Two-Stage Stochastic Program for Supply Chain Network Design under Facility Disruptions

Abstract

A supply chain disruption is an unanticipated event that disrupts the flow of materials in a supply chain. Any given supply chain disruption could have a significant negative impact on the entire supply chain. Supply chain network designs usually consider two stage of decision process in a business environment. The first stage deals with strategic levels, such as to determine facility locations and their capacity, while the second stage considers in a tactical level, such as production quantity, delivery routing. Each stage’s decision could affect the other stage’s result, and it could not be determined individual. However, supply chain network designs often fail to account for supply chain disruptions. In this paper, this paper proposed a two-stage stochastic programming model for a four-echelon global supply chain network design problem considering possible disruptions at facilities. A modified simulated annealing (SA) algorithm is developed to determine the strategic decision at the first stage. The comparison of traditional supply chain network decision framework shows that under disruption, the stochastic solutions outperform the traditional one. This study demonstrates the managerial viability of the proposed model in designing a supply chain network in which disruptive events are proactively accounted for.

1. Introduction

Supply chain network design constitutes a crucial decision-making process that affects organizational performance, especially given the highly volatile nature of a business environment characterized by various sources of risk and uncertainty. For example, a disruptive event in one country can result in the disruption of an entire supply chain. The massive floods that took place in Thailand in 2011 offer a case study of global supply chain disruption, especially in the electronic and automotive industries. Western Digital, Sony, and Honda were all obliged to stop production due to the direct flooding of their factories, while Toyota suffered from an indirect impact due to the disruption experienced by key suppliers [1] A proactive supply chain network design decision framework is, therefore, vital for all enterprises [2,3,4].

In this study, a two-stage stochastic programming model in which the possibility of a facility disruption is accounted for is formulated to solve a supply chain network design problem. Compared to other methods such as deterministic model or Mixed Integer Program (MIP), two-stage stochastic programming is more appropriate for decision making under uncertainty in a business environment where two steps of decision processes are involved. The first stage deals with strategic levels, such as to determine facility locations and their capacity, while the second stage considers at a tactical level, such as production quantity, delivery routing. Each stage’s decision could affect the other stage’s result, and it could not be determined individually. For that reason, the two-stage stochastic programming is selected as it is closed to business decision process.

The problem consists of a four-echelon, single-product supply chain network environment, which consists of suppliers, manufacturers, distribution centers (DCs), and retailers. These facilities are located around the world. The locations of the manufacturers and retailers are known, whereas the suppliers and the locations of DCs are decisions to be made, and the objective function is to maximize the expected profit under facility disruptions. A two-stage stochastic programing is used. In the first stage, the model selects the suppliers and DC locations. In the second stage, the model determines the purchased, produced, and transported quantities throughout the network in a way that takes disruptive scenarios into account. The main contributions of the study to the supply chain network design decision literature are twofold. First, the location, production, purchasing, and transportation decision variables are considered simultaneously in a four-echelon network design under disruptive scenario conditions. Second, this paper proposed a simulated annealing (SA) meta-heuristic algorithm solution approach for finding good quality solutions to large-scale problems in a reasonable time.

The remainder of this paper is organized as follows. Section 2 provides a literature review of stochastic programming models constructed with the goal of addressing supply chain network design problems. In Section 3, a two-stage stochastic programming model is formulated. Section 4 discusses a solution technique. Section 5 presents a numerical example to illustrate the usefulness of the proposed model and compare the result with those obtained using a traditional supply chain network design. Section 6 provide a conclusion and recommendations for future research.

2. Literature Review

This section provides the review of stochastic programming models for supply chain network design problems with a focus on disruption. For a comprehensive review of a supply chain network design problem, interested readers are referred to [3,5,6]. Ref. [3] focuses on analyzing the ripple effect on supply chain design, while ref [5] focuses on analyzing supply chain design and planning with both disruptions and recovery. Ref [6] provides the landscape of supply chain network design literature by reviewed 1310 publications from 1999 to 2019 in 20 journals and proceeding related to supply chain network design. The authors analyzed data using bibliometric analysis approach.

Disruption due to man-made or natural disasters becomes a critical factor in supply chain network design literature. Ref. [7] formulated a two-stage stochastic programming model considering uncertainty in demand, supply, processing, transportation, shortage capacity expansion cost, and facility disruption. The model is to minimize fixed cost and expected transportation cost. Ref. [8] proposed a mixed integer programming (MIP) model and a continuum approximation (CA) model for the reliable uncapacitated fixed charge location problem (RUFL) under normal and failure scenarios. The work of ref. [8] is extended from ref. [9] by assuming that the failure of each facility is independent. Ref. [10] solved a supply chain network design problem under operation and disruption risk with delivery lead-time sensitive customers. The facility’s capacity varies randomly due to disruptions. Ref. [11] emphasized on supply disruption and also incorporated sustainability as another objective. For disruptive scenarios, both partial and complete disruption can occur at suppliers’ facility. Three mitigation strategies were evaluated to hedge against disruption risks: adding extra production capacity to plant, utilizing multiple sourcing instead of single source, and having back up supplier, assuming that the back-up supplier does not affect by disruption. Ref. [12] also focuses on a resilient supplier network and optimal order allocation under disruption risks. Ref. [13] applied disruption management in a multi-product imperfect production system to obtain the optimal batch size for multi-item in the recovery time window under profit maximization objective. Ref. [14] proposed a two-stage stochastic model for supply chain network design under disruption subject to time and cost of supply chain recovery as resilience measures. The model determines optimal location, allocation, inventory, and order-size decisions. Ref. [15] solved a supply chain network design for blood supply chain to provide location-allocation decisions under facility disruption and uncertainty in a disaster situation. The authors defined two types of facilities: temporary facilities which prone to partial disruption, while blood centers are safe from disruptions due to a larger investment compare the former ones. Disruption on temporary facilities are assumed as partial failed. Disaster situation affects the ability of blood donation at temporary facilities. Probability of disruptive scenarios are predefined.

