CHAPTER 8 DISCUSSIONS

Transcription

1 153 CHAPTER 8 DISCUSSIONS This chapter discusses the developed models, methodologies to solve the developed models, performance of the developed methodologies and their inferences. 8.1 MULTI-PERIOD FIXED CHARGE MODELS The multi-period fixed charge models are an extension of the general MPDP problem in which a fixed cost is incurred for every shipment between the source and the destination. Since the multi-period fixed charge models belong to FCT category, they are more difficult to solve due to the presence of fixed costs. The complexity of the problem is further increased when time dependent inventories and backorder/subcontract are included in the model. This limits the usage of the conventional FCT solution procedures. The other practical points addressed in this thesis are the simultaneous consideration of the inventory at the suppliers location and customers location, admission of backorder/subcontract during excess demand period and consideration of production cost while integrating production and distribution. These novelties have been included in the proposed multi-period fixed charge models. Hence the models considered in this thesis are more complex than the general MPDP and FCT problems.

2 154 The review of literature on related papers of simple problem specific heuristics, SAA from neighbourhood based heuristics and GA from population based heuristics show that they are proven tools for NP-hard problems. In the light of the above, this thesis considers EVC heuristic, SAA based heuristic and GA based heuristic as the tools to evolve the solutions for all the multi-period fixed charge models under study. The capability of the proposed EVC, SAA and GA heuristics in handling the four multi-period fixed charge models is analysed by comparing their outputs with the lower bound ( ) and LINGO solution ( ). Forty test problems (given in Appendix I) are considered for the performance comparison. The complete statistics (shown in Tables 8.1 to 8.4 and Figures 8.1 to 8.4) and inferences are as follows:

3 155 Table 8.1 Percentage deviations of LINGO with lower bound and proposed heuristics with lower bound and LINGO in Model I Test Problem No. Size m * n * T Total cost of distribution Percentage deviation = Z1-Z Z2 EVC SAA 2 *100 GA Z EVC Z (SAA) Z (GA) 1. 2 * 2 * 2 9,190 8,484 9,190 9,190 9, * 3 * 2 17,625 16,813 18,275 17,925 17, * 4 * 2 14,915 14,132 16,275 14,985 14, * 5 * 2 30,850 30,021 31,580 30,850 30, * 6 * 2 28,720 27,450 30,039 28,720 28, * 7 * 2 18,320 18,214 25,636 18,320 18, * 2 * 2 8,200 8,157 8,600 8,200 8, * 3 * 2 11,290 11,237 13,390 11,290 11, * 4 * 2 15,010 15,000 16,755 15,010 15, * 5 * 2 18,690 17,991 22,986 18,690 18, * 6 * 2 24,300 23,674 25,400 24,300 24, * 7 * 2 25,780 25,731 32,525 25,780 25, * 2 * 2 10,155 10,106 10,475 10,155 10, * 3 * 2 21,620 20,695 21,960 21,620 21, * 4 * 2 20,421 20,347 23,320 20,421 20, * 5 * 2 20,460 20,310 22,970 20,460 20, * 6 * 2 24,805 21,178 26,510 24,805 24, * 2 * 2 14,285 14,010 15,555 14,285 14, * 3 * 2 21,180 21,019 22,590 21,180 21, * 4 * 2 24,283 23,206 26,780 24,283 24, * 5 * 2 20,528 19,542 24,864 20,528 20, * 6 * 2 17,444 16,834 20,525 17,444 17, * 2 * 3 14,530 14,511 14,530 14,530 14, * 3 * 3 25,100 25,051 26,025 25,100 25, * 4 * 3 22,655 21,916 25,515 22,655 22, * 2 * 3 11,500 11,479 13,240 11,500 11, * 3 * 3 17,340 17,340 18,580 17,960 17, * 4 * 3 23,165 20,773 23,165 22,640 22, * 5 * 3 23,820 21,542 24,142 22,260 22, * 2 * 3 15,435 14,608 16,210 15,435 15, * 3 * 3 22,650 21,299 25,720 22,650 22, * 4 * 3 28,390 27,415 28,390 27,830 27, * 5 * 3 26,120 24,266 26,960 24,805 24, * 2 * 3 21,200 21,140 26,520 21,200 21, * 3 * 3 28,610 27,668 32,038 28,610 28, * 2 * 4 17,975 17,275 18,725 17,975 17, * 3 * 4 38,725 38,369 43,845 38,725 38, * 4 * 4 20,860 20,208 25,180 20,860 20, * 2 * 4 12,360 12,069 13,100 12,390 12, * 3 * 4 13,875 13,112 14,595 13,525 13,

