Solving the facility location and fixed charge solid transportation problem

Gbeminiyi John Oyewole; Olufemi Adetunji

doi:10.3934/jimo.2020034

Article Contents

2021, Volume 17, Issue 4: 1557-1575. Doi: 10.3934/jimo.2020034

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Solving the facility location and fixed charge solid transportation problem

Gbeminiyi John Oyewole^, and
Olufemi Adetunji

Department of Industrial and Systems Engineering, University of Pretoria, Pretoria 0002, South Africa

^* Corresponding author: Gbeminiyi John Oyewole
^* Corresponding author: Gbeminiyi John Oyewole

Received: April 30, 2019

Revised: July 31, 2019

Early access: February 2020

Published: July 2021

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Abstract

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.

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

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Figure 1. Schematic representation of FCSLTP

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Figure 2. Procedure for computing the FCSTP-EQ

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Figure 3. FCSLTP and FCSTP-EQ

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Figure 4. Solution time of FCSLTP and FCSTP

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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

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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{)}\ } $

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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

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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%

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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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