Ref. [16] studied a network of single-product, multi-echelon supply chain, which consists of three manufacturing facilities, three DCs, and nine retailers. Disruptions of manufacturing facilities, DCs, and their connecting links are considered. The authors used path-through-facility formulation instead of defining flow variables among facilities so that disruption at different connecting links can be considered simultaneously. Demand at each retailer is assumed to follow a normal distribution. In addition, they defined four possible scenarios for the production disruptions, i.e., all production facilities are usable, the second production facility is impaired, the third production facility is impaired, and both the second and third facilities are impaired. The proposed model was applied to an agri-food case study. Ref. [17] formulated a mixed integer, non-linear model for designing a global medical device manufacturer. The authors proposed six resilience strategies to mitigate the risk of correlated disruptions in supply chain facilities. Ref. [18] formulated a mixed-integer, non-linear programming model for a bio-fuel supply chain network considering the failure of links among the multi-modal facilities, which includes feedstock suppliers, intermodal hubs, bio-refinery facilities, and markets. The authors used a spatial model to estimate the disruption probability of the intermodal links based on a historical event of disasters. This paper assumed that a link’s disruption probability is approximately equivalent to the probability of a disaster. The objective function is to maximize post-disaster connectivity and minimize transportation costs between the origin locations to the markets where bio-fuel is sold. The model is solved using a generalized Benders decomposition algorithm. Many of recent work consider mitigation strategies for creating robustness or resiliency in supply chain network design problem. Ref. [19] formulated a multi-objective optimization model for designing resilient supply chain networks under a case of mergers and acquisition activity. They evaluated the trade-off between minimizing total network cost and maximizing overall supply chain network connectivity. The disruption was examined for the post-acquisition supply chain network, which may prone to vulnerability based on demand-weight connectivity. Facility disruption was also included in this study using some probability of disruption. Ref. [20] focuses on the transportation disruption from plant to a distribution center, with four mitigation strategies in order to find the best one based on service level and total cost. These mitigation strategies are (1) the risk acceptance strategy, (2) the redundant stock strategy, (3) the flexible route strategy, and (4) the redundant-flexible strategy. The authors concluded that the solutions are varied depends on the budget that managers are willing to deploy to achieve service level. In these studies, the disruption of facility or transportation was predefined. Ref. [21] uses mixed integer programming to create a supply chain network. Then, simulation approach is used to find risk mitigation strategies when a supply chain is exposed to random disruptions. The results shown that robustness is suited under mild to moderate disruptions, while a combination of risk management strategies for robustness and resiliency is essential to cope with severe disruptions. For the performance indicators for supply chain resilience, interested readers are referred to ref. [22].

Types (random or predefined) and levels (partial or complete) of disruptions are another characteristic of disruption considered in literature. Ref. [23] solved a reliable stochastic supply chain network problem considering random disruptions at the DCs and the transportation mode. The authors allowed partial disruption at DC. The authors applied the conditional value-at-risk approach to control the risk of model. The model was solved using an exact solution method and a hybrid between tabu search and simulated annealing algorithms. Ref. [24] formulated a two-stage stochastic programming model for a four-echelon bio-fuel supply chain considering disruption at intermodal hub. The model was solved using an accelerated generalized Benders decomposition algorithm. Refs. [25,26] formulated a two-stage stochastic programming model considering disruption at suppliers. Ref. [27] studied a single-product, two-echelon supply chain network consisting of one manufacturer and one retailer. The authors considered randomness in demand at retailer, randomness in production yield at manufacturer, and disruption at manufacturer. The main objective was to evaluate the effects of various uncertainties on the optimal decisions of order quantity placed at the manufacturer, reserved quantity at the backup supplier, and unit wholesale price for the manufacturer. The authors also evaluated the coordinating condition between centralized and decentralized models. In terms of disruption, the authors assumed a fixed value of probability. Once disruption occurs, a manufacturer cannot ship anything to a retailer. Ref. [10] proposed a multi-period supply chain network design considering random partial disruption at facilities (manufacturing plants and warehouses). In this paper, each customer site is firstly assigned to a specific warehouse. Then, under disruptive scenarios that generated by a simulation approach, the model determines fortify facility and alternate sourcing decisions to obtain a resilient supply chain network. The authors formulated a multi-stage stochastic model and demonstrated with a glass supply chain network. Ref. [28] solved a supply chain network problem under completed disruption at facility and uncertain demand. For a collection work of a stochastic programming models for supply chain management, interested readers are referred to ref. [29]. The author focuses on management of disrupted flows in customer-driven supply chains. The key decision was the allocation of material flows (parts and finished products) among suppliers and production facilities. The allocation of parts to suppliers and allocation of demands to production facilities are computed based on before, during, and after disruptive scenarios in the network. The probability of disruption scenarios is mostly predefined with partial level of disruption.

Table 1. Stochastic programming models for supply chain network design problem under disruptions.