4 156 Table 8.2 Percentage deviations of LINGO with lower bound and proposed heuristics with lower bound and LINGO in Model II Test Problem No. Size m * n * T Total cost of distribution Percentage deviation = Z1-Z Z2 EVC SAA 2 *100 GA Z EVC Z (SAA) Z (GA) 1. 2 * 2 * 2 9,010 8,034 9,010 9,010 9, * 3 * 2 17,625 16,813 18,275 17,925 17, * 4 * 2 14,915 14,132 16,275 14,985 14, * 5 * 2 29,690 28,452 31,553 29,690 29, * 6 * 2 28,720 27,450 30,039 28,720 28, * 7 * 2 17,450 17,005 22,691 17,450 17, * 2 * 2 7,875 7,127 8,600 7,875 7, * 3 * 2 11,290 11,237 13,390 11,290 11, * 4 * 2 15,010 15,000 16,755 15,010 15, * 5 * 2 18,690 17,991 22,986 18,690 18, * 6 * 2 22,940 20,200 26,311 22,940 22, * 7 * 2 25,780 25,731 32,525 25,780 25, * 2 * 2 10,155 10,106 10,475 10,155 10, * 3 * 2 21,220 20,008 27,125 21,220 21, * 4 * 2 20,421 20,347 23,320 20,421 20, * 5 * 2 17,400 16,010 25,018 17,400 17, * 6 * 2 24,805 21,178 26,510 24,805 24, * 2 * 2 14,285 14,010 15,555 14,285 14, * 3 * 2 21,180 21,019 22,590 21,180 21, * 4 * 2 24,283 23,206 26,780 24,283 24, * 5 * 2 20,150 20,025 24,517 20,150 20, * 6 * 2 17,444 16,834 20,525 17,444 17, * 2 * 3 14,630 14,015 14,980 14,630 14, * 3 * 3 23,750 22,005 23,750 23,750 23, * 4 * 3 21,485 20,408 26,111 21,485 21, * 2 * 3 11,550 11,005 12,000 11,550 11, * 3 * 3 17,430 17,340 20,990 17,430 17, * 4 * 3 23,320 20,850 28,121 21,485 21, * 5 * 3 24,010 21,010 28,598 21,630 21, * 2 * 3 15,130 14,012 17,540 15,130 15, * 3 * 3 22,540 22,402 26,081 22,540 22, * 4 * 3 28,390 27,415 28,390 27,830 27, * 5 * 3 26,450 22,120 30,807 23,700 23, * 2 * 3 20,750 20,003 23,355 20,750 20, * 3 * 3 28,150 27,283 30,877 28,150 28, * 2 * 4 17,885 17,106 17,885 17,885 17, * 3 * 4 38,100 36,750 42,065 38,100 38, * 4 * 4 19,360 18,605 22,640 19,360 19, * 2 * 4 11,505 11,000 12,900 11,505 11, * 3 * 4 13,035 12,040 16,990 13,035 13,

