A novel separate chance-constrained programming model to design a sustainable medical ventilator supply chain network during the Covid-19 pandemic

Amin Reza Kalantari Khalil Abad; Farnaz Barzinpour; Seyed Hamid Reza Pasandideh

doi:10.3934/jimo.2021234

Article Contents

2023, Volume 19, Issue 2: 1395-1425. Doi: 10.3934/jimo.2021234

This issue Previous Article The effect of rebate value and selling price-dependent demand for a four-level production manufacturing system Next Article Robust Markowitz: Comprehensively maximizing Sharpe ratio by parametric-quadratic programming

A novel separate chance-constrained programming model to design a sustainable medical ventilator supply chain network during the Covid-19 pandemic

1.
Department of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran
2.
Department of Industrial Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran

^* Corresponding author: Seyed Hamid Reza Pasandideh
^* Corresponding author: Seyed Hamid Reza Pasandideh

Received: July 2021

Revised: October 2021

Published: February 2023

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Abstract

Providing new models or designing sustainable networks in recent studies represents a growing trend. However, there is still a gap in the simultaneous modeling of the three dimensions of sustainability in the electronic medical device supply chain (SC). In this paper, a novel hybrid chance-constrained programming and cost function model is presented for a green and sustainable closed-loop medical ventilator SC network design. To bring the problem closer to reality, a wide range of parameters including all cost parameters, demands, the upper bound of the released $ co_2 $, and the minimum percentage of the units of product to be disposed and collected from a customer and to be dismantled and shipped from DCs are modeled as uncertain along with the normal probability distribution. The problem was first formulated into the framework of a bi-objective stochastic mixed-integer linear programming (MILP) model; then, it was reformulated into a tri-objective deterministic mixed-integer nonlinear programming (MINLP) one. In order to model the environmental sustainability dimension, in addition to handling the total greenhouse gas emissions, the total waste products were also controlled. The efficiency and applicability of the proposed model were tested in an Iranian medical ventilator production and distribution network. For sensitivity analyses, the effect of some critical parameters on the values of the objective functions was carefully examined. Finally, valuable managerial insights into the challenges of companies during the COVID-19 pandemic were presented. Numerical results showed that with the increase in the number of customers in the COVID-19 crisis, social responsibility could improve cost mean by up to 8%.

Keywords:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Significant growth in the demand for ventilator in New York City in 2020.¹

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Figure 2. The underlying configuration of integrated closed-loop medical ventilator SC network

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Figure 3. Research roadmap

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Figure 4. The necessity of the simultaneous reduction of the mean and variance of the costs to increase the decision -making confidence

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Figure 5. Geographical locations of the selected facilities for solution

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Figure 6. Schematic view of the connection and transportation methods between facilities

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Figure 7. The effect of increasing the demand of all customers on different objective functions and CPU time

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Figure 8. The effect of the increasing the upper bound of $co_2$ released on different objective functions and CPU time

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Figure 9. The effect of increasing the upper bound of waste product on different objective functions and CPU time

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Figure 10. Comparison between the expected value of cost in the models with and without the SR indicator

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Figure 11. Comparison between the variance of cost in the models with and without the SR

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Figure 12. The growing trend of the expected value of costs with increasing the number of customers

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Figure 13. The growing trend of the cost variance with increasing the number of customers

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Table 1. The assumption of the hypotheses that have been considered in previous studies and the gaps covered by this research

