A new model to design the suppliers portfolio in newsvendor problem based on product reliability

Azam Modares; Nasser Motahari Farimani; Vahideh Bafandegan Emroozi

doi:10.3934/jimo.2022124

Article Contents

2023, Volume 19, Issue 6: 4112-4151. Doi: 10.3934/jimo.2022124

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A new model to design the suppliers portfolio in newsvendor problem based on product reliability

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: December 2021

Revised: May 2022

Early access: July 28, 2022

Published online: July 28, 2022

This work was supported in part by: Research Deputy of Ferdowsi University of Mashhad, under Grant 57596.

Abstract / Introduction Full Text(HTML) Figure(18) / Table(24) Related Papers Cited by

Abstract

Suppliers' selection problem has always been daunting challenges in the Newsvendor problem. Furthermore, since the failures in the supplier's products cause irreparable damage to the retailer, it is necessary to consider the reliability of products in ordering suppliers' products. This paper develops the Newsvendor model by considering the impactful criteria in supplier selection and product reliability so that the total cost of the chain is minimized in a multi-product and multi-period model with multiple suppliers. While multiple criteria decision making (MCDM) accounts for multiple criteria and their tradeoffs, its application in Newsvendor model is not considered. This paper applies the Bayesian best worst method (BWM), as one of the MCDM methods, for ranking criteria and the fuzzy technique for order of preference by similarity to ideal solution (TOPSIS) for prioritizing the suppliers. Then, the obtained weights are plugged into the model as the inputs of the designed model. A case study with real data in the electronic supply chain is considered. To validate the results obtained by the proposed method, genetic algorithm (GA) and particle swarm optimization (PSO) are leveraged to solve the proposed model. Finally, the efficiency of the designed model is verified through a case study.

Keywords:

Mathematics Subject Classification: 90B06, 90B05.

Citation:

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Figure 1. BWM Bayesian hierarchical model

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Figure 2. General steps of research

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Figure 3. The confidence of criteria to one another using Bayesian BWM

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Figure 4. The normalized weights of suppliers

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Figure 5. Mean $ S\big/N $ ratios of the parameters of GA

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Figure 6. Mean $ S\big/N $ ratios of the parameters of PSO

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Figure 7. The the structure of the chromosomes in GA

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Figure 8. Convergence diagram of the best implementation of the GA

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Figure 9. Convergence diagram of the best implementation of the PSO

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Figure 10. The balance between objective functions for $ \gamma_1 $ different

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Figure 11. The balance between objective functions for $ \gamma_2 $ different

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Figure 12. The balance between objective functions for $ \gamma_3 $ different

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Figure 13. changes in the average demand for the first objective function in first period

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Figure 14. changes in the average demand for the first objective function in first period

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Figure 15. changes in the average demand for the second objective function in first period

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Figure 16. changes in the average demand for the second objective function in first period

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Figure 17. changes in the average demand for the third objective function in first period

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Figure 18. changes in the average demand for the third objective function in first period

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Table 1. Some studies on newsvendor problem

			Supplier selection criteria		Costs

[16]	Single	No	No	Cost	Yes	No	No	No
[29]	Single	No	No	Cost	Yes	No	No	No
[62]	Single	No	No	Cost	Yes	No	No	No
[57]	Single	No	No	Cost	Yes	No	No	No
[59]	Single	No	No	Cost	Yes	No	No	No
[12]	Single	No	No	Cost	Yes	No	No	No
[35]	Single	No	No	Cost	Yes	No	No	No
[14]	Multiple	Budget and retailer of capacity	No	The cost and sustainability factors	Yes	Yes	No	No
[26]	Multiple	Budget and retailer of capacity	No	The cost and the service level	Yes	Yes	No	No
Present study	Multiple	Budget and retailer of capacity supplier of capacity lead time and balance constraints	Yes	cost	Yes	Yes	Yes	Yes

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Table 2. Weight of the criteria

Criteria	Weight
Price	0.1855
Quality	0.2250
Service after-sale	0.1520
Collaboration history	0.1151
Delivery	0.1497
Distance	0.0824
Reliability	0.0903

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Table 3. The normalized values of the weights

suppliers	$ D+ $	$ D- $	$ CI $	Normalized weight	$ Raking $
$ 1 $	0.161	0.118	0.423	0.192	3
$ 2 $	0.246	0.384	0.135	0.061	4
$ 3 $	0.04	0.235	0.853	0.387	1
$ 4 $	0.063	0.242	0.793	0.36	2