5 157 Table 8.3 Percentage deviations of LINGO with lower bound and proposed heuristics with lower bound and LINGO in Model III Test Problem No. Size m * n * T Total cost of distribution Percentage deviation = Z1-Z Z2 EVC SAA 2 *100 GA Z EVC Z (SAA) Z (GA) 1. 2 * 2 * 2 13,365 12,659 13,365 13,365 13, * 3 * 2 22,500 21,688 23,150 22,800 22, * 4 * 2 21,305 20,522 22,665 21,375 21, * 5 * 2 45,350 44,521 46,080 45,350 45, * 6 * 2 43,420 42,150 44,739 43,420 43, * 7 * 2 30,120 30,014 37,436 30,120 30, * 2 * 2 11,955 11,912 12,355 11,955 11, * 3 * 2 17,390 17,337 19,490 17,390 17, * 4 * 2 21,600 21,590 23,345 21,600 21, * 5 * 2 31,310 30,611 35,606 31,310 31, * 6 * 2 38,220 37,594 39,320 38,220 38, * 7 * 2 45,500 45,451 52,245 45,500 45, * 2 * 2 15,695 15,646 16,015 15,695 15, * 3 * 2 37,390 36,465 37,730 37,390 37, * 4 * 2 38,061 37,987 40,960 38,061 38, * 5 * 2 35,260 35,110 37,770 35,260 35, * 6 * 2 39,335 35,708 41,040 39,335 39, * 2 * 2 21,845 21,570 23,115 21,845 21, * 3 * 2 36,360 36,199 37,770 36,360 36, * 4 * 2 45,533 44,456 48,030 45,533 45, * 5 * 2 34,538 33,552 38,874 34,538 34, * 6 * 2 26,074 25,464 29,155 26,074 26, * 2 * 3 21,405 21,386 21,405 21,405 21, * 3 * 3 32,150 32,101 33,075 32,150 32, * 4 * 3 32,255 31,516 35,115 32,255 32, * 2 * 3 16,920 16,899 18,660 16,920 16, * 3 * 3 27,170 27,170 28,510 27,790 27, * 4 * 3 32,105 29,713 32,105 31,580 31, * 5 * 3 37,630 35,352 37,952 36,070 36, * 2 * 3 23,755 22,928 24,530 23,755 23, * 3 * 3 38,610 37,259 41,680 38,610 38, * 4 * 3 46,230 45,255 46,230 45,670 45, * 5 * 3 44,140 42,286 44,980 42,825 42, * 2 * 3 38,890 38,830 44,210 38,890 38, * 3 * 3 48,740 47,798 52,168 48,740 48, * 2 * 4 25,355 24,655 26,105 25,355 25, * 3 * 4 49,845 49,489 54,965 49,845 49, * 4 * 4 26,710 26,058 31,030 26,710 26, * 2 * 4 18,290 17,999 19,030 18,320 18, * 3 * 4 19,495 18,732 20,215 19,145 19,

6 158 Table 8.4 Percentage deviations of LINGO with lower bound and proposed heuristics with lower bound and LINGO in Model IV Test Problem No. Size m * n * T Total cost of distribution Percentage deviation = Z1-Z Z2 EVC SAA 2 *100 GA Z EVC Z (SAA) Z (GA) 1. 2 * 2 * 2 13,185 12,209 13,185 13,185 13, * 3 * 2 22,500 21,688 23,150 22,800 22, * 4 * 2 21,305 20,522 22,665 21,375 21, * 5 * 2 44,190 42,952 46,053 44,190 44, * 6 * 2 43,420 42,150 44,739 43,420 43, * 7 * 2 29,250 28,805 34,491 29,250 29, * 2 * 2 11,630 10,882 12,355 11,630 11, * 3 * 2 17,390 17,337 19,490 17,390 17, * 4 * 2 21,600 21,590 23,345 21,600 21, * 5 * 2 31,310 30,611 35,606 31,310 31, * 6 * 2 36,860 34,120 40,231 36,860 36, * 7 * 2 45,500 45,451 52,245 45,500 45, * 2 * 2 15,695 15,646 16,015 15,695 15, * 3 * 2 36,990 35,778 42,895 36,990 36, * 4 * 2 38,061 37,987 40,960 38,061 38, * 5 * 2 32,200 30,810 39,818 32,200 32, * 6 * 2 39,335 35,708 41,040 39,335 39, * 2 * 2 21,845 21,570 23,115 21,845 21, * 3 * 2 36,360 36,199 37,770 36,360 36, * 4 * 2 45,533 44,456 48,030 45,533 45, * 5 * 2 34,160 34,035 38,527 34,160 34, * 6 * 2 26,074 25,464 29,155 26,074 26, * 2 * 3 21,505 20,890 21,855 21,505 21, * 3 * 3 30,800 29,055 30,800 30,800 30, * 4 * 3 31,085 30,008 35,711 31,085 31, * 2 * 3 16,970 16,425 17,420 16,970 16, * 3 * 3 27,260 27,170 28,410 27,260 27, * 4 * 3 32,260 29,790 37,061 30,425 30, * 5 * 3 37,820 34,820 42,408 35,440 35, * 2 * 3 23,450 22,332 25,860 23,450 23, * 3 * 3 38,500 38,362 42,041 38,500 38, * 4 * 3 46,230 45,255 46,230 45,670 45, * 5 * 3 44,470 40,140 48,827 41,720 41, * 2 * 3 38,440 37,693 41,045 38,440 38, * 3 * 3 48,280 47,413 51,007 48,280 48, * 2 * 4 25,265 24,486 25,265 25,265 25, * 3 * 4 49,220 47,870 53,185 49,220 49, * 4 * 4 25,210 24,455 28,490 25,210 25, * 2 * 4 17,435 16,930 18,830 17,435 17, * 3 * 4 18,655 17,660 22,610 18,655 18,