Year	Scholars	Flow	goals	Decision variables			Sustainability dimensions			Uncertainty modeling method	Uncertain parameters	Case study
Year	Scholars	Flow	goals	location/allocation	production technology	transportation mode	economical	Environmental	social	Uncertainty modeling method	Uncertain parameters	Case study
2014	Pasandideh et al. [32]	Direct	M	√			√			Hybrid chance-constraint and cost function	costs, demand, production and set-up times	General
2015	Alshamsi and Diabat [2]	Reverse	S	√		√	√	√	√	-	-	Washing machines and tumble dryers
2016	Nurjanni et al. [29]	Direct-reverse	M	√		√	√	√		-	-	General
2016	Keyvanshokooh et al. [18]	Direct-reverse	M	√			√	√	√	Robust stochastic programming approach	demand and returns based on market conditions	General
2018	Fathollahi Fard and Hajiaghaei Keshteli [7]	Direct-reverse	M	√			√	√		two-stage stochastic scenario based	production, manufacturing costs, assigning the cost of costumers to distribution centers, demands and return rates	General
2018	Tsao et al. [45]	Direct	M	√	√		√	√	√	fuzzy programming	demand, costs, carbon emissions, job opportunities, and the detrimental effects	General
2018	Gonela et al. [12]	Direct	M	√			√	√	√	multi-objective stochastic MILP programming	electricity con-version rate, biomass yield rate, and coal excavation rate	Electricity generation
2019	Resat and unusal [36]	Direct-reverse	M	√		√	√	√	√	-	-	Packaging
2019	Chalmardi and Vallejo [4]	Direct	M	√			√	√	√	-	-	General
2020	Yakavenka et al. [48]	Direct	M	√		√	√	√	√	-	-	Perishable food
2020	Kalantari Khalil Abad and Pasandideh [17]	Direct-reverse	S	√		√	√	√		two-stage stochastic scenario based	demand and carbon cap	General
2020	Mohammadi et al. [24]	Direct-reverse	M	√			√	√	√	multi-stage stochastic programming	demand and return products	plastic
2021	Lotfi et al. [20]	Direct	S	√			√	√		two-stage robust stochastic programming	Costs, $co_2$ emission, energy consumption	General
2021	Mohtashami et al. [25]	Direct	M	√			√	√	√	-	-	Biodiesel
2021	Sadrnia et al. [37]	Direct-reverse	M	√			√	√	√	-	-	General
2021	Pahlevan et al. [31]	Direct-reverse	M	√			√	√	√	-	-	Aluminum
2021	Nayeri et al. [27]	Direct	M	√		√	√	√	√	fuzzy robust stochastic approach	The demand, costs, the capacity of facilities, environmental impacts, job opportunities, and the rates of remained capacity at disrupted facilities	Water heater
-	Current research	Direct-reverse	M	√	√	√	√	√	√	hybrid chance-constraint programming and cost function	production, reproduction, holding, disassembly, collecting and transportation costs, demands, upper bound of $co_2$ released, and the minimum percentage of the units of product to be disposed and collected from a customer and to be dismantled and shipped from a DC	Medical ventilator (ICU and portable)

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Table 2. Sets and indexes of the model

Indices		Statement
$ i $	:	Set of manufacturing plants, $ i\in \{1,2,\dots,I\} $
$ j $	:	Set of warehouses $ j\in \{1,2,\dots,J\} $
$ k $	:	Set of customers (university of medical S sciences) $ k\in \{1,2,\dots,K\} $
$ l $	:	Set of DCs $ l\in \{1,2,\dots,L\} $
$ m $	:	Set of transportation modes from manufacturing plants $ m\in\{1,2,\dots,M\} $
$ n $	:	Set of transportation modes from warehouses $ n\in \{1,2,…,N\} $
$ o $	:	Set of transportation modes from customers $ o\in \{1,2,\dots,O\} $
$ q $	:	Set of transportation modes from DCs $ q\in \{1,2,\dots,Q\} $
$ t $	:	Set of production technologies $ t\in \{1,2,\dots,T\} $

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Table 3. Parameters of the model

