Scientific Programming

Volume 2016, Article ID 3848520, 15 pages

http://dx.doi.org/10.1155/2016/3848520

## Supply Chain Network Design under Demand Uncertainty and Supply Disruptions: A Distributionally Robust Optimization Approach

School of Business Administration, Northeastern University, No. 195, Chuangxin Road, Hunan New District, Shenyang, Liaoning 110169, China

Received 24 May 2016; Revised 6 August 2016; Accepted 22 September 2016

Academic Editor: Xiaofeng Xu

Copyright © 2016 Ruozhen Qiu and Yizhