731],{},"Der Versand kann nur dann kostenfrei sein, ",[109,16431,16433,16454],{"className":16432},[112],[109,16434,16436],{"className":16435},[116],[118,16437,16438],{"xmlns":120},[122,16439,16440,16452],{},[125,16441,16442,16448,16450],{},[128,16443,16444,16446],{},[131,16445,1067],{},[131,16447,862],{},[652,16449,4554],{},[648,16451,657],{},[138,16453,7396],{"encoding":140},[109,16455,16457,16512],{"className":16456,"ariaHidden":146},[145],[109,16458,16460,16463,16503,16506,16509],{"className":16459},[150],[109,16461],{"className":16462,"style":1082},[154],[109,16464,16466,16469],{"className":16465},[159],[109,16467,1067],{"className":16468,"style":1089},[159,163],[109,16470,16472],{"className":16471},[168],[109,16473,16475,16495],{"className":16474},[172,173],[109,16476,16478,16492],{"className":16477},[177],[109,16479,16481],{"className":16480,"style":893},[181],[109,16482,16483,16486],{"style":1104},[109,16484],{"className":16485,"style":190},[189],[109,16487,16489],{"className":16488},[194,195,196,197],[109,16490,862],{"className":16491,"style":905},[159,163,197],[109,16493,206],{"className":16494},[205],[109,16496,16498],{"className":16497},[177],[109,16499,16501],{"className":16500,"style":213},[181],[109,16502],{},[109,16504],{"className":16505,"style":1646},[691],[109,16507,4554],{"className":16508},[1682],[109,16510],{"className":16511,"style":1646},[691],[109,16513,16515,16518],{"className":16514},[150],[109,16516],{"className":16517,"style":4642},[154],[109,16519,657],{"className":16520},[159],", wenn die Ausgaben bei Lieferant ",[109,16523,16525,16538],{"className":16524},[112],[109,16526,16528],{"className":16527},[116],[118,16529,16530],{"xmlns":120},[122,16531,16532,16536],{},[125,16533,16534],{},[131,16535,862],{},[138,16537,862],{"encoding":140},[109,16539,16541],{"className":16540,"ariaHidden":146},[145],[109,16542,16544,16547],{"className":16543},[150],[109,16545],{"className":16546,"style":947},[154],[109,16548,862],{"className":16549,"style":905},[159,163]," den Wert ",[109,16552,16554,16573],{"className":16553},[112],[109,16555,16557],{"className":16556},[116],[118,16558,16559],{"xmlns":120},[122,16560,16561,16571],{},[125,16562,16563],{},[267,16564,16565,16567,16569],{},[131,16566,6831],{},[131,16568,862],{},[275,16570,6870],{},[138,16572,7487],{"encoding":140},[109,16574,16576],{"className":16575,"ariaHidden":146},[145],[109,16577,16579,16582],{"className":16578},[150],[109,16580],{"className":16581,"style":987},[154],[109,16583,16585,16588],{"className":16584},[159],[109,16586,6831],{"className":16587,"style":6990},[159,163],[109,16589,16591],{"className":16590},[168],[109,16592,16594,16631],{"className":16593},[172,173],[109,16595,16597,16628],{"className":16596},[177],[109,16598,16600,16611],{"className":16599,"style":310},[181],[109,16601,16602,16605],{"style":7236},[109,16603],{"className":16604,"style":190},[189],[109,16606,16608],{"className":16607},[194,195,196,197],[109,16609,862],{"className":16610,"style":905},[159,163,197],[109,16612,16613,16616],{"style":325},[109,16614],{"className":16615,"style":190},[189],[109,16617,16619],{"className":16618},[194,195,196,197],[109,16620,16622],{"className":16621},[159,197],[109,16623,16625],{"className":16624},[159,338,197],[109,16626,6870],{"className":16627},[159,197],[109,16629,206],{"className":16630},[205],[109,16632,16634],{"className":16633},[177],[109,16635,16637],{"className":16636,"style":1044},[181],[109,16638],{}," erreichen. Für ",[109,16641,16643,16664],{"className":16642},[112],[109,16644,16646],{"className":16645},[116],[118,16647,16648],{"xmlns":120},[122,16649,16650,16662],{},[125,16651,16652,16658,16660],{},[128,16653,16654,16656],{},[131,16655,1067],{},[131,16657,862],{},[652,16659,4554],{},[648,16661,650],{},[138,16663,7792],{"encoding":140},[109,16665,16667,16722],{"className":16666,"ariaHidden":146},[145],[109,16668,16670,16673,16713,16716,16719],{"className":16669},[150],[109,16671],{"className":16672,"style":1082},[154],[109,16674,16676,16679],{"className":16675},[159],[109,16677,1067],{"className":16678,"style":1089},[159,163],[109,16680,16682],{"className":16681},[168],[109,16683,16685,16705],{"className":16684},[172,173],[109,16686,16688,16702],{"className":16687},[177],[109,16689,16691],{"className":16690,"style":893},[181],[109,16692,16693,16696],{"style":1104},[109,16694],{"className":16695,"style":190},[189],[109,16697,16699],{"className":16698},[194,195,196,197],[109,16700,862],{"className":16701,"style":905},[159,163,197],[109,16703,206],{"className":16704},[205],[109,16706,16708],{"className":16707},[177],[109,16709,16711],{"className":16710,"style":213},[181],[109,16712],{},[109,16714],{"className":16715,"style":1646},[691],[109,16717,4554],{"className":16718},[1682],[109,16720],{"className":16721,"style":1646},[691],[109,16723,16725,16728],{"className":16724},[150],[109,16726],{"className":16727,"style":4642},[154],[109,16729,650],{"className":16730},[159]," ist die Zeile trivial erfüllt.",[66,16733,7864],{"id":7863},[11,16735,16736,16737,16739,16740,16742],{},"Hier können Sie das MILP aus seiner gezippten ",[61,16738,7870],{},"-Datei laden und mit CP-SAT lösen. Beachten Sie, dass die Leistung von CP-SAT recht empfindlich auf die Zahl der ",[61,16741,63],{}," sowie auf das Zeitlimit reagieren kann.",[7876,16744,16745],{"className":7878,"code":7879,"language":7880,"meta":7881,"style":7881},[61,16746,16747,16761,16767,16771,16775,16783,16805,16809,16821,16829,16839,16853,16861,16865,16885,16907],{"__ignoreMap":7881},[109,16748,16749,16751,16753,16755,16757,16759],{"class":7886,"line":7887},[109,16750,7891],{"class":7890},[109,16752,7895],{"class":7894},[109,16754,7898],{"class":7890},[109,16756,7901],{"class":7894},[109,16758,7904],{"class":7890},[109,16760,7907],{"class":7894},[109,16762,16763,16765],{"class":7886,"line":7910},[109,16764,7898],{"class":7890},[109,16766,7915],{"class":7894},[109,16768,16769],{"class":7886,"line":7918},[109,16770,7922],{"emptyLinePlaceholder":7921},[109,16772,16773],{"class":7886,"line":7925},[109,16774,7929],{"class":7928},[109,16776,16777,16779,16781],{"class":7886,"line":7932},[109,16778,7935],{"class":7894},[109,16780,4554],{"class":7890},[109,16782,7940],{"class":7894},[109,16784,16785,16787,16789,16791,16793,16795,16797,16799,16801,16803],{"class":7886,"line":7943},[109,16786,7946],{"class":7894},[109,16788,1067],{"class":7890},[109,16790,7554],{"class":7951},[109,16792,7955],{"class":7954},[109,16794,7958],{"class":7951},[109,16796,7961],{"class":7954},[109,16798,7964],{"class":7951},[109,16800,3167],{"class":7894},[109,16802,7969],{"class":7951},[109,16804,7972],{"class":7894},[109,16806,16807],{"class":7886,"line":7975},[109,16808,7922],{"emptyLinePlaceholder":7921},[109,16810,16811,16813,16815,16817,16819],{"class":7886,"line":7980},[109,16812,7983],{"class":7894},[109,16814,4554],{"class":7890},[109,16816,7988],{"class":7894},[109,16818,7991],{"class":7951},[109,16820,7994],{"class":7894},[109,16822,16823,16825,16827],{"class