Ref	Disruption		Decision Variables
	Facility	Link	Location	Production Quantity	Purchasing Quantity	Distribution Quantity
[16]	V	V	V			V
[23]	V	V	V			V
[24]	V		V			V
[27]	V	V			V
[26]	V				V
[17]	V		V	V	V	V
[18]		V	V			V
[10]	V		V	V		V
[14]	V		V			V
[19]	V			V	V	V
[29]	V			V	V
[11]	V		V		V
[12]	V		V		V
[13]	V			V	V
[20]		V				V
[28]	V		V		V	V
[15]	V		V
This paper	V		V	V	V	V

summarizes the literature by (i) the location of disruption, e.g., facility and transportation link, that each study has been considered; and (ii) the decision variables that the proposed model has been covered. From the review, most of supply chain network design problems considered location and distribution quantity decisions. This paper focus on facility disruption. In terms of decisions, the proposed model solves the location, production, purchasing, and distribution decisions, simultaneously.

Table 2. Stochastic programming models for supply chain network design problem.

Ref	Stage of Supply Chain				Type of Disruption		Level of Disruption
	Supplier	Manufacturer	DC	Link	Random	Premeditate	Partial	Complete
[16]		V		V		V		V
[23]			V	V	V		V
[24]			V		V			V
[27]		V		V		V		V
[26]	V					V	V	V
[17]	V	V	V			V	V
[18]				V	V			V
[10]		V	V		V		V
[14]			V			V		V
[19]	V	V	V			V	V
[29]	V	V				V	V
[11]	V	V				V	V	V
[12]	V					V		V
[13]		V				V		V
[28]	V		V					V
[20]		V	V	V		V	V
[15]		V	V			V	V
This paper	V	V	V		V			V

summarizes the literature by (i) the stage of supply chain where the disruption occurs, e.g., supplier, manufacturer, distribution centers (DC), and transportation link; (ii) type of disruption, e.g., random or premeditate; and (iii) level of disruption, e.g., partial disruption and complete disruption. The proposed model considers random disruption risk, which means that disruption may occur at any stage of the supply chain network, including supplier, manufacturers, and DC, depend upon the random parameter values. In addition, this paper focus on evaluating an effect of a complete disruption.

3. Model Formulation

3.1. Problem Description

This paper considers a four-echelon supply chain network consisting of suppliers, manufacturers, DCs, and retailers, as illustrated in Figure 1. The locations of the candidate suppliers, manufacturers, candidate DCs, and retailers are drawn from ref. [30]. A disruption, which may occur in any region, leads to a disruption affecting the suppliers, manufacturers, and DCs located in the disrupted region. Given the possibility of a facility disruption, the goal of the decision maker is to select suppliers and DCs such that the expected profit of the supply chain is maximized.

The following assumptions are made in this study:

(1)

Given that the problem deals with strategic decision to design a supply chain network, therefore this paper assumes that demands at the retailers are deterministic for observing the effect of disruption easily.

(2)

A disruption may occur at any supplier, manufacturer, or DC. To incorporate disruption into the model and measure its impact, this paper generates scenarios based on the probability distribution of the disruption by setting the binary parameters from one to zero. If the disruption occurs at any facility, it will be completely disrupted. In addition, this paper assumes that the probabilities of disruption occurs at each facility are independent.

(3)

The set of all possible disruptive scenarios is finite.

3.2. Mathematical Model

In this section, a two-stage stochastic program for the supply chain network design problem is provided. The model involves decision making in two stages: In the first stage, the locations and the sizes of the DCs, as well as the suppliers for the manufacturers must be determined before a disruption has taken place. In the second stage, the flows within the supply chain network and the amount of lost sales are determined to compensate for the effects of the disruptive event. The objective of the model is to maximize the expected total profit under disruptive scenario conditions.

A two-stage stochastic program for the problem is given as follows:

Sets:

• S = Set of suppliers
• M = Set of manufacturers
• W = Set of DCs
• C = Set of retailers
• L = Set of DC capacities
• R = Set of countries
• K = Set of disruptive scenarios

Indices:

• s = Index of suppliers, s ∈ S
• m = Index of manufacturers, m ∈ M w = Index of DCs, w ∈ W
• c = Index of retailers, c ∈ C
• l = Index of DC capacities, l ∈ L
• r = Index of countries, r ∈ R
• k = Index of disruptive scenarios, k ∈ K

Parameters:

• cap_m = Production capacity at manufacturer m
• cap_s = Capacity at supplier s
• $ca{p}_w^l$ = Capacity at DC w of size l
• d_c = Demand for products at retailer c
• msm = Minimum transportation quantity from suppliers to manufacturers
• $f_w^l$ = Fixed cost of opening a DC w of capacity l
• pm_sm = Purchasing cost per unit of material from supplier s by manufacturer m
• tr_mw = Transportation cost per unit from manufacturer m to DC w
• tr_wc = Transportation cost per unit from DC w to retailer c
• pc_m = Production cost for a product at manufacturer m
• np = Price of a product
• ls_c = Lost sales cost at retailer c
• p_r = Probability of disruption occurs at country r

Random Parameters:

${\alpha }_{sk}=\{\begin{array}{c}1\\ 0\end{array}$	if supplier s operates in scenario k (with probability 1 − pc)
	if a disruption occurs at supplier s in scenario k (with probability pc)
${\delta }_{wk}=\{\begin{array}{c}1\\ 0\end{array}$	if DC w operates in scenario k (with probability 1 − pc)
	if a disruption occurs at DC w in scenario k (with probability pc)
${\beta }_{mk}=\{\begin{array}{c}1\\ 0\end{array}$	if manufacturer m operates in scenario k (with probability 1 − pc)
	if a disruption occurs at manufacturer m in scenario k (with probability pc)

p_k = The probability of a disruptive scenario k

First-stage Decision Variables:

