Research Article

# A Hybrid Method for Modeling and Solving Supply Chain Optimization Problems with Soft and Logical Constraints

## Table 8

The results of numerical examples for both methods.
(a)
 DHIP MILP_T () 165180 10 818 (788) 1367 570715 35 987 (948) 1376 897015 71 1264 (1220) 1381 1564990 145 1536 (1492) 1381 2619310 201 1696 (1652) 1381 2963030 254 2080 (2036) 1381 3860300 290 2240 (2196) 1381 4786960 320 2512 (2468) 1381 5964200 321 2779 (2736) 1380 7397970 367 2939 (2896) 1380
(b)
 Mathematical programming MILP () 165180 84 80117 (79204) 47426 570715 752 80117 (79204) 50626 897015 7248 80117 (79204) 53826 1613453 600 80117 (79204) 57026 2699720 600 80117 (79204) 60226 3003310 600 80117 (79204) 63426 4062950 600 80117 (79204) 66626 5092400 600 80117 (79204) 69826 6543450 600 80117 (79204) 73026 8213450 600 80117 (79204) 76226
(c)
 DHIP HM () 193590 17 829 (800) 1384 641315 45 999 (960) 1394 1000770 82 1276 (1232) 1399 1677290 198 1548 (1504) 1399 2638270 234 1708 (1664) 1399 3046250 298 2092 (2045) 1399 3970820 311 2252 (2208) 1399 4854680 345 2524 (2480) 1399 6034960 401 2791 (2748) 1398 7475980 453 2951 (2908) 1398
(d)
 DHIP HM () 165180 11 821 (792) 1371 570715 38 991 (952) 1381 897015 74 1268 (1224) 1386 1564990 151 1540 (1496) 1386 2631090 204 1700 (1656) 1386 2998950 260 2084 (2040) 1386 3945430 296 2244 (2200) 1386 4796250 321 2516 (2472) 1386 6004750 378 2783 (2740) 1385 7652490 421 2943 (2900) 1385
(e)
 DHIP HM () 165180 9 801 (772) 1266 570715 31 971 (922) 1276 897015 67 1240 (1196) 1381 1571600 134 1504 (1460) 1381 2634620 198 1664 (1620) 1381 3000310 234 2032 (1988) 1381 4027740 287 2192 (2148) 1381 4916740 311 2456 (2412) 1381 5978220 318 2715 (2672) 1380 7475980 332 2875 (2832) 1380
: the optimal value of Fc (objective function).
: time for finding solution (in seconds).
The feasible value of the objective function after the time .
Calculation was stopped after 600 s.
(): the number of decision variables (integer decision variables).
: the number of constraints.
MILP: the implementation in the MP-based environment - MILP model.
MILP_T: the implementation in the declarative hybrid implementation framework (DHIF) MILP model after transformation.
HM: implementation in the declarative hybrid implementation framework (DHIF) model after transformation.