":7886,"line":7997},[109,16824,8000],{"class":7894},[109,16826,8003],{"class":7954},[109,16828,7994],{"class":7894},[109,16830,16831,16833,16835,16837],{"class":7886,"line":8008},[109,16832,63],{"class":7954},[109,16834,8013],{"class":7890},[109,16836,8016],{"class":7954},[109,16838,8019],{"class":7928},[109,16840,16841,16843,16845,16847,16849,16851],{"class":7886,"line":8022},[109,16842,8025],{"class":7894},[109,16844,1067],{"class":7890},[109,16846,8030],{"class":7951},[109,16848,8033],{"class":7954},[109,16850,7554],{"class":7951},[109,16852,7994],{"class":7894},[109,16854,16855,16857,16859],{"class":7886,"line":8040},[109,16856,8043],{"class":7894},[109,16858,4554],{"class":7890},[109,16860,8048],{"class":7894},[109,16862,16863],{"class":7886,"line":8051},[109,16864,7922],{"emptyLinePlaceholder":7921},[109,16866,16867,16869,16871,16873,16875,16877,16879,16881,16883],{"class":7886,"line":8056},[109,16868,8059],{"class":7954},[109,16870,1374],{"class":7894},[109,16872,1067],{"class":7890},[109,16874,8066],{"class":7951},[109,16876,8069],{"class":7954},[109,16878,8072],{"class":7894},[109,16880,8075],{"class":7954},[109,16882,7554],{"class":7951},[109,16884,7994],{"class":7894},[109,16886,16887,16889,16891,16893,16895,16897,16899,16901,16903,16905],{"class":7886,"line":8082},[109,16888,8059],{"class":7954},[109,16890,1374],{"class":7894},[109,16892,1067],{"class":7890},[109,16894,8091],{"class":7951},[109,16896,8069],{"class":7954},[109,16898,8096],{"class":7894},[109,16900,8099],{"class":7890},[109,16902,8075],{"class":7954},[109,16904,7554],{"class":7951},[109,16906,7994],{"class":7894},[109,16908,16909,16911,16913,16915,16917,16919,16921,16923,16925,16927,16929,16931],{"class":7886,"line":8108},[109,16910,8059],{"class":7954},[109,16912,1374],{"class":7894},[109,16914,1067],{"class":7890},[109,16916,8117],{"class":7951},[109,16918,8069],{"class":7954},[109,16920,8122],{"class":7894},[109,16922,8125],{"class":7890},[109,16924,8075],{"class":7954},[109,16926,8130],{"class":7951},[109,16928,8033],{"class":7954},[109,16930,8135],{"class":7951},[109,16932,7994],{"class":7894},[8139,16934],{":lines":8141,"label":16935},"Ausgabe",[66,16937,8146],{"id":8145},[11,16939,16940],{},"Im Folgenden bauen wir das Modell mit Pyomo nach (um zu zeigen, wie das geht) und schicken es dann zum Lösen an Quicopts API-Service.",[7876,16942,16943],{"className":7878,"code":8152,"language":7880,"meta":7881,"style":7881},[61,16944,16945,16955,16965,16975,16979,16983],{"__ignoreMap":7881},[109,16946,16947,16949,16951,16953],{"class":7886,"line":7887},[109,16948,7898],{"class":7890},[109,16950,8161],{"class":7894},[109,16952,7904],{"class":7890},[109,16954,8166],{"class":7894},[109,16956,16957,16959,16961,16963],{"class":7886,"line":7910},[109,16958,7891],{"class":7890},[109,16960,8173],{"class":7894},[109,16962,7898],{"class":7890},[109,16964,8178],{"class":7894},[109,16966,16967,16969,16971,16973],{"class":7886,"line":7918},[109,16968,7891],{"class":7890},[109,16970,8185],{"class":7894},[109,16972,7898],{"class":7890},[109,16974,8190],{"class":7894},[109,16976,16977],{"class":7886,"line":7925},[109,16978,7922],{"emptyLinePlaceholder":7921},[109,16980,16981],{"class":7886,"line":7932},[109,16982,8199],{"class":7928},[109,16984,16985,16987,16989],{"class":7886,"line":7943},[109,16986,7935],{"class":7894},[109,16988,4554],{"class":7890},[109,16990,8208],{"class":7894},[11,16992,16993,16994,12162,17047,17081,17082,17085,17086,17120],{},"Als Nächstes führen wir die