$x_w^l=\{\begin{array}{c}1\\ 0\end{array}$	if DC w operates with size l
	otherwise
$y_s=\{\begin{array}{c}1\\ 0\end{array}$	if supplier s is selected
	otherwise

Second-stage Decision Variables:

QSMsmk =	Quantity of raw material purchased from supplier s by manufacturer m in scenario k
QMWmwk =	Quantity of products shipped from manufacturer m to DC w in scenario k
QWCwck =	Quantity of products shipped from DC w to retailer c in scenario k
LDck =	Quantity of sales lost at retailer c in scenario k

3.2.1. Objective Function

The objective of the model is to maximize the expected supply chain profit (Z), which is the difference between the expected revenue and the total cost. The objective function consists of first stage cost and second-stage profit (cost and revenue). The first-stage cost is the fixed cost of opening DCs, and the second stage value is the difference between the expected revenue and operation costs, which includes purchasing cost, production cost, transportation cost between manufacturers and DCs, transportation cost between DCs and retailers, and lost sales cost. The objective function is given in Equation (1).

maximize

$$\begin{array}{ll}Z=-{\displaystyle {\displaystyle \sum }_{w\in W}}{\displaystyle {\displaystyle \sum }_{l\in L}}{f}_w^l{x}_w^l& +{\displaystyle {\displaystyle \sum }_{k\in K}}{p}_k[np\left({\displaystyle {\displaystyle \sum }_{w\in W}}{\displaystyle {\displaystyle \sum }_{c\in C}}QW{C}_{wck}\right)-\left({\displaystyle {\displaystyle \sum }_{s\in S}}{\displaystyle {\displaystyle \sum }_{m\in M}}p{m}_{sm}QS{M}_{smk}\right)\\ & -\left({\displaystyle {\displaystyle \sum }_{m\in M}}{\displaystyle {\displaystyle \sum }_{w\in W}}t{r}_{mw}QM{W}_{mwk}+{\displaystyle {\displaystyle \sum }_{w\in W}}{\displaystyle {\displaystyle \sum }_{c\in C}}t{r}_{wc}QW{C}_{wck}\right)\\ & -{\displaystyle {\displaystyle \sum }_{m\in M}}p{c}_m\left({\displaystyle {\displaystyle \sum }_{w\in W}}QM{W}_{mwk}\right)-{\displaystyle {\displaystyle \sum }_{c\in C}}l{s}_{c}L{D}_{ck}]\end{array}$$

(1)

3.2.2. Constraints

Supplier capacity:

Constraint (2) ensures that the quantity of raw materials supplied by the suppliers to all the manufacturers should be less than or equal to the suppliers’ capacity in each scenario.

$${\displaystyle \sum }_{m\in M}QS{M}_{smk}\le ca{p}_s{\alpha }_{sk}{y}_s,\forall s\in S, k\in K.$$

(2)

Inter-stage flow:

Constraint (3) ensures that the shipment between the supplier and the manufacturer follows the minimum requirement and does not exceed the supplier’s capacity in each scenario.

$$msm·{\alpha }_{sk}{y}_s\le QS{M}_{smk}, \forall s\in S, m\in M, k\in K.$$

(3)

Production capacity:

Constraint (4) ensures that the sum of the products transported to the DCs from manufacturer m is less than or equal to the manufacturer’s capacity in each scenario.

$${\displaystyle \sum }_{m\in W}QM{W}_{mwk}\le ca{p}_m{\beta }_{mk, } \forall m\in M, k\in K.$$

(4)

Material flow between suppliers and manufacturers:

Constraint (5) ensures that the quantity of raw material flowing into manufacturer m is equal to the quantity of products flowing out of that manufacturer to the DCs in each scenario.

$${\displaystyle \sum }_{s\in S}QS{M}_{smk}={\displaystyle \sum }_{w\in W}QM{W}_{mwk}, \forall m\in M, k\in K.$$

(5)

DC capacity:

Constraint (6) ensures that the quantity of products flowing into DC w does not exceed its storage capacity in each scenario. Constraint (7) ensures that one or no capacity level is selected for each DC.

$${\displaystyle \sum }_{m\in M}QM{W}_{mwk}\le {\displaystyle \sum }_{l\in L}ca{p}_w{\delta }_{wk}{x}_w^l,\forall w\in W, k\in K.$$

(6)

$${\displaystyle \sum }_{l\in L}{x}_w^l\le 1, \forall w\in W$$

(7)

Product flows between DCs and retailers:

Constraint (8) ensures that the quantity of products flowing into DC w is equal to the quantity of products flowing out of that DC to the retailers in each scenario.

$${\displaystyle \sum }_{m\in M}QM{W}_{mwk}={\displaystyle \sum }_{c\in C}QM{C}_{wck}\forall , w\in W, k\in K.$$

(8)

Demand requirement:

Constraint (9) represents the demand satisfaction constraints. The total quantity of products flowing into retailer c and the lost sales at the retailer c should be equal to the demand at that retailer in each scenario.

$${\displaystyle \sum }_{w\in W}QM{C}_{wck}+L{D}_{ck}=d_c, \forall c\in C, k\in K.$$

(9)

Non-negativity:

$$QS{M}_{smk},QM{W}_{mwk},QW{C}_{wck},L{D}_{ck}\ge 0$$

(10)

