Article Contents
Article Contents

# Solving the facility location and fixed charge solid transportation problem

• * Corresponding author: Gbeminiyi John Oyewole
• In this paper, a new variant of the Solid Transportation Problem (STP) that incorporates both facility location and Fixed Charge Solid Transportation Problem (FCSTP) is presented with significant applications in logistics. It integrates decisions of diverse planning horizons: operational, tactical and strategic. The problem is termed Fixed Charge Solid Location and Transportation Problem (FCSLTP). Benchmark data obtained from the literature was extended for experimentation purposes. Solution to the FCSLTP was obtained using CPLEX commercial optimization solver. A Lagrange Relaxation Heuristic (LRH) was developed as an alternative solution for users not possibly having access to CPLEX. We further defined an equivalent FCSLTP in the main paper and termed this as FCSTP-EQ. The FCSTP-EQ was compared to our FCSLTP to investigate possible cost savings with both formulations. Results obtained showed CPLEX outperforming the Lagrange relaxation heuristic developed both in the upper bound and lower bound generation for the problem sizes considered. Additionally, the cost savings obtained using the FCSLTP was consistently better than the FCSTP-EQ. The upper bound generation capability of Lagrange relaxation could possibly be improved by using better search methods such as metaheuristics. Under certain conditions, the FCSTP could feasibly be used as a starting solution to solve the FCSLTP.

Mathematics Subject Classification: Primary: 90B06, 90B10; Secondary: 90C11.

 Citation:

• Figure 1.  Schematic representation of FCSLTP

Figure 2.  Procedure for computing the FCSTP-EQ

Figure 3.  FCSLTP and FCSTP-EQ

Figure 4.  Solution time of FCSLTP and FCSTP

Table 1.  Problem sizes and number of instances used for experimentation

 Problem Size No. Problem Size $\boldsymbol{m}\boldsymbol{\times }\boldsymbol{n}\boldsymbol{\times }\boldsymbol{a}$ No of instances 1 2 3 4 5 6 7 8 9 10 5$\mathrm{\times}$5$\mathrm{\times}$2 5$\mathrm{\times}$8$\mathrm{\times}$2 7$\mathrm{\times}$10$\mathrm{\times}$2 8$\mathrm{\times}$8$\mathrm{\times}$2 10$\mathrm{\times}$10$\mathrm{\times}$3 10$\mathrm{\times}$20$\mathrm{\times}$3 15$\mathrm{\times}$30$\mathrm{\times}$4 20$\mathrm{\times}$20$\mathrm{\times}$5 25$\mathrm{\times}$38$\mathrm{\times}$8 35$\mathrm{\times}$42$\mathrm{\times}$9 5 5 5 5 5 5 5 5 5 5

Table 2.  Parameter distribution used for experimentation

 Parameter Distribution ${\boldsymbol{S}}_{\boldsymbol{i}}$ U(200,400)${\boldsymbol{D}}_{\boldsymbol{j}}$ U(50,100)${\boldsymbol{T}}_{\boldsymbol{r}}$ U(800, 1800)${\boldsymbol{c}}_{\boldsymbol{ijr}}$ U(20,150)${\boldsymbol{H}}_{\boldsymbol{ijr}}$ U(200,600)${\boldsymbol{F}}_{\boldsymbol{i}}\boldsymbol{=}\boldsymbol{U}\left(\boldsymbol{0},\boldsymbol{90}\right)\boldsymbol{+\ }\sqrt{{\boldsymbol{S}}_{\boldsymbol{i}}}\boldsymbol{\ }\boldsymbol{U}\boldsymbol{(}\boldsymbol{100},\boldsymbol{110}\boldsymbol{)}$ ${\boldsymbol{M}}_{\boldsymbol{ijr}}\boldsymbol{=}{\boldsymbol{\mathrm{min}} \boldsymbol{(}{\boldsymbol{S}}_{\boldsymbol{i}}\boldsymbol{,\ }{\boldsymbol{D}}_{\boldsymbol{j}},{\boldsymbol{T}}_{\boldsymbol{r}}\boldsymbol{)}\ }$