Parameters		Explanation
$\widetilde{fp_{i,t}}$	:	Fixed cost for establishing the manufacturing plant $i\in I$ with production technology $t\in T$ with the mean $\mu fp_{i,t}$ and the variance $\sigma^2 fp_{i,t}$
$\widetilde{fw_j }$	:	Fixed cost for establishing the warehouse $j\in J$ with the mean $\mu fw_j$ and the variance $\sigma^2 fw_j$
$\widetilde{fd_l}$	:	Fixed cost for establishing DC $l\in L$ with the mean $\mu fd_l$ and the variance $\sigma^2 fd_l$
$\widetilde{vp_{i,t} } $	:	Unit variable cost for producing a unit product with the technology $t\in T$ in the manufacturing plant $i\in I$ with the mean $\mu vp_{i,t}$ and the variance $\sigma^2 vp_{i,t}$
$\widetilde{vh_{j,t}} $	:	Unit variable cost for handling a unit of product with the technology $t\in T$ in the warehouse $j\in J$ with the mean $\mu vh_{j,t}$ and the variance $\sigma^2 vh_{j,t}$
$\widetilde{vc_{k,t}} $	:	Unit variable cost for collecting a unit of product with the technology $t\in T$ to be disposed from the customer $k\in K$ with the mean $\mu vc_{k,t}$ and the variance $\sigma^2 vc_{k,t}$
$\widetilde{vd_{l,t}} $	:	Unit variable cost for disassembling a unit of product with the technology $t\in T$ to be disposed in the DC $l\in L$ with the mean $\mu vd_{l,t}$ and the variance $\sigma^2 vd_{l,t}$
$\widetilde{vr_{i,t} } $	:	Unit variable cost for reproducing a unit product with the technology $t\in T$ in the manufacturing plant $i\in I$ with the mean $\mu vr_{i,t}$ and the variance $\sigma^2 vr_{i,t}$
$\widetilde{tpw_{i,j,t}^m } $	:	Unit transportation cost for products with the technology $t\in T$ from the manufacturing plant $i\in I$ to the warehouse $j\in J$ with the transportation mode $m\in M$, the mean $\mu tpw_{i,j,t}^m$ and the variance $\sigma^2 tpw_{i,j,t}^m$
$\widetilde{twc_{j,k,t}^n } $	:	Unit transportation cost for products with the technology $t\in T$ from the warehouse $j\in J$ to the customer $k\in K$ with the transportation mode $n\in N$, the mean $\mu twc_{j,k,t}^n$ and the variance $\sigma^2 twc_{j,k,t}^n$
$\widetilde{tcd_{k,l,t}^o } $	:	Unit transportation cost for products with the technology $t\in T$ from the customer $k\in K$ to DC $l\in L$ with the transportation mode $o\in O$, the mean $\mu tcd_{k,l,t}^o$ and the variance $\sigma^2 tcd_{k,l,t}^o$
$\widetilde{tdp_{l,i,t}^q } $	:	Unit transportation cost for products with the technology $t\in T$ from the DC $l\in L$ to the manufacturing plant $i\in I$ with the transportation mode $v\in V$, the mean $\mu tdp_{l,i}^v$ and the variance $\sigma^2 tdp_{l,i}^v$
${ep_{i,t}} $	:	The rate of the released $co_2$ to produce unit of product with the technology $t\in T$ in the manufacturing plant $i\in I$
$ew_{j,t}$	:	The rate of the released $co_2$ to handle and storage unit of product with the technology $t\in T$ in the warehouse $j\in J$
$ed_{l,t}$	:	The rate of the released $co_2$ to disassemble unit of product with the technology $t\in T$ to be disposed in the DC $l\in L$
$er_{i,t}$	:	The rate of the released $co_2$ to remanufacture unit of product with the technology $t\in T$ to be dismantled in the manufacturing plant $i\in I$
$etp_m$	:	$co_2$ released by the transportation mode $m\in M$ to forward a unit of product from a manufacturing plant to a warehouse for a unit distance
$etw_n$	:	$co_2$ released by the transportation mode $n\in N$ to forward a unit of product from a warehouse to a customer for a unit distance
$etc_o $	:	$co_2$ released by the transportation mode $o\in O$ to collect a unit disposal from a customer to a DC for a unit distance
$etd_q $	:	$co_2$ released by the transportation mode $q\in Q$ to ship a unit of product to be dismantled from a DC to a manufacturing plant for a unit distance
$cp_{i,t} $	:	Maximum production capacity of the manufacturing plant $i\in I$ for products with the technology $t\in T$
$cw_{j,t}$	:	Maximum storage and handling, and the processing capacity of the warehouse $j\in J$ for products with the technology $t\in T$
$cd_{l,t}$	:	Maximum disassembly capacity of DC $l\in L$ for products with the technology $t\in T$
$cr_{i,t}$	:	Maximum reproduction capacity of the manufacturing plant $i\in I$ for products with the technology $t\in T$