4. Solution Methodology

The proposed model in Section 3.2 is typically a large-scale problem. Depending on the number of facilities and disruptive scenarios, the problem may turn out to be intractable to a direct approach in practice, even using the state-of-the-art software packages. The IBM CPLEX Optimizer is one of the most powerful and well-known optimization solvers for solving linear programming, Mixed Integer Program (MIP), quadratic programming and quadratic constrained programming problems. Initial attempt at solving the model was performed by using CPLEX Optimizer with a sample size of 10. (Please refer to the example problem in Section 5). The computation took 2195.03 s, 1,237,645 MIP simplex iterations, and 1269 branch-and-bound nodes to obtain an optimal solution. However, an attempt at solving the model with a sample size of 20 was unsuccessful. Therefore, a simulated annealing (SA) meta-heuristic algorithm is proposed to find the first-stage solution, and the second-stage solution is obtained by re-optimizing the associated second-stage problem from a given first-stage solution. The Monte Carlo sampling technique is employed to construct a smaller sample average problem instead of solving the problem with all possible scenarios. This technique is known as the sample average approximation (SAA) technique [31]. As the first-stage solution is obtained outside the model, the objective value of the given first-stage solution instead of the optimal objective value of the problem is approximated. However, by continuously improving the quality of the solution, the proposed approximation is likely to get closer to the true optimal objective value. The estimate of the objective function value is obtained by solving the second-stage sample average problem with the state of the art solver CPLEX. Implementation also employs information feedback to improve the chances of reaching the global optimum in the SA algorithm. The general framework of the algorithm is depicted in Figure 2.

4.1. Simulated Annealing (SA)

Simulated annealing (SA) is a probabilistic technique for solving the global optimization problem [32]. The technique takes its inspiration from annealing in metallurgy, a thermal process involving heating and controlled cooling of a solid so that the particles will arrange themselves in the ground state. During the cooling process, the internal energy of the system changes from the current state to the next state according to some probabilities. This feature is emulated in the SA algorithm, which allows SA to accept worse solutions in order to prevent it from becoming stuck at a local optimum.

The following parameters are defined for the proposed algorithm:

• r = Index of iterations
• c_r = Control parameter at iteration r
• c₀ = Initial value of the control parameter
• c_min = Minimum value of the control parameter
• a, b = First-stage solution vectors
• T = Number of solutions evaluated in each iteration
• N = Sample size (number of sampled scenarios)
• ${\widehat{\mathrm{Z}}}_{\mathrm{N}}$= Estimate of the objective function value by using sample size N
• P = Probability vector of the first-stage variables (x and y)
• U = Random number in [0, 1)
• ρ = Coefficient of the control parameter update
• λ = Coefficient of the probabilities update

The scheme of the proposed algorithm is given in Algorithm 1. The parameters ρ, λ, c₀, T, and N, and the probability vector P are initialized to some prespecified values in the Initialize() function. The function Generate() generates a feasible first-stage solution vector from the probability vector P. The algorithm uses an iterative procedure to search for a better solution. Through the algorithm, new solutions are generated by using the Generate() function, and their objective values are compared to the objective value of the current solution a. A new solution b will be accepted if its objective value is larger than the current objective value, or if the probability of acceptance $\exp \left(\frac{{\widehat{\mathrm{Z}}}_{\mathrm{N}}\left(\mathrm{b}\right)-{\widehat{\mathrm{Z}}}_{\mathrm{N}}\left(\mathrm{a}\right)}{{\mathrm{c}}_{\mathrm{r}}}\right)$ is greater than a random generated number U. The control parameter c plays the role of temperature control in the cooling process, which controls the probability of a solution being accepted in the algorithm. The value of c_r decreases by a factor of ρ in each iteration, which reduces the chance that a worse solution will be accepted. Note that ${\widehat{Z}}_N\left(b\right)-{\widehat{Z}}_N\left(a\right)$ is negative in the expression, so that the probability of acceptance is less than 1. The function Update() updates the probability vector from the current solution vector. The algorithm terminates when c_r is less than a given value c_min.

4.2. Solution Construction

The algorithm maintains a probability vector P for the first-stage solution. This probability vector serves as information storage for the characteristic of the solution space. The basic idea is to increase the probability of the potential DC locations and suppliers being selected in the first-stage solution. The probabilities are initialized to a prespecified value during the initialization process and are updated during the course of the algorithm, which will be further discussed in Section 4.4. The first-stage solution is randomly generated according to the probability associated with each element in the solution vector. The elements in the solution vector comprise all the elements from the first-stage decision variables x and y. Therefore, a feasible first-stage solution is constructed in two phases: DC selection and supplier selection.

DC selection:

As only one DC capacity can be built at any DC location, to prevent multiple elements from being selected at the same location, the DC selection is generated in two steps. First, a DC location w is selected randomly according to the probability given by

$$p_w=\max \left(p_w^1,p_w^2,\dots ,p_w^L\right),$$

(11)

for w ∈ W, where $p_w^1$, $p_w^2$, ..., $p_w^L$ are the elements of the probability vector P associated with DC location w. Second, if DC w is selected, then its capacity is randomly selected from one of the capacity sizes $ca{p}_w^1$, l ∈ L according to the probability $p_w^1$, $p_w^2$, ..., $p_w^L$ of the DC location w.