Table 3.  Mean values for best lower bound and upper bound computation per solution method

 Problem Size No. Problem Size $\boldsymbol{m}\boldsymbol{\times }\boldsymbol{n}\boldsymbol{\times }\boldsymbol{a}$ Total Problem Instances mean ${\boldsymbol{LB}}_{\boldsymbol{LRH}}$ (best) mean ${\boldsymbol{UB}}_{\boldsymbol{LRH}}$ (best) mean ${\boldsymbol{LB}}_{\boldsymbol{CPLEX}}$ (best) mean ${\boldsymbol{UB}}_{\boldsymbol{CPLEX}}$ (best) 1 5$\mathrm{\times}$5$\mathrm{\times}$2 5 10879.80 18505.79 17859.01 17860.29 2 5$\mathrm{\times}$8$\mathrm{\times}$2 5 17322.20 28333.22 25534.45 26333.42 3 8$\mathrm{\times}$8$\mathrm{\times}$2 5 15736.80 29614.83 25534.45 25667.43 4 7$\mathrm{\times}$10$\mathrm{\times}$2 5 22063.60 35494.88 33925.34 33925.34 5 10$\mathrm{\times}$10$\mathrm{\times}$3 5 18764.20 34423.95 29758.67 29758.67 6 10$\mathrm{\times}$20$\mathrm{\times}$3 5 39061.60 62065.47 58664.97 58813.86

Table 4.  Mean Gap% of each solution method using the best mean lower bound (CPLEX)

 Problem Size No. Problem Size $\boldsymbol{m}\boldsymbol{\times }\boldsymbol{n}\boldsymbol{\times }\boldsymbol{a}$ mean ${\boldsymbol{LB}}_{\boldsymbol{CPLEX}}$ (best) Gap% LRH Gap % CPLEX 1 5$\mathrm{\times}$5$\mathrm{\times}$2 17859.01 3.62% 0.007% 2 5$\mathrm{\times}$8$\mathrm{\times}$2 25534.45 7.72% 0.1% 3 8$\mathrm{\times}$8$\mathrm{\times}$2 25534.45 15.98% 0.05% 4 7$\mathrm{\times}$10$\mathrm{\times}$2 33925.34 4.63% 0.00% 5 10$\mathrm{\times}$10$\mathrm{\times}$3 29758.67 15.68% 0.00% 6 10$\mathrm{\times}$20$\mathrm{\times}$3 58664.97 5.8% 0.03%

Table 5.  Comparison between the FCSTP EQ and FCSLTP using CPLEX under 9000secs computation time

 Problem No Problem Size $\boldsymbol{m}\boldsymbol{\times }\boldsymbol{n}\boldsymbol{\times }\boldsymbol{a}$ Total no. of Instances $\boldsymbol{FCSTP}\boldsymbol{\ }$ EQ mean $\boldsymbol{FCSLTP}$ mean Cost Difference % Cost Difference 1 5$\mathrm{\times}$5$\mathrm{\times}$2 5 18480.39 17860.29 620.10 3% 2 5$\mathrm{\times}$8$\mathrm{\times}$2 5 28333.22 26333.42 1999.80 8% 3 8$\mathrm{\times}$8$\mathrm{\times}$2 5 29064.07 25667.43 3396.64 13% 4 7$\mathrm{\times}$10$\mathrm{\times}$2 5 35807.10 33925.34 1881.76 6% 5 10$\mathrm{\times}$10$\mathrm{\times}$3 5 32147.15 29758.67 2388.48 8% 6 7 8 9 10 10$\mathrm{\times}$20$\mathrm{\times}$3 15$\mathrm{\times}$30$\mathrm{\times}$4 20$\mathrm{\times}$20$\mathrm{\times}$5 25$\mathrm{\times}$38$\mathrm{\times}$8 35$\mathrm{\times}$42$\mathrm{\times}$9 5 5 5 5 5 62797.43 85778.98 61498.56 106532.31 120932.51 58813.86 77653.71 50054.12 89098.73 96508.73 3983.57 8125.27 11444.44 17433.58 24,423.78 7% 10% 23% 20% 25%
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