$\xi p_{i,j}^m$	:	Transportation rate from the manufacturing plant $i\in I$ to the warehouse $j\in J$ with the transportation mode $m\in M$
$\xi w_{j,k}^n$	:	Transportation rate from the warehouse $j\in J$ to the customer $k\in K$ with the transportation mode $n\in N$
$\xi c_{k,l}^o$	:	Transportation rate cost for collecting the unit of product from the customer $k\in K$ to the DC $l\in L$ with the transportation mode $o\in O$
$\xi d_{l,i}^q$	:	Transportation rate from the DC $l\in L$ to the manufacturing plant $i\in I$ with the transportation mode $q\in Q$
$dpw_{i,j}$	:	Distance between the manufacturing plant $i\in I$ and the warehouse $j\in J$
$dwc_{j,k}$	:	Distance between the warehouse $j\in J$ and the customer $k\in K$
$dcd_{k,l}$	:	Distance between the customer $k\in K$ and the DC $l\in L$
$ddp_{l,i}$	:	Distance between the DC $l\in L$ and the manufacturing plant $i\in I$
$\tilde{\delta}$	:	Minimum percentage of the units of product to be disposed to be collected from a customer with the mean $\mu \delta$ and the variance $\sigma \delta$
$\tilde{\delta'}$	:	Minimum percentage of the units of product to be dismantled to be shipped from a DC with the mean $\mu \delta'$ and the variance $\sigma^2 \delta'$
$\widetilde{dem_{k,t}} $	:	Demand of the customer $k\in K$ for products with technology $t\in T$ with the mean $\mu dem_{k,t}$ and the variance $\sigma^2 dem_{k,t}$
$ \widetilde{UBE}$	:	The upper bound of the emission capacity of $co_2$ released, which is determined by the government and regulatory bodies with the mean $\mu UBE$ and the variance $\sigma UBE$
$UBD$	:	The upper bound of the number of products to be disposed
$1-\sigma,1-\omega,1-\xi$	:	The chance of rejecting a solution that does not satisfy the constraint
$Z_{1-\sigma},Z_{1-\omega},Z_{1- \xi}$	:	The lower critical point of the standard normal distribution used for a $( 1-\sigma\%, 1-\omega\%, 1- \xi\%)$ chance constraint on the solution obtained
$fjp_{i,t}$	:	The number of fixed job opportunities (i.e., job opportunities which are independent of the production capacity like managerial positions) created by the manufacturing plant $i\in I$ with the technology $t\in T$
$fjw_j$	:	The number of fixed job opportunities (i.e., job opportunities which are independent of the production capacity like managerial positions) created by the warehouse $j\in J$
$fjd_l$	:	The number of fixed job opportunities (i.e., job opportunities which are independent of the production capacity like managerial positions) created by the DC $l\in L$
$vjp_{i,t}$	:	The number of variable job opportunities (i.e., job opportunities which vary by production capacity like manufacturing line workers) created through producing at the manufacturing plant $i\in I$ with the technology $t\in T$
$vjh_j$	:	The number of variable job opportunities created through handling at the warehouse $j\in J$
$vjd_l$	:	The number of variable job opportunities created through disassembling at the DC $l\in L$
$vjr_{i,t}$	:	The number of variable job opportunities created through remanufacturing at the manufacturing plant $i\in I$ with the technology $t\in T$
$ph_t$	:	Average fraction of the potentially hazardous products when the technology $u\in U$ is used
$eldp_{i,t}$	:	The lost days caused from work's damages during the establishment of the technology $u\in U$ at the manufacturing plant $i\in I$
$eldw_j$	:	The lost days caused from the work's damages during the establishment of the warehouse $j\in J $
$eldd_l$	:	The lost days caused from the work's damages during the establishment of the DC $l\in L$
$vldp_{i,t}$	:	The lost days caused from the work's damages during production at the manufacturing plant $i\in I$ with the technology $u$
$vldh_j$	:	The lost days caused from the work's damages during handling at the warehouse $j\in J$
$vldd_l$	:	The lost days caused from the work's damages during disassembling at the DC $l\in L$
$vldr_{i,u}$	:	The lost days caused from the work's damages during remanufacturing at the manufacturing plant $i\in I$ with the technology $u\in U$
$wj$	:	Weighting the factor of the total number of the produced job opportunities
$wp$	:	Weighting the factor of the total number of the potentially hazardous products
$wl$	:	Weighting the factor of the total number of lost days caused from the work's damages