Algorithm 1 Simulated Annealing Algorithm

Initialized(ρ, λ, c0, cmin,T, N, P) Generate (a) r:=1 cr:= c0 repeat   for t:=1 to T do    Generate (b)    if ${\widehat{Z}}_N\left(b\right)-{\widehat{Z}}_N\left(a\right)$ then      a:=b   else if $exp\left(\frac{{\widehat{Z}}_N\left(b\right)-{\widehat{Z}}_N\left(a\right)}{c_r}\right)>U$ then      a:=b   end if   end for   Update(P)   cr:= ρ ∙ cr   r:= r + 1   until cr < cmin

Supplier selection:

The chances of a supplier being selected are determined by its probability in the probability vector P.

4.3. Monte Carlo Sampling and Solution Evaluation

In practice, the total number of scenarios can be extremely large, which causes the algorithm to take a significant amount of time to solve the model. A sampling technique is often used to reduce the number of scenarios in order to make the problem size manageable. The Monte Carlo sampling technique is used in this study to draw N samples from the set of original scenarios K. The objective function value Z in Equation (1) is estimated by substituting the first-stage solution obtained from SA to the model in Section 3.2, and then constructing the sample average second-stage problem by using the Monte Carlo sampling technique: i.e., the random vector is replaced in the model with the sampled scenarios and the probability p_k is changed to 1/N in the objective function. The resulting sample average problem is then solved by using CPLEX.

After maximizing the expected profit using the sampling method, any feasible (first-stage) solution will provide a statistical lower bound of the objective value to the original problem [31].

4.4. Information Update

Information update is employed to improve the chances of finding better solutions. As the algorithm progresses, more information is gathered through the evaluation of the first-stage solutions, and this information is used to update the probability vector P. The information update is done by utilizing the candidate solution, i.e., a solution accepted in the current iteration that potentially has the best objective value. For a given candidate solution a^∗, the probability of an element j in the probability vector P either increases or decreases according to the following rules:

$$P_j=\{\begin{array}{c}{P}_j^{\lambda } \mathrm{if}{a}_j^{*}=1;\\ \lambda P_j \mathrm{if}{a}_j^{*}=0;\end{array}$$

(12)

The parameter λ is the coefficient for the probabilities update, which has a value between (0, 1). From Equation (12), it is not hard to see that, if a DC or supplier j is selected in the candidate solution ($a_j^{*}=1$), then its associated probability increases because both P_j and λ are less than 1. Otherwise, the probability decreases by multiplying λ to its previous value. The use of a single parameter λ to both increase and decrease the probability value provides a simpler procedure for implementation and more importantly for parameter tuning in the experiment.

5. Illustrative Example

5.1. Supply Chain Network Design without Disruption

For the purpose of illustration, a traditional supply chain network design problem primarily focuses on profit maximization in ref. [30]. A supply chain must fulfill the demand of 100 retailers. The solution of the problem consists of selecting 13 suppliers, 5 manufacturers, and 3 DCs from the respective sets of available facilities. It is assumed that these facilities are located in six different regions according to the six continents of the world: Africa, Asia, Europe, North America, Australia, and South America.

Table 3. Facility locations of a traditional network design.

Region	Supplier	Manufacturer	DC
Region 1	S16	M3	-
Region 2	S8, S10, S11, S15, S17	M1, M2	-
Region 3	S6, S7, S9, S14	M4	W8
Region 4	S4, S13	M5	W12, W13
Region 5	-	-	-
Region 6	S1	-	-
Total	13	5	3

summarizes the locations of the suppliers, manufacturers, and DCs in the different regions, and Figure 3 presents the geographical locations of those facilities.

A traditional supply chain network design from ref. [30] has a centralized supply-based characteristic. It selects low-cost suppliers and DCs in order to minimize fixed costs. In addition, the model also selects the facilities that are located in closed proximity to each other in order to minimize transportation costs.

5.2. Supply Chain Network Design with Disruption Consideration

5.2.1. Disruptive Scenarios

To estimate the probability of disruption for each facility, the number of disasters reported is referred in the country where a facility is located during 1900–2018 from the International Disaster Database, Center for Research on the Epidemiology of Disaster CRED (http://www.emdat.be/) [33]. The probability of disruption for each country is presented in

Table 4. Disruption occurrence probability of countries.

Country	Suppliers	Manufacturers	DCs	Probability of Occurrence
Argentina	S11	-	-	0.017
Australia	S14	-	W13	0.035
Austria	-	-	W4	0.007
Belgium	S13	-	W12	0.010
Brazil	S7	-	-	0.034
Canada	S4	-	W6	0.020
Cayman Islands	S16	M3	W17	0.001
China	-		W19	0.133
France	S17		W20, W25	0.023
Germany	S5	M1	W7	0.011
India	S10	-	-	0.105
Indonesia	S18	-	W21	0.073
Italy	S3	-	W5	0.022
Japan	-	-	W14	0.051
Korea (the Republic of)	S6	-	W8	0.017
Malaysia	-	-	W15	0.012
Mexico	-	-	W11	0.038
Netherlands (the)	S15	-	W16	0.005
Pakistan	-	-	W18	0.031
Philippines (the)	S20	M4, M5	W23, W24	0.092
Russian Federation (the)	S19	-	W22	0.024
South Africa	S12	-	-	0.015
Sweden	S2	-	W2	0.002
Taiwan (Province of China)	-	-	W3	0.016
Thailand	S1	-	W1	0.021
Turkey	S9	-	-	0.024
United Kingdom of Great Britain and Northern Ireland (the)	S8	M2	W9	0.013
United States of America (the)	-	-	W10	0.147
Total				1.000

. The total number of possible disruptive scenarios depends on the number of facilities. However, based on the sample average approximation technique, disruptive scenarios are sampling with a sample size of N. Therefore, p_k is equal to 1/N. These probabilities will be used later in Section 5.2.2 to sample the disruptive scenarios.

