A vendor-managed inventory model based on optimal retailers selection and reliability of supply chain

Azam Modares; Nasser Motahari Farimani; Vahideh Bafandegan Emroozi

doi:10.3934/jimo.2022078

Article Contents

2023, Volume 19, Issue 5: 3075-3106. Doi: 10.3934/jimo.2022078

This issue Previous Article Parameterized S-type M-eigenvalue inclusion intervals for fourth-order partially symmetric tensors and its applications Next Article Statistical analysis of a linear regression model with restrictions and superfluous variables

A vendor-managed inventory model based on optimal retailers selection and reliability of supply chain

1.
Ph.D. Candidate - Operations Research, Department of Management, Faculty of Economics and Administrative Sciences, Ferdowsi University of Mashhad, Mashhad, Iran
2.
Associate Professor, Department of Management, Faculty of Economics and Administrative Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

^*Corresponding author: Nasser Motahari Farimani
^*Corresponding author: Nasser Motahari Farimani

Received: November 2021

Revised: March 2022

Early access: May 2022

Published: May 2023

This work was supported in part by: Research Deputy of Ferdowsi University of Mashhad, under Grant No. 57130

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Abstract

One of the most common and successful approaches for the integrated supply chain management (SCM) is the vendor-managed inventory (VMI). In VMI, a vendor takes control of the inventory decisions for retailers. To establish a long-run relationship between the vendor and the retailer, it is necessary to consider two impactful factors: the vendors and retailers' reliability, and the optimal selection of retailers. For this purpose, the redundancy allocation problem (RAP), as an effective technique for increasing the reliability of vendors, is used in this paper. Also, the reliability of retailers as well as the reliability of the relationship between retailers and vendors are considered. In the retailer selection, process decisions on impactful criteria are simultaneously considered. For the retailers' selection, one analysis hierarchical process (AHP) is performed for each vendor, and the weights of retailers are obtained. Then, the obtained weights are plugged into the model as the inputs of the designed model. Since the developed model is a non-differentiable, non-convex, and mixed-integer function, genetic algorithm (GA) and particle swarm optimization (PSO) are leveraged to solve the formulated model. Finally, the efficiency of the presented method is verified through a case study with data collected from the electronic supply chain.

Keywords:

Mathematics Subject Classification: Primary: 90B06, 90B05.

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Figure 1. General steps of research

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Figure 2. The framework of obtaining the weight of the retailers

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Figure 3. The weight of retailers

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Figure 4. The structure of the inventory management system

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Figure 5. Inventory levels for vendors

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Figure 6. Series-parallel system

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Figure 7. Mean $ S\big/N $ ratios of the parameters of GA (model by objective function 2 and constraints)

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Figure 8. Mean $ S\big/N $ ratios of the parameters of GA

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Figure 9. Mean $ S\big/N $ ratios of the parameters of PSO

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Figure 10. Chromosome representation with objective function 2 (solution encoding)

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Figure 11. Chromosome representation with the GA approach (solution encoding)

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Figure 12. Convergence diagram of the best implementation of the PSO

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Figure 13. Convergence diagram of the best implementation of the GA

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Table 1. Some studies on VMI problems