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Table 4. Binary and continuous decision variables

Decision variables		Description
$YA_{i,t}$	:	1 if the manufacturing plant $i\in I$ with the production technology $t\in T$ is established, otherwise is 0
$YB_j$	:	1 if the warehouse $j\in J$ is established, otherwise is 0
$YD_l$	:	1 if DC $l\in L$ is established, otherwise is 0
$XA_{i,j,t}^m$	:	The number of the unit product shipped from the manufacturing plant $i\in I$ with technology $t\in T$ to the warehouse $j\in J$ with the transportation mode $m\in M$
$XB_{j,k,t}^n $	:	The number of the unit product shipped from the warehouse $j\in J$ with technology $t\in T$ to the customer $k\in K$ with the transportation mode $n\in N$
$XC_{k,l,t}^0$	:	he number of the unit product to be disposed and collected from the customer $k\in K$ with technology $t\in T$ to the DC $l\in L$ with the transportation mode $o\in O$
$XD_{l,i,t}^q$	:	The number of the unit product to be dismantled and shipped from the DC $l\in L$ with technology $t\in T$ to the manufacturing plant $i\in I$ with the production technology $t\in T$ and the transportation mode $q\in Q$

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Table 5. The mean and variance of the uncertain demands

University of Medical Sciences(customer zones)	Mean of the stochastic demand (unit per month)	Variance of the stochastic demand
$ c_1 $ (Tehran)	500	400
$ c_2 $ (Markazi)	260	90
$ c_3 $ (Esfahan)	150	40
$ c_4 $ (Fars)	210	160
$ c_5 $ (Yazd)	160	40
$ c_6 $ (Tabriz)	200	90
$ c_7 $ (Mazandaran)	180	250
$ c_8 $ (Mashahd)	170	250

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Table 6. Mean and variance of the fixed costs, the number of fixed opportunities created by establishing the manufacturing plants and the number of variable jobs created through manufacturing and remanufacturing

Factories(i)	Production technology(m)	Mean of the Stochastic fixed cost (million Rials)	Variance of the Stochastic fixed cost (million Rials)$^2$	Number of fixed job opportunities	Number of variable job opportunities created through manufacturing	Number of variable job opportunities created through remanufacturing
$f_1$ (Tehran province)	ICU	120000	169000	70	15	10
$f_1$ (Tehran province)	MRI	123000	144000	60	17	9
$f_2$ (Tabriz province)	ICU	143000	121000	78	17	9
$f_2$ (Tabriz province)	MRI	143000	100000	73	16	12
$f_3$ (Yazd province)	ICU	132000	100000	78	16	10
$f_3$ (Yazd province)	MRI	170000	121000	79	18	9

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Table 7. Mean and variance of the fixed costs, the number of fixed opportunities created by establishing warehouses and the number of variable jobs created through handling

Medical equipment storage centers (j)	Mean of the Stochastic fixed cost (million Rials)	Variance of the Stochastic fixed cost(million Rials)$^2$	Number of fixed job opportunities	Number of variable job opportunities created through handling
$w_1$ (Hamedan province)	112000	121000	50	13
$w_2$ (Qom province)	140000	144000	55	14
$w_3$ (Esfahan province)	112000	100000	58	12

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Table 8. Mean and variance of the fixed costs, the number of fixed opportunities created by establishing DCs and the number of variable jobs created through disassembling

Disassembly centers (l)	Mean of the Stochastic fixed cost (million Rials)	Variance of the Stochastic fixed cost(million Rials)$^2$	Number of fixed job opportunities	Number of variable job opportunities created through disassembling
$d_1$ (Tehran province)	70000	49000	39	15
$d_2$ (Ardabil province)	75000	36000	32	13
$d_3$ (Yazd province)	72000	36000	42	15

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Table 9. Coefficient confidence of the chance-constraints

	Environmental constraints ($\alpha$)	Customer demands constraints ($\beta$)	Return flow establishing constraints ($\gamma$)
Coefficient confidence	0.95	0.99	0.95
the lower critical point of the standard normal distribution	1.645	1.96	1.645

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