5.2.2. Computational Results

To solve the problem, the algorithm is coded in Microsoft Visual C# and solved using ILOG Concert Technology with CPLEX 12.51 on a PC with an AMD FX Processor at 4.0 GHz and 8.0 GB RAM. In order to test the effectiveness of the proposed method, the results are compared from the heuristic to the optimal objective value obtained from CPLEX by solving a small sampled version of the model with sample size N = 10. The same sampled problem is solved for 10 replications for the heuristic. The results are provided in

Table 5. Comparison of objective values obtained from CPLEX and simulated annealing (SA) algorithm with sample size 10.

Replication	Optimal (CPLEX)	SA	Difference	Difference (%)
1		12,259,408.30	234,270.30	1.88
2		12,178,013.94	315,664.66	2.53
3		12,309,582.24	184,096.36	1.47
4		12,178,665.69	315,012.91	2.52
5		12,324,980.65	168,697.95	1.35
6	12,493,678.60	12,035,623.28	458,055.32	3.67
7		12,370,727.82	122,950.78	0.98
8		12,364,367.23	129,311.37	1.04
9		12,097,164.50	396,514.10	3.17
10		12,350,310.94	143,367.66	1.15
Average		12,246,884.46	246,794.14	1.98

. From the results, the largest optimality gap is only 3.67%. Thus, the proposed method yields acceptable solutions.

Table 6. Objective values obtained from SA with different sample sizes.

Replication	N = 20	N = 50	N = 100
1	12,926,743.17	12,936,504.50	12,418,851.37
2	12,418,814.86	12,537,833.87	12,125,486.15
3	12,919,320.51	12,464,742.63	12,274,587.18
4	12,546,105.51	12,504,353.57	12,059,285.24
5	12,288,548.63	12,366,851.17	12,412,839.32
6	12,846,196.02	12,621,856.67	11,995,844.01
7	12,952,681.61	12,346,809.29	12,403,648.33
8	12,797,331.73	12,415,321.97	12,493,808.13
9	12,882,444.22	12,549,969.87	11,953,054.86
10	12,006,506.97	11,823,100.12	12,280,738.77
Average	12,658,469.32	12,456,734.37	12,241,814.33
Standard Deviation	327,145.91	278,571.02	195,392.62

presents the supply chain profit that resulted from the first-stage of the proposed model for various numbers of disruptive scenarios (sample size N = 20, 50, and 100) generated from Monte Carlo sampling. For each sample size N, the model is run for 10 replications. The average time taken to solve the sample problem for each sample size is 31.30, 80.45, and 162.24 s respectively. The last two rows in Table 6 are the average and standard deviation, respectively, of the supply chain profit from the 10 replications. As the number of disruptive events (N) increases, the variance in the supply chain profit decreases.

Table 7. Objective values evaluated for different solutions without disruption.

Replication	Solution from Original Design	Solutions	from SA with Sample Size
		N = 20	N = 50	N = 100
1		13,096,522.68	13,051,966.74	13,184,154.53
2		12,920,227.45	13,155,981.66	13,180,699.57
3		12,942,537.45	13,003,074.42	13,057,226.19
4		13,171,893.86	13,089,908.03	13,073,942.27
5	13,248,679.26	13,065,948.69	12,983,342.65	13,007,254.34
6		13,149,727.22	12,932,378.48	13,193,867.37
7		13,027,280.15	12,974,604.06	13,074,024.28
8		13,154,910.60	13,168,145.63	13,149,428.82
9		13,103,227.15	13,070,412.66	13,091,222.72
10		13,149,634.13	13,123,011.48	13,060,618.83
Average		13,078,190.94	13,055,282.58	13,107,243.89
Standard Deviation		89,409.60	80,567.36	64,753.36
Mitigation cost		170,488.32	193,396.68	141,435.37

presents a comparison of the supply chain profit values under normal conditions (no disruption) between the traditional design as proposed in ref. [30] and the proposed model, which is solved by the SA heuristic approach. The traditional design offers a profit of $13,248,679.26. This value implies the upper-bound of the supply chain profit under normal conditions. The average supply chain profit values obtained by the SA approach at N = 20, 50, and 100 are $13,078,190.94, $13,055,282.58, and $13,107,243.89, respectively. The differences between these values and the upper-bound value implies mitigation costs due to supply chain re-design, which are $170,488.32 (1.30%), $193,396.68 (1.48%), and $141,435.37 (1.08%), respectively.

Table 8. Objective values evaluated for different solutions under disruption using a large sample size of 3000.

Replication	Solution from Original Design	Solutions	from SA with Sample Size
		N = 20	N = 50	N = 100
1	11,950,373.24	11,621,036.69	12,125,437.22	12,009,805.94
2	11,838,016.30	11,851,151.56	11,947,473.61	11,981,417.92
3	11,831,140.24	11,448,720.92	12,049,300.08	12,004,185.16
4	11,780,590.92	12,045,964.41	11,935,091.85	11,830,527.34
5	11,789,481.46	12,239,861.49	12,065,857.69	12,118,408.82
6	11,895,626.39	11,982,653.16	12,273,107.33	12,027,116.72
7	11,886,468.32	11,991,857.49	11,834,090.51	11,922,442.47
8	11,880,323.77	11,894,812.90	11,923,011.41	11,960,271.74
9	11,838,575.11	12,107,121.93	12,013,970.80	11,982,746.14
10	11,801,826.73	11,990,353.93	11,905,938.10	12,314,238.09
Average	11,849,242.25	11,917,353.45	12,007,327.86	12,015,116.03
Standard Deviation	53,573.30	231,904.94	127,375.64	128,360.49
Mitigation benefit		68,111.20	158,085.61	165,873.78

presents a comparison of the supply chain profit values in disruptive scenarios between the traditional design in ref. [30] and this proposed model. The results are based on evaluating the existing solutions with a larger sample size, i.e., of 3000, to provide more accurate estimates. When a disruption has taken place, the traditional design has an average profit value of $11,849,242.25. In other words, the average disruption impact is $1,399,437.01 (10.56%) in terms of the supply chain profit. On the other hand, the supply chain network solutions from the proposed model, when a disruption has taken place, yield higher average profit values. The differences between these values and the value obtained with the traditional design implies that the proposed model renders an average mitigation benefit of $68,111.20 (0.57%), $158,085.61 (1.33%), and $165,873.78 (1.40%).

Although the average mitigation cost (based on the results from 10 replications) outweighs the average mitigation benefit, in practice the best solution can be chosen (N = 100, replication #10) which has the mitigation cost of $114,201.25 ($13,248,679.26 $13,134,478.01) and the mitigation benefit of $464,995.84 ($12,314,238.09 $11,849,242.25). The result confirms that the proposed model performs better than the traditional design under disruption. The network configuration is shown in

Table 9. Facility locations of the best solution.

Region	Supplier	Manufacturer	DC
Region 1	S16	M3	W17
Region 2	S5, S8, S10, S11, S12, S15, S17, S18	M1, M2	-
Region 3	S6, S7, S9, S14, S19	M4	W3, W21
Region 4	S4, S13, S20	M5	-
Region 5	S2	-	-
Region 6	S1, S3	-	-
Total	20	5	3

. All 20 suppliers are selected when disruptions are considered in the design. In other words, the model suggests having backup suppliers to mitigate the negative impact due to disruption. For DCs, the model suggests implementing decentralized strategy to minimize the disruptive impact.

To compare the quality of the solution from the original design with those from the two-stage program under disruption, Dunnett’s test with simultaneous comparisons at a 95% confidence level was conducted between the result from the original design and those from the sample size at N = 20, 50, and 100. The result from the original design was used as a control. The test results are provided in Figure 4. The proposed method with a sample size of 100 provides better solutions than the original design does, as all of the profit differences are greater than 0. In addition, a comparison of the results of all three sample sizes shows evidence to clearly indicate that sample size has a strong impact on the solution. Therefore, it is possible to obtain a usable solution with a larger sample size when solving practical problems.

From the illustrative example, supply chain disruption in the supply chain network design decision improves the resilience of a supply chain network. The proposed model is applicable for solving the supply chain network model with facility disruption taken into account.

6. Conclusions

In this paper, a two-stage stochastic programming model is proposed to solve a supply chain network design problem subjected to facility disruption. The first stage of the model is solved to determine supplier selection decision and DC location decision using a modified simulated annealing (SA) heuristic approach. The second stage was then solved to determine the product flows between facilities using a sample average approximation (SAA) approach. Facility disruption scenarios were generated using the Monte Carlo sampling method and then incorporated into the model. The expected supply chain profit of the supply chain network solutions obtained from the proposed model compared with the one from the traditional design in ref. [30]. The Dunnett Simultaneous comparison at 95% CI are applied to compare the profit difference of the solutions from the proposed model to the traditional design. The results show that in a disruptive scenario the supply chain network solutions provided better profit than did the traditional design. Furthermore, a managerial implication is also discussed in the demonstrated example. The results show that having backup suppliers to mitigate the negative impact due to disruption. For DCs, the model suggests implementing decentralized strategy to minimize the disruptive impact.

This paper contributes to the literature on supply chain network design problems by considering facility disruption in a decision-making process. The proposed solution framework also provided good-quality solutions in an efficient way to solve the problem specified in this study.

In this study, once a disruption has taken place, it will affect 100% of the facility’s capacity. Future work may relax this assumption by modeling risk such that the severity and duration of the disruption are random parameters. In addition, stochastic parameters (demand, cost, and facility capacities), multiple-product, and relation between the raw material consumption rates and the product should be considered so that the model is rendered more practical than the initial version presented herein.

In terms of solution technique, the Monte Carlo sampling technique constructs the sample average problem in order to obtain an estimate of the objective function value. However, more sophisticated sampling techniques could be used to reduce the variance of the estimate. Ref. [34] reported promising results on the Latin-Hypercube sampling technique for stochastic programs, which could be an interesting avenue whereby future research could improve the quality of the solutions.

Furthermore, it would be interesting to explore alternate solutions for supply chain network design. Typical robustness and resilience supply chain network solutions consider proactive and reactive redundancies deployment would result in expensive system to cope with uncertainty. A Low-Certainty-Need (LCN) supply chain concept as an analogy to level strategy in Sales and Operations Planning to maintain a stable production system behavior rather than the chase strategy that change the system behavior according to changes in the system environment [35]. The future research could consider how robust, and resilient supply chain design and planning can be integrated with the principle of efficiency; how to design and implement robust and resilient supply chain focusing on the supply chain’s ability to efficiently operate regardless of environmental changes.