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Table 4. Notations

sets
$ I $	The set of products indexed by $ i $
$ J $	The set of vendors indexed by j
$ K $	The set of vehicles indexed by $ k $
$ T $	The set of periods indexed by $ t $
Decision variables
$ q_{ijt} $	Amount of $ i^{th} $ product ordered to $ j^{th} $ vendor in $ t^{th} $ period.
$ x_{ijt} $	Binary variable taking the value of 1 when $ j^{th} $ vendor is selected to supply.
	$ i^{th} $ product in $ t^{th} $ period; otherwise, it is equal to 0.
$ q_{ijkt} $	Amount of $ i^{th} $ product ordered to $ j^{th} $ vendor by $ k^{th} $ vehicle in $ t^{th} $ period.
Parameters
$ c_{ijt} $	Ordering cost of $ j^{th} $ vendor for $ i^{th} $ product in $ t^{th} $ period.
$ h_{it} $	Unit holding cost of $ i^{th} $ product in $ t^{th} $ period for the retailer.
$ w_{j} $	Weight of $ j^{th} $ vendor (see Table 2).
$ B_t $	Maximum budget in $ t^{th} $ period for the retailer.
$ D_{it} $	The demand of $ i^{th} $ product in $ t^{th} $ period for the retailer.
$ D_{ijt} $	The demand of $ i^{th} $ product for $ j^{th} $ vendor in $ t^{th} $ period.
$ P_{it} $	The selling price of $ i^{th} $ product in $ t^{th} $ period.
$ p_{ij} $	Production rate of $ i^{th} $ product for $ j^{th} $ vendor.
$ V{S_{it}} $	Unit salvages value of $ i^{th} $ product in $ t^{th} $ period for the retailer.
$ C{S_{it}} $	Unit the shortage cost of $ i^{th} $ product in $ t^{th} $ period for the retailer.
$ C'_{ijt} $	The purchase price of $ j^{th} $ vendor for $ i^{th} $ product in $ t^{th} $ period.
$ Maxca{p_{it}} $	Maximum holding capacity for $ i^{th} $ product in $ t^{th} $ period for the retailer.
$ MaxB_{it} $	Maximum purchase capacity for $ i^{th} $ product in $ t^{th} $ period for the retailer.
$ MaxV_{ik} $	Maximum vehicle capacity of $ k^{th} $ vehicle to transporting $ i^{th} $ product to the retailer.
$ F{t_{ijt}} $	Production and delivery time of $ j^{th} $ vendor for $ i^{th} $ product in $ i^{th} $ period.
$ L{T_{it}} $	Delivery leads time for $ i^{th} $ product in $ i^{th} $ period for the retailer.
$ O_{ijt} $	Maximum production capacity of $ j^{th} $ vendor for $ i^{th} $ product in $ t^{th} $period.
$ \lambda '_{ijt} $	The failure rate of $ i^{th} $ product ordered to $ j^{th} $ vendor in $ i^{th} $ period.
$ \lambda _{kt} $	The failure rate of $ k^{th} $ vehicle in $ i^{th} $ period.
$ \pi_{ij} $	The minimum period in which $ i^{th} $ product purchased from $ j^{th} $ vendor not failed.
$ \tau _k $	The minimum period in that $ k^{th} $ vehicle has not failed.

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Table 5. The average and variance related to the distribution of demand

Product	$ Demand\, \, Distribution $
$ Product1 $	$ N(17000, \, 500) $
$ Product2 $	$ N(11000, \, 400) $
$ Product3 $	$ N(14000, \, 700) $

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

Solutions of models	$ Z_{1} ^{*} $	$ Z_{2} ^{*} $	$ Z_{3} ^{*} $
GA	43000000	25726	106151
PSO	41567000	22345	965443

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Table 7. Controllable factors and their levels

	Parameter	Notation	Level			Optimal level
	Parameter	Notation	Level 1	Level 2	Level 3	Optimal level
GA	Popsize	$ A $	100	150	200	100
	$ p_{c} $	B	0.4	0.5	0.8	0.4
	$ p_{m} $	C	0.1	0.3	0.4	0.3
PSO	$ C_{1} $	A	1	1.5	2	1.5
	$ C_{2} $	B	1	1.5	2	1
	$ W $	C	0.8	0.8	0.95	0.9