7 159 Figure 8.1 Percentage deviations of LINGO with lower bound and proposed heuristics with lower bound and LINGO in Model I Figure 8.2 Percentage deviations of LINGO with lower bound and proposed heuristics with lower bound and LINGO in Model II

8 160 Figure 8.3 Percentage deviations of LINGO with lower bound and proposed heuristics with lower bound and LINGO in Model III Figure 8.4 Percentage deviations of LINGO with lower bound and proposed heuristics with lower bound and LINGO in Model IV

9 LINGO SOLUTIONS LINGO solver, an Operations Research software tool, is very useful to solve hard optimization problems. It solves the problems by using branch and bound manager. In this thesis, the mathematical formulations of all the four models have been solved using LINGO to find its capability of providing optimal solutions and time taken for solution convergence. This is tested with forty test problems of different dimensions. The outcomes reveal the following: LINGO s modeling language enables the users to express their problems in a natural manner that is very similar to standard mathematical notation. It categorizes the multi-period fixed charge problems under PINLP class. Based on the performance analysis (chapter 4), LINGO is capable to solve small size test problems to global optimal. The computational solver times are reasonable and are in the range of seconds. But LINGO is not capable of providing solutions to 5 large problem instances and solver came to an end after certain time and resulted in failure reply (i.e. could not find global optimum). Hence, those 5 problems were solved to local optimal by LINGO. This may be due to the solver methodology used in LINGO. LINGO generally uses branch and bound methodology in solver execution. In branch and bound methodology, the computational time grows along with problem size. Hence the

10 162 computational solver time grows exponentially with the test problem dimensions. Eventually the discussion concludes that LINGO is suitable for small size multi-period fixed charge test problems and is generally not suitable to handle and solve large sized real time NP-hard problems. LINGO is also used to analyse the deviation among the four multiperiod fixed charge problems (Model I to Model IV) using the results of the sample problem solved using LINGO. In Model I, the total distribution cost is the sum of transportation costs, backorder penalty cost and inventory holding costs. In Model II, the total distribution cost is the sum of transportation costs, subcontracting costs and inventory holding costs. Though the subcontracting cost was assumed to be equal to backorder penalty cost in Model II, the distribution schedule and total cost are not equal. The reasons are: In backorder case, the backorder increases the total demand in the forthcoming period. Consequently it creates a distribution schedule for the current and forthcoming periods according to the new demand pattern. But in subcontract case, the excess demand is fulfilled by subcontract in the same period itself. Nothing is carried over to the forthcoming period. Eventually the subcontract creates a distribution schedule for the current and forthcoming periods according to the existing demand pattern. This is clearly shown in Model I distribution schedule and Model II distribution schedule. Since the Models III and IV are the extensions of Model I and Model II respectively, the total distribution cost comparison between Model III and Model IV provides the variations due to the addition of production cost in the objective function.

11 EVC HEURISTIC SOLUTIONS EVC heuristic is one of the problem specific heuristic to solve multi-period fixed charge problems. The outcomes reveal the following: EVC heuristic provides a good foundation for solving small size multi-period fixed charge problems. It converts non-linear multi-period problems into relaxed LP models. Since the relaxed models belong to LP category, it solves the entire test problems in short run time compared with other heuristics. It provides approximate and lower bound value to the problems. Generally the approximate solutions are inferior to the optimal solutions of the problems. Hence the optimal solution of the problem is expected to lie in between the lower bound and approximate solution. Sometimes the approximate solution may be equal to optimal solution. This is observed from the test problem instances solved by EVC heuristic (given in chapter 5). In 4 test problems EVC performs equally with in all the four models. The average percentage deviations of EVC heuristic with lower bound values in Model I, Model II, Model III and Model IV are 14.12%, 20.38%, 8.89% and 12.29% respectively. The average percentage deviations of EVC heuristic with LINGO solutions in Model I, Model II, Model III and Model IV are 10.45%, 14.04%, 6.61% and 8.52% respectively.