Study	Retailer	Vendor	Distributor	Objectives	Constraints	Main contribution
$ \begin{array}{c} \rm{{Astantiet.}}, \rm{{ al. (2022)}} \end{array} $	Single	Single	No	Cost	No	Designing a new model by considering the effect of defect rate, deterioration, and carbon emissions in terms of CO2.
$ \begin{array}{c} \rm{{Lotfi, et. al.}} \rm{{(2022)}} \end{array} $	Multiple	Multiple	NO	Cost	Yes	Appling data-driven robust optimization in health care SC to cope uncertainty and disruption
$ \begin{array}{c} \rm{{Kusuma et.}} \rm{{al.( 2022)}} \end{array} $	Multiple	Single	Single	Cost	No	Eliminating the exclusive relationship between vendor and customer.
$ \begin{array}{c} \rm{{Shang et.}} \rm{{al. (2022)}} \end{array} $	Multiple	Single	Multiple	Cost	Yes	Designing a location-inventory routing problem with both deterministic demand and uncertain demand.
$ \begin{array}{c} \rm{{Gharaei et.}} \rm{{al. (2018)}} \end{array} $	Multiple	Single	NO	Cost	Yes	Integrating QC and environmental policies with SCs and inventory systems
$ \begin{array}{c} \rm{{Liu et. al.}} \rm{{(2020) }} \end{array} $	Multiple	Multiple	NO	Cost	Yes	Applying VMI to blood products distribution, and designing decomposition algorithm to solve problems
$ \begin{array}{c} \rm{{Wettasinghe}} \rm{{et. al.}} \rm{{(2020)}} \end{array} $	Single	Single	NO	Cost	Yes	Incorporating the inventory positioning decision into the VMI model to determine the optimal system base stock level, delivery quantity to retailer and cycle length.
$ \begin{array}{c} \rm{{Wei et. al.}} \rm{{(2020)}} \end{array} $	Multiple	Single	NO	Cost	No	Considering stochastic learning effect in VMI and RMI
$ \begin{array}{c} \rm{{Poursoltan}} \rm{{et. al. (2020)}} \end{array} $	Single	Single	NO	Cost	Yes	Considering the manufacturing and remanufacturing process and percentage of defective products randomly
$ \begin{array}{c} \rm{{Najafnejhad }} \rm{{et. al. (2021)}} \end{array} $	Multiple	Single	NO	Cost	Yes	Considering upper inventory limits as decision variables
$ \begin{array}{c} \rm{{Keshavarz-}} \rm{{Ghorbani}} \rm{{et. al. (2021)}} \end{array} $	Multiple	Single	NO	Cost	Yes	Investigating and analyzing the system behavior under the learning curve and Optimizing a two-level closed-loop VMI.
$ \begin{array}{c} \rm{{Karbasi-}} \rm{{bonab, et al.}} \rm{{(2018)}} \end{array} $	Multiple	Multiple	NO	$ \begin{array}{c} 1. \rm{Cost}\\2. \rm{warehouse\;space} \end{array} $		Minimizing the total supply chain cost and optimizing the warehouse space
$ \begin{array}{c} \rm{{Sadeghi,}} \rm{{et al. (2013)}} \end{array} $	Multiple	Multiple	NO	Cost	Yes	Finding the order quantities along with the number of shipments received by retailers and vendors such that the total inventory cost of the chain is minimized.
$ \begin{array}{c} \rm{{Sadeghi,}} \rm{{et al. (2014)}} \end{array} $	Multiple	Single	Yes	$ \begin{array}{c} 1. \rm{Cost}\\2. \rm{RAP} \end{array} $	Yes	Maximizing the system reliability of producing items within constraints using the RAP approach.
Karimi & Niknamfar (2017)	Multiple	Single	No	$ \begin{array}{c} \rm{Carbon}\;\rm{emission}\;\rm{cost}\;\rm{RAP} \end{array} $	Yes	Maximizing the system reliability of using the RAP approach and minimizing the carbon emission cost
$ \begin{array}{c} \rm{{Present}}\;\rm{{study}} \end{array} $	Multiple	Multiple	No	$ \begin{array}{c} 1. \rm{Cost}\\2. \rm{{ RAP}}\\3. \rm{Selection}\;\rm{of\;retailers} \end{array} $	Yes	Considering the cost, reliability of vendors and retailers, and retailers' selection, simultaneously.

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Table 2. The weight of retailers to vendors using AHP

Retailer/Vendor	$ j=1 $	$ j=2 $	$ j=3 $
$ k=1 $	0.113	0.134	0.144
$ k=2 $	0.041	0.052	0.056
$ k=3 $	0.105	0.089	0.089
$ k=4 $	0.086	0.067	0.069
$ k=5 $	0.061	0.084	0.084
$ k=6 $	0.102	0.094	0.088
$ k=7 $	0.082	0.091	0.062
$ k=8 $	0.093	0.085	0.085
$ k=9 $	0.073	0.058	0.078
$ k=10 $	0.077	0.065	0.076

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Table 3. Notations used in designed model

Sets
$ J $	The set of vendors indexed by$ j $;
$ K $	The set of retailers indexed by$ k $;
$ S $	The set of subsystems indexed by$ s $;

Decision variables
$ q_{jk} $	Amount of product ordered to $ j^{th} $ vendor by $ k^{th} $ retailer.
$ x_{jk} $	Binary variable taking the value of 1 when $ k^{th} $ retailer allocated to $ j^{th} $ vendor; otherwise, it is equal to 0.
$ M_{sj} $	The number redundant machine of $ s^{th} $ subsystem to $ j^{th} $ vendor.
$ q_{j} $	Amount of product ordered by $ j^{th} $ vendor $ (q_{j} =\sum _{k=1}^{K}q_{jk} ) $.
$ b_{j} $	The maximum shortage for $ j^{th} $ vendor