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Table 8. The results of the suggested model by GA

product	vendor	lpmetric
		$ q_{ijt} $		$ x_{ijt} $
		t=1	t=2	t=1	t=2
product1	vendor1	0	15700	0	1
product2		12000	13000	1	1
product3		17000	0	1	0
product1	vendor2	0	0	0	0
product2		17000	0	1	0
product3		0	17000	0	1
product1	vendor3	17000	0	1	0
product2		13000	16000	1	1
product3		10000	13000	1	1
product1	vendor4	0	0	0	0
product2		0	13000	0	1
product3		16000	12000	1	1

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Table 9. The results of the suggested model by PSo

product	vendor	PSO
		$ q_{ijt} $		$ x_{ijt} $
		t=1	t=2	t=1	t=2
product1	vendor1	0	14300	0	1
product2		11000	12700	1	1
product3		16800	0	1	0
product1	vendor2	0	0	0	0
product2		16100	0	1	0
product3		0	15800	0	1
product1	vendor3	16000	0	1	0
product2		11800	15400	1	1
product3		9500	11900	1	1
product1	vendor4	0	0	0	0
product2		0	12400	0	1
product3		15100	10900	1	1

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Table 10. The results of solving using LP-metric for $ \gamma_1 $ different

$ \gamma_1 $	$ Z_1 $	$ Z_2 $	$ Z_3 $
1	1	0.833	0.871
0.9	1	0.833	0.871
0.8	1	0.833	0.871
0.7	0.914	0.869	0.886
0.6	0.914	0.869	0.886
0.5	0.914	0.869	0.886
0.4	0.914	0.869	0.886
0.3	0.895	0.902	0.899
0.2	0.895	0.902	0.899
0.1	0.856	1	0.928
0	0.856	1	0.928

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Table 11. The results of solving using LP-metric for $ \gamma_2 $ different

$ \gamma_2 $	$ Z_1 $	$ Z_2 $	$ Z_3 $
1	0.905	1	0.871
0.9	0.905	1	0.871
0.8	0.907	0.902	0.928
0.7	0.907	0.902	0.928
0.6	0.907	0.869	0.935
0.5	0.907	0.869	0.935
0.4	0.907	0.869	0.935
0.3	0.91	0.833	0.935
0.2	0.91	0.833	0.935
0.1	0.914	0.782	0.937
0	0.914	0.782	0.937

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Table 12. The results of solving using LP-metric for $ \gamma_3 $ different

$ \gamma_3 $	$ Z_1 $	$ Z_2 $	$ Z_3 $
1	0.902	0.833	1
0.9	0.902	0.833	1
0.8	0.902	0.833	1
0.7	0.902	0.869	0.934
0.6	0.902	0.869	0.934
0.5	0.907	0.869	0.928
0.4	0.907	0.869	0.928
0.3	0.907	0.902	0.871
0.2	0.907	0.902	0.871
0.1	0.883	0.902	0.871
0	0.883	0.902	0.871

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Table 13. The value of change in the first objective function

average Demand	t=1			t=2
average Demand	product1	product2	product3	product1	product2	product3
-20	-6.9	-5.2	-7.5	-7.3	-5.4	-8
-10	-3.6	-2.5	-4.3	-3.4	-2.5	-4.5
0	0	0	0	0	0	0
10	5.5	1.8	6.3	6	2	6.5
20	12.2	4.8	13.8	12.6	5	14.5

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Table 14. The value of change in the second objective function

Average Demand	t=1			t=2
Average Demand	product1	product2	product3	product1	product2	product3
-20	-12.24	-6.38	-9.56	-6.12	-9.56	-11.16
-10	-6.12	-3.2	-4.78	-3.06	-4.78	-5.58
0	0	0	0	0	0	0
10	6.12	3.2	4.78	3.06	4.78	5.58
20	12.24	6.38	9.56	6.12	9.56	11.16

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Table 15. The value of change in the third objective function

Average Demand	t=1			t=2
Average Demand	product1	product2	product3	product1	product2	product3
-20	-1.12	-3.09	-3.256	-1.18	-3.17	-3.48
-10	-0.5	-1.54	-1.633	-0.5	-1.58	-1.74
0	0	0	0	0	0	0
10	0.61	1.86	1.802	0.67	1.65	1.84
20	1.25	3.2	3.407	1.23	3.42	3.85

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Table 16. Data related to Ordering cost