12 164 Eventually the discussion concludes that the EVC heuristic is suitable for all multi-period fixed charge benchmark problems, but it provides only approximate solutions. It also provides lower bound value of the problems. 8.4 SAA BASED HEURISTIC SOLUTIONS The performance of SAA based heuristic is as follows: Model I: In 5 test problems, SAA provides solutions better than. In 31 test problems SAA performs equally with and in 4 test problems SAA provides solutions inferior (< 3.5%) to. Model II: In 5 test problems, SAA provides solutions better than. In 33 test problems SAA performs equally with and in 2 test problems SAA provides solutions inferior (< 1.7%) to. Model III: In 5 test problems, SAA provides solutions better than. In 31 test problems SAA performs equally with and in 4 test problems SAA provides solutions inferior (< 2.28%) to. Model IV: In 5 test problems, SAA provides solutions better than. In 33 test problems SAA performs equally with and in 2 test problems SAA provides solutions inferior (< 1.33%) to. The average percentage deviations of SAA based heuristic with lower bound values in Model I, Model II, Model III and Model IV are 3.48%, 4.87%, 2.21% and 3.09% respectively.

13 165 The average percentage deviations of SAA based heuristic with LINGO solutions in Model I, Model II, Model III and Model IV are -0.36%, -0.87%, -0.22% and -0.54% respectively. Eventually the discussion concludes that the SAA based heuristic is a suitable tool for solving multi-period fixed charge problems in a reasonable computational time. 8.5 GA BASED HEURISTIC SOLUTIONS The performance of GA based heuristic is as follows: Model I: In 5 test problems, GA provides solutions better than. In 34 test problems GA performs equally with and in 1 test problem GA provides solutions inferior (< 0.25%) to. Model II: In 4 test problems, GA provides solutions better than and in the other 36 test problems GA performs equally with. Model III: In 5 test problems, GA provides solutions better than. In 34 test problems GA performs equally with and in 1 test problem GA provides solutions inferior (< 0.164%) to. Model IV: In 4 test problems, GA provides solutions better than and in the other 36 test problems GA performs equally with.

14 166 The average percentage deviations of GA based heuristic with lower bound values in Model I, Model II, Model III and Model IV are 3.34%, 4.82%, 2.12% and 3.05% respectively. The average percentage deviations of GA based heuristic with LINGO solutions in Model I, Model II, Model III and Model IV are -0.58%, -0.96%, -0.37% and -0.61% respectively. Eventually the discussion concludes that the GA based heuristic is a suitable tool for solving multi-period fixed charge problems in a reasonable computational time. The computational experience of the proposed approaches with respect to forty different test problems (m*n*t) are: The proposed LINGO generally uses branch and bound methodology in solver execution. In branch and bound methodology, the computational time grows along with problem size. Hence, the outcomes of test problems reveal that LINGO is not capable of providing optimal solution in large problem instances and the solver time grows exponentially with the problem dimensions. The outcomes of the proposed EVC heuristic reveal that the EVC heuristic provides approximate solutions to all the test problems in a quick computational/solver time, but are inferior to lower bound and LINGO solutions. The proposed SAA heuristic generates better solutions to all the test problems in a reasonable computational time and it grows linearly with problem size. The computational time of SAA mainly depends on parameters such as, initial temperature, final temperature, cooling rate, problem based string length and counters. The average computational solver time of SAA heuristic is considerably lower than LINGO.

15 167 The proposed GA heuristic also generates better solutions to all the test problems in a reasonable computational time and it grows linearly with problem size. The computational time of GA mainly depends on parameters such as, population size, problem based number of generation (100+(m*T*(n*T)) and problem based chromosome length. The average computational solver time of GA heuristic is considerably lower than LINGO. Figure 8.5 Computational/Solver time comparisons among LINGO and EVC, SAA and GA heuristics Figure 8.5 shows the computational time comparison of the proposed heuristics. The same computational time trends (reasonable in SAA and GA and exponential in LINGO) will be experienced while implementing the proposed heuristics to large scale problems. Finally, the computational/solver time comparisons reveal that the proposed SAA and GA based heuristics provides good solutions in a reasonable computational time to the real time multi-period fixed charge problems.