Parameters
$ A_{jk} $	Ordering cost of $ k^{th} $ retailer supplied by $ j^{th} $ vendor.
$ a_{j} $	Setup cost of $ j^{th} $the vendor per unit time.
$ H_{jk} $	The unit holding cost of $ k^{th} $ retailer's product supplied by $ j^{th} $ vendor per unit time.
$ h_{j} $	Holding cost of $ j^{th} $ vendor's products per unit time.
$ d_{jk} $	The expected demand of $ k^{th} $retailer supplied by $ j^{th} $ vendor per unit time.
$ d_{j} $	The expected demand of $ j^{th} $ vendor per unit time.
$ R_{sj} $	Reliability of $ j^{th} $ vendor production in $ s^{th} $ subsystem.
$ R_{k} $	Reliability of $ k^{th} $ retailer.
$ R_{jk} $	Reliability of relationship between $ k^{th} $ retailer and $ j^{th} $ vendor.
$ \lambda _{k} $	The failure rate of $ k^{th} $ retailer activity (parameter of exponential distribution).
$ \lambda _{jk} $	The failure rate of relationship between $ k^{th} $ retailer and $ j^{th} $vendor (parameter of exponential distribution)
$ v_{k} $	Final order quantity $ k^{th} $retailer from all vendors for the product.
$ p_{j} $	Production rate of $ j^{th} $ vendor.
$ \pi_{j} $	Linear backorder cost per unit of $ j^{th} $ vendor.
$ \hat{\pi }_{j} $	Fixed backorder cost per unit of $ j^{th} $ vendor (time-independent).
$ F_{jk} $	Product transportation cost from $ j^{th} $ vendor to $ k^{th} $ retailer.
$ S_{jk} $	Safety stock of $ j^{th} $ vendor for $ k^{th} $ retailer.
$ f $	Space required storing one unit of the product.
$ \rm{Cap}_{j} $	Maximum of space capacity for $ j^{th} $ vendor.
$ \rm{Sho}_{j} $	Maximum of permissible shortage for $ j^{th} $ vendor.
$ C_{sj} $	The number of $ s^{th} $ the subsystem for $ j^{th} $ vendor.
$ \rm{bud}_{j} $	Available budget to purchasing and installation machines for $ j^{th} $ vendor.
$ u_{jk} $	Minimum of total order quantity for $ j^{th} $ vendor by $ k^{th} $ retailer.
$ w_{jk} $	Weight of $ k^{th} $retailer to $ j^{th} $ vendor.

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Table 4. Controllable factors and their levels (model by objective function 2 and constraints)

	Parameter	Notation	Level			Optimal level
			Level 1	Level 2	fLevel 3
GA	Popsize	$ A $	50	80	100	100
	$ p_{c} $	B	0.4	0.5	0.6	0.6
	$ p_m $	C	0.05	0.1	0.2	0.2

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Table 5. Controllable factors and their levels

	Parameter	Notation	Level			Optimal level
			Level 1	Level 2	Level 3
GA	Popsize	$ A $	50	100	150	100
	$ p_{c} $	B	0.4	0.5	0.8	0.4
	$ p_{m} $	C	0.1	0.2	0.3	0.3
PSO	$ C_{1} $	A	1	1.5	2	2
	$ C_{2} $	B	1	1.5	2	2
	$ W $	C	0.7	0.8	0.9	0.9

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Table 6. The results of the objective functions using GA and PSO

Solution of model	$ Z_{1} ^{*} $	$ Z_{2} ^{*} $	$ Z_{3} ^{*} $
GA	300988.2	2.72	4559
PSO	298305.4	2.72	4540

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Table 7. The results of the suggested model using the GA and PSO method