$ C_{ijt} $		t = 1	t = 2
vendor1	Product1	100000	130000
	Product2	120000	115000
	Product3	110000	110000
vendor2	Product1	120000	105000
	Product2	120000	115000
	Product3	110000	110000
vendor3	Product1	110000	120000
	Product2	125000	128000
	Product3	120000	115000
vendor4	Product1	115000	100000
	Product2	120000	115000
	Product3	130000	125000

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Table 17. Data related to maximum production capacity

$ O_{ijt} $		t = 1	t = 2
vendor1	Product1	17000	20000
	Product2	17000	16000
	Product3	15000	17000
vendor2	Product1	21000	16000
	Product2	17000	17000
	Product3	18000	18000
vendor3	Product1	16000	17000
	Product2	17500	16000
	Product3	15500	19000
vendor4	Product1	14500	17500
	Product2	19000	16000
	Product3	18000	18000

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Table 18. Data related to the purchase price

$ C'_{ijt} $		t = 1	t = 2
vendor 1	Product1	35000	40000
	Product2	38000	38000
	Product3	60000	60000
vendor 2	Product1	40000	40000
	Product2	45000	45000
	Product3	55000	55000
vendor 3	Product1	44000	44000
	Product2	45000	45000
	Product3	50000	50000
vendor 4	Product1	40000	40000
	Product2	55000	60000
	Product3	50000	50000

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Table 19. Data related to production and delivery time

$ F{t_{ijt}} $		t = 1	t = 2
vendor1	Product1	18	18
	Product2	20	20
	Product3	15	15
vendor2	Product1	20	20
	Product2	15	15
	Product3	20	20
vendor3	Product1	20	20
	Product2	10	10
	Product3	12	12
vendor4	Product1	10	10
	Product2	20	20
	Product3	14	14

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Table 20. Data related to the failure rate of $ i^{th} $ product ordered to $ j^{th} $ vendor

$ {\lambda '_{ijt}} $		t = 1	t = 2
vendor1	Product1	0.1	0.16
	Product2	0.44	0.46
	Product3	0.31	0.33
vendor2	Product1	0.15	0.16
	Product2	042	035
	Product3	025	028
vendor3	Product1	0.18	0.20
	Product2	0.38	0.3
	Product3	0.18	0.15
vendor4	Product1	0.21	0.17
	Product2	0.38	0.34
	Product3	0.27	0.22

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Table 21. Data related to the failure rate of the $ k^{th} $ vehicle in $ t^{th} $ period

$ {\lambda _{kt}} $	t = 1	t = 2
Vehicle 1	0.4	0.34
Vehicle 2	0.34	0.40
Vehicle 3	0.21	0.28
Vehicle 1	0.42	0.34
Vehicle 2	0.42	0.35
Vehicle 3	0.28	0.27
Vehicle 1	0.37	0.38
Vehicle 2	0.32	0.3
Vehicle 3	0.16	0.15
Vehicle 1	0.43	0.42
Vehicle 2	0.38	0.30
Vehicle 3	0.27	0.21

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Table 22. Data related to delivery lead time

$ LT _{it} $	t = 1	t = 2
Product 1	24	34
Product 2	72	18
Product 3	12	72

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Table 23. Data related to maximum vehicle capacity

Max $ V _{ik} $	Machine 1	Machine 2	Machine 3	Machine 4	Machine 5
vendor 1	2700	1900	1400	1800	1590
vendor 2	2500	1810	1350	1700	1600
vendor 3	2600	1900	1380	1760	1580
vendor 4	2500	1850	1400	1700	1620

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Table 24. Data related to holding cost, the shortage cost, the salvage value, the selling price, maximum warehouse capacity, maximum purchase capacity

Parameters		t = 1	t = 2
$ {h_{it}} $	Product1	600	600
	Product2	550	550
	Product3	650	650
$ C{S_{it}} $	Product1	6000	6000
	Product2	5000	5000
	Product3	5000	4500
$ V{S_{it}} $	Product1	3000	3000
	Product2	1500	2500
	Product3	1500	2000
$ Maxca{p_{it}} $	Product1	300	330
	Product2	280	250
	Product3	250	300
$ {P_{it}} $	Product1	53000	53000
	Product2	45000	45000
	Product3	67000	67000
$ Max{B_{it}} $	Product1	20000	20000
	Product2	45000	45000
	Product3	45000	45000

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