Entscheidungsvariablen ",[109,16995,16997,17014],{"className":16996},[112],[109,16998,17000],{"className":16999},[116],[118,17001,17002],{"xmlns":120},[122,17003,17004,17012],{},[125,17005,17006,17008,17010],{},[131,17007,133],{"mathvariant":1317},[652,17009,654],{"separator":146},[131,17011,578],{"mathvariant":1317},[138,17013,8232],{"encoding":140},[109,17015,17017],{"className":17016,"ariaHidden":146},[145],[109,17018,17020,17023,17032,17035,17038],{"className":17019},[150],[109,17021],{"className":17022,"style":8242},[154],[109,17024,17026],{"className":17025},[159],[109,17027,17029],{"className":17028},[159],[109,17030,133],{"className":17031,"style":1570},[159,1569],[109,17033,654],{"className":17034},[687],[109,17036],{"className":17037,"style":692},[691],[109,17039,17041],{"className":17040},[159],[109,17042,17044],{"className":17043},[159],[109,17045,578],{"className":17046},[159,1569],[109,17048,17050,17063],{"className":17049},[112],[109,17051,17053],{"className":17052},[116],[118,17054,17055],{"xmlns":120},[122,17056,17057,17061],{},[125,17058,17059],{},[131,17060,374],{"mathvariant":1317},[138,17062,3184],{"encoding":140},[109,17064,17066],{"className":17065,"ariaHidden":146},[145],[109,17067,17069,17072],{"className":17068},[150],[109,17070],{"className":17071,"style":3157},[154],[109,17073,17075],{"className":17074},[159],[109,17076,17078],{"className":17077},[159],[109,17079,374],{"className":17080},[159,1569]," ein, denen wir oben bereits begegnet sind. Beachten Sie, dass ",[56,17083,17084],{},"Nebenbedingungstyp (1)"," bereits in der Pyomo-Deklaration von ",[109,17087,17089,17102],{"className":17088},[112],[109,17090,17092],{"className":17091},[116],[118,17093,17094],{"xmlns":120},[122,17095,17096,17100],{},[125,17097,17098],{},[131,17099,578],{"mathvariant":1317},[138,17101,3220],{"encoding":140},[109,17103,17105],{"className":17104,"ariaHidden":146},[145],[109,17106,17108,17111],{"className":17107},[150],[109,17109],{"className":17110,"style":3157},[154],[109,17112,17114],{"className":17113},[159],[109,17115,17117],{"className":17116},[159],[109,17118,578],{"className":17119},[159,1569]," enthalten ist:",[7876,17122,17123],{"className":7878,"code":8343,"language":7880,"meta":7881,"style":7881},[61,17124,17125,17129,17143,17147,17151,17171,17175,17179,17183,17191,17195,17199,17215,17219,17223,17233],{"__ignoreMap":7881},[109,17126,17127],{"class":7886,"line":7887},[109,17128,8350],{"class":7928},[109,17130,17131,17133,17135,17137,17139,17141],{"class":7886,"line":7910},[109,17132,8355],{"class":7894},[109,17134,4554],{"class":7890},[109,17136,8360],{"class":7894},[109,17138,8364],{"class":8363},[109,17140,4554],{"class":7890},[109,17142,8369],{"class":7894},[109,17144,17145],{"class":7886,"line":7918},[109,17146,7922],{"emptyLinePlaceholder":7921},[109,17148,17149],{"class":7886,"line":7925},[109,17150,8378],{"class":7928},[109,17152,17153,17155,17157,17159,17161,17163,17165,17167,17169],{"class":7886,"line":7932},[109,17154,8383],{"class":7894},[109,17156,4554],{"class":7890},[109,17158,8388],{"class":7894},[109,17160,8364],{"class":8363},[109,17162,4554],{"class":7890},[109,17164,8395],{"class":7894},[109,17166,8398],{"class":8363},[109,17168,8401],{"class":7890},[109,17170,8404],{"class":7894},[109,17172,17173],{"class":7886,"line":7943},[109,17174,8409],{"class":7894},[109,17176,17177],{"class":7886,"line":7975},[109,17178,7922],{"emptyLinePlaceholder":7921},[109,17180,17181],{"class":7886,"line":7980},[109,17182,8418],{"class":7928},[109,17184,17185,17187,17189],{"class":7886,"line":7997},[109,17186,8423],{"class":7894},[109,17188,4554],{"class":7890},[109,17190,8428],{"class":7894},[109,17192,17193],{"class":7886,"line":8008},[109,17194,7922],{"emptyLinePlaceholder":7921},[109,17196,17197],{"class":7886,"line":8022},[109,17198,8437],{"class":7928},[109,17200,17201,17203,17205,17207,17209,17211,17213],{"class":7886,"line":8040},[109,17202,8442],{"class":7894},[109,17204,4554],{"class":7890},[109,17206,8447],{"class":7894},[109,17208,8450],{"class":7890},[109,17210,8453],{"class":7894},[109,17212,8456],{"class":7890},[109,17214,8459],{"class":7894},[109,17216,17217],{"class":7886,"line":8051},[109,17218,7922],{"emptyLinePlaceholder":7921},[109,17220,17221],{"class":7886,"line":8056},[109,17222,8468],{"class":7928},[109,17224,17225,17227,17229,17231],{"class":7886,"line":8082},[109,17226,8450],{"class":7890},[109,17228,8475],{"class":7894},[109,17230,8456],{"class":7890},[109,17232,8480],{"class":7894},[109,17234,17235,17237,17239,17241,17243,17245,17247,17249,17251,17253,17255,17257,17259,17261,17263,17265,17267,17269,17271,17273,17275,17277],{"class":7886,"line":8108},[109,17236,8485],{"class":7894},[109,17238,1067],{"class":7890},[109,17240,8490],{"class":7951},[109,17242,8069],{"class":7954},[109,17244,1346],{"class":7894},[109,17246,8075],{"class":7954},[109,17248,7554],{"class":7951},[109,17250,8501],{"class":7894},[109,17252,8450],{"class":7890},[109,17254,8506],{"class":7894},[109,17256,8456],{"class":7890},[109,17258,8511],{"class":7894},[109,17260,8450],{"class":7890},[109,17262,8453],{"class":7894},[109,17264,8456],{"class":7890},[109,17266,8520],{"class":7894},[109,17268,1396],{"class":7890},[109,17270,8525],{"class":7894},[109,17272,8450],{"class":7890},[109,17274,8506],{"class":7894},[109,17276,8456],{"class":7890},[109,17278,8534],{"class":7894},[11,17280,17281],{},"Als Nächstes definieren wir die übrigen Nebenbedingungen, danach die Zielfunktion:",[7876,17283,17284],{"className":7878,"code":8540,"language":7880,"meta":7881,"style":7881},[61,17285,17286,17290,17336,17340,17344,17376,17380,17384,17416,17420,17424,17458,17462,17466,17490,17518,17548,17556,17576,17616,17624,17644,17648,17652],{"__ignoreMap":7881},[109,17287,17288],{"class":7886,"line":7887},[109,17289,8547],{"class":7928},[109,17291,17292,17294,17296,17298,17300,17302,17304,17306,17308,17310,17312,17314,17316,17318,17320,17322,17324,17326,17328,17330,17332,17334],{"class":7886,"line":7910},[109,17293,8552],{"class":7894},[109,17295,4554],{"class":7890},[109,17297,8557],{"class":7894},[109,17299,8560],{"class":8363},[109,17301,8401],{"class":7890},[109,17303,8565],{"class":7894},[109,17305,8568],{"class":7954},[109,17307,8571],{"class":7894},[109,17309,8574],{"class":7890},[109,17311,8577],{"class":7894},[109,17313,8580],{"class":7890},[109,17315,8583],{"class":7894},[109,17317,8574],{"class":7890},[109,17319,8588],{"class":7894},[109,17321,8450],{"class":7890},[109,17323,8453],{"class":7894},[109,17325,8456],{"class":7890},[109,17327,8597],{"class":7894},[109,17329,8600],{"class":7890},[109,17331,8603],{"class":7894},[109,17333,8574],{"class":7890},[109,17335,8608],{"class":7894},[109,17337,17338],{"class":7886,"line":7918},[109,17339,7922],{"emptyLinePlaceholder":7921},[109,17341,17342],{"class":7886,"line":7925},[109,17343,8617],{"class":7928},[109,17345,17346,17348,17350,17352,17354,17356,17358,17360,17362,17364,17366,17368,17370,17372,17374],{"class":7886,"line":7932},[109,17347,8622],{"class":7894},[109,17349,4554],{"class":7890},[109,17351,8627],{"class":7894},[109,17353,8560],{"class":8363},[109,17355,8401],{"class":7890},[109,17357,8634],{"class":7894},[109,17359,8568],{"class":7954},[109,17361,8639],{"class":7894},[109,17363,8450],{"class":7890},[109,17365,8506],{"class":7894},[109,17367,8456],{"class":7890},[109,17369,8648],{"class":7894},[109,17371,8651],{"class":7890},[109,17373,8654],{"class":7954},[109,17375,7994],{"class":7894},[109,17377,17378],{"class":7886,"line":7943},[109,17379,7922],{"emptyLinePlaceholder":7921},[109,17381,17382],{"class":7886,"line":7975},[109,17383,8665],{"class":7928},[109,17385,17386,17388,17390,17392,17394,17396,17398,17400,17402,17404,17406,17408,17410,17412,17414],{"class":7886,"line":7980},[109,17387,8670],{"class":7894},[109,17389,4554],{"class":7890},[109,17391,8675],{"class":7894},[109,17393,8560],{"class":8363},[109,17395,8401],{"class":7890},[109,17397,8682],{"class":7894},[109,17399,8568],{"class":7954},[109,17401,8687],{"class":7894},[109,17403,8450],{"class":7890},[109,17405,8453],{"class":7894},[109,17407,8456],{"class":7890},[109,17409,8696],{"class":7894},[109,17411,8699],{"class":7890},[109,17413,8654],{"class":7954},[109,17415,7994],{"class":7894},[109,17417,17418],{"class":7886,"line":7997},[109,17419,7922],{"emptyLinePlaceholder":7921},[109,17421,17422],{"class":7886,"line":8008},[109,17423,8712],{"class":7928},[109,17425,17426,17428,17430,17432,17434,17436,17438,17440,17442,17444,17446,17448,17450,17452,17454,17456],{"class":7886,"line":8022},[109,17427,8717],{"class":7894},[109,17429,4554],{"class":7890},[109,17431,8722],{"class":7894},[109,17433,8560],{"class":8363},[109,17435,8401],{"class":7890},[109,17437,8729],{"class":7894},[109,17439,8568],{"class":7954},[109,17441,8571],{"class":7894},[109,17443,8574],{"class":7890},[109,17445,8577],{"class":7894},[109,17447,8450],{"class":7890},[109,17449,8453],{"class":7894},[109,17451,8456],{"class":7890},[109,17453,8746],{"class":7894},[109,17455,8699],{"class":7890},[109,17457,8751],{"class":7894},[109,17459,17460],{"class":7886,"line":8040},[109,17461,7922],{"emptyLinePlaceholder":7921},[109,17463,17464],{"class":7886,"line":8051},[109,17465,8760],{"class":7928},[109,17467,17468,17470,17472,17474,17476,17478,17480,17482,17484,17486,17488],{"class":7886,"line":8056},[109,17469,8765],{"class":7894},[109,17471,4554],{"class":7890},[109,17473,8770],{"class":7894},[109,17475,8568],{"class":7954},[109,17477,8775],{"class":7894},[109,17479,8574],{"class":7890},[109,17481,8780],{"class":7894},[109,17483,8450],{"class":7890},[109,17485,8506],{"class":7894},[109,17487,8456],{"class":7890},[109,17489,8789],{"class":7894},[109,17491,17492,17494,17496,17498,17500,17502,17504,17506,17508,17510,17512,17514,17516],{"class":7886,"line":8082},[109,17493,8794],{"class":7890},[109,17495,8797],{"class":7954},[109,17497,8800],{"class":7894},[109,17499,8574],{"class":7890},[109,17501,8577],{"class":7894},[109,17503,1396],{"class":7890},[109,17505,8809],{"class":7894},[109,17507,8574],{"class":7890},[109,17509,8588],{"class":7894},[109,17511,8450],{"class":7890},[109,17513,8453],{"class":7894},[109,17515,8456],{"class":7890},[109,17517,8822],{"class":7894},[109,17519,17520,17522,17524,17526,17528,17530,17532,17534,17536,17538,17540,17542,17544,17546],{"class":7886,"line":8108},[109,17521,8794],{"class":7890},[109,17523,8797],{"class":7954},[109,17525,8831],{"class":7894},[109,17527,8834]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dann schicken wir es ab:",[7876,17676,17677],{"className":7878,"code":9015,"language":7880,"meta":7881,"style":7881},[61,17678,17679,17683,17707],{"__ignoreMap":7881},[109,17680,17681],{"class":7886,"line":7887},[109,17682,9022],{"class":7928},[109,17684,17685,17687,17689,17691,17693,17695,17697,17699,17701,17703,17705],{"class":7886,"line":7910},[109,17686,9027],{"class":7894},[109,17688,4554],{"class":7890},[109,17690,9032],{"class":7894},[109,17692,9035],{"class":8363},[109,17694,4554],{"class":7890},[109,17696,9040],{"class":7954},[109,17698,3167],{"class":7894},[109,17700,9045],{"class":8363},[109,17702,4554],{"class":7890},[109,17704,9050],{"class":7954},[109,17706,7994],{"class":7894},[109,17708,17709,17711],{"class":7886,"line":7918},[109,17710,8059],{"class":7954},[109,17712,9059],{"class":7894},[8139,17714],{":lines":9062,"label":16935},[11,17716,17717],{},"Sehen wir uns die Qualität der Lösung an:",[7876,17719,17720],{"className":7878,"code":9068,"language":7880,"meta":7881,"style":7881},[61,17721,17722,17744],{"__ignoreMap":7881},[109,17723,17724,17726,17728,17730,17732,17734,17736,17738,17740,17742],{"class":7886,"line":7887},[109,17725,8059],{"class":7954},[109,17727,1374],{"class":7894},[109,17729,1067],{"class":7890},[109,17731,9081],{"class":7951},[109,17733,8069],{"class":7954},[109,17735,8096],{"class":7894},[109,17737,8099],{"class":7890},[109,17739,8075],{"class":7954},[109,17741,7554],{"class":7951},[109,17743,7994],{"class":7894},[109,17745,17746,17748,17750,17752,17754,17756,17758,17760,17762,17764],{"class":7886,"line":7910},[109,17747,8059],{"class":7954},[109,17749,1374],{"class":7894},[109,17751,1067],{"class":7890},[109,17753,9104],{"class":7951},[109,17755,8069],{"class":7954},[109,17757,9109],{"class":7894},[109,17759,8099],{"class":7890},[109,17761,8075],{"class":7954},[109,17763,7554],{"class":7951},[109,17765,7994],{"class":7894},[8139,17767],{":lines":9120,"label":16935},[11,17769,17770,17771,17774,17775,17778,17779,17783],{},"Quicopt hat in weniger als ",[56,17772,17773],{},"zwei Sekunden"," eine ",[56,17776,17777],{},"bessere Lösung"," gefunden! CP-SAT hat 60 s auf acht CPUs investiert (womit es manchmal tatsächlich gelingt, dieselbe Lösung wie Quicopt zu finden). Das ",[24,17780,17782],{"href":40,"rel":17781},[28],"Notebook"," auf unserem GitHub enthält zusätzliche Nachweise, dass die Lösung des Quicopt-Solvers in der ursprünglichen MILP-Formulierung korrekt ist.",[9138,17785,9140],{},{"title":7881,"searchDepth":7910,"depth":7910,"links":17787},[17788,17793,17794],{"id":9213,"depth":7910,"text":9214,"children":17789},[17790,17791,17792],{"id":9217,"depth":7918,"text":9218},{"id":10302,"depth":7918,"text":10303},{"id":12089,"depth":7918,"text":12090},{"id":7863,"depth":7910,"text":7864},{"id":8145,"depth":7910,"text":8146},"Wir tauchen tief in das sogenannte „Internet-Shopping-Optimierungsproblem“ ein, auch bekannt als „Lieferantenauswahl und Bestellaufteilung“. Es tritt praktisch immer dann auf, wenn eine große Zahl von Artikeln bei einer großen Zahl von Lieferanten bestellt werden muss. Da es um ganzzahlige Mengen geht, wird es üblicherweise als gemischt-ganzzahliges lineares Problem (MILP) formuliert. Leider gelten MILPs als NP-schwer – das heißt mehr oder weniger, dass sie sehr schwer zu lösen sein können.",{},"\u002Fuse-cases\u002Finternet-shopping.de",{"title":9161,"description":17795},"use-cases\u002Finternet-shopping.de","nED0GB2BeXN6KXpUjuuGRe43mBAJh0PXFQz_6Rob0c8",1793877539754]