GA						PSO
Vendor	Decision variables				Retailer	Vendor	Decision variables				Retailer
	$ b_{j} $	$ q_{j} $	$ q_{jk} $	$ x_{jk} $			$ b_{j} $	$ q_{j} $	$ q_{jk} $	$ x_{jk} $
			0	0	1				0	0	1
			0	0	2				0	0	2
			0	0	3				0	0	3
			0	0	4				0	0	4
Vendor1	1530	8934	2300	1	5	Vendor1	1480	8702	2300	1	5
			4000	1	6				4000	1	6
			2634	1	7				2402	1	7
			0	0	8				0	0	8
			0	0	9				0	0	9
			0	0	10				0	0	10
			0	0	1				0	0	1
			0	0	2				0	0	2
			2498	1	3				2498	1	3
			2200	1	4				2200	1	4
Vendor2	1620	9098	0	0	5	Vendor2	1620	9098	0	0	5
			0	0	6				0	0	6
			0	0	7				0	0	7
			2300	1	8				2300	1	8
			0	0	9				0	0	9
			2100	0	10				2100	1	10
			2340	1	1				2340	1	1
			2200	1	2				2200	1	2
			0	0	3				0	0	3
			0	0	4				0	0	4
Vendor3	1480	7915	0	0	5	Vendor3	1480	7915	0	0	5
			0	0	6				0	0	6
			0	0	7				0	0	7
			0	0	8				0	0	8
			3375	1	9				3375	1	9
			0	0	10				0	0	10

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Table 8. Optimal number of machine using GA and PSO

$ M_{Sj} $	Machine 1	Machine 2	Machine 3	Machine 4	Machine 5
Vendor 1	1	2	1	2	1
Vendor 2	1	3	1	1	2
Vendor 3	2	3	1	2	1

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Table 9. Data related to obtain weighted of the AHP method, holding cost each of the retailer, ordering cost, demand of the retailers, reliability, the failure rate of retailers

$ {{ H}}_{{ jk}} $
1	15	13	14	12	18	17	19	14	12	17
2	17	18	15	20	20	18	19	17	16	19
3	18	13	15	10	13	12	10	15	17	13
$ {{ w}}_{{ jk}} $
1	0.113	0.041	0.105	0.086	0.061	0.102	0.082	0.093	0.073	0.077
2	0.134	0.052	0.089	0.067	0.084	0.094	0.091	0.085	0.058	0.065
3	0.144	0.056	0.089	0.069	0.084	0.088	0.062	0.085	0.078	0.076
$ {{ F}}_{{ jk}} $
1	10	10	12	14	13	14	10	11	12	13
2	10	11	10	13	12	11	11	12	13	11
3	13	12	11	12	13	13	10	11	12	10
$ {{ d}}_{{ jk}} $
1	2530	2768	4325	3021	3452	2876	3041	2870	3056	2690
2	3456	3405	2345	2900	3425	3409	4072	3023	3456	1025
3	3400	2313	3645	3965	3956	4357	3456	2047	3056	4076
$ {{ \alpha }}_{{ jk}} $
f1	0.85	0.86	0.78	0.7	0.78	0.75	065	074	0.83	0.78
2	0.84	0.82	0.75	0.68	0.70	0.8	0.84	0.76	0.83	0.78
3	0.76	0.74	0.82	0.75	0.67	0.78	0.76	0.83	0.80	0.76
$ {{ A}}_{{ jk}} $
1	25	16	85	84	39	60	45	32	66	56
2	42	60	91	54	65	97	94	78	65	76
3	74	65	64	90	80	67	72	64	55	60
$ {{ u}}_{{ jk}} $
1	2500	2500	2300	2300	2300	2200	2300	2500	2300	2200
2	2300	2300	2200	2200	2300	2300	2400	2300	2300	2100
3	2340	2200	2350	2300	2350	2400	2300	2200	2300	2400
$ {{ v}}_{{ k}} $
Vendors	3500	3000	3000	6000	4000	3500	4000	3000	3500	4500
$ {{\lambda }}_{{ k}} $
Vendors	0.2	0.2	0.3	0.2	0.4	0.2	0.3	0.2	0.4	0.3

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Table 10. Data related to holding cost, ordering, shortage, maximum production capacity and shortage capacity, maximum available budget

Parameter	Vendor 1	Vendor 1	Vendor 1
$ a_{j} $	1000	1100	1050
$ h_{j} $	9	10	7
$ p_{j} $	30000	28000	25000
$ \pi _{j} $	5	7	6
$ \hat{\pi }_{j} $	2	3	2
$ \text{Cap}_{j} $	64000	58000	78000
$ \text{sho}_{j} $	750	750	750
$ \text{bud}_{j} $	300000	400000	430000

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Table 11. Data related to reliability of vendor in subsystems, The purchasing cost of the machine for each of the retailer

$ C_{sj} $	Vendor 1	Vendor 2	Vendor 3
Subsystem 1	15000	17000	11000
Subsystem 2	12000	9000	18000
Subsystem 3	16000	18000	12000
Subsystem 4	15000	9000	10000
Subsystem 5	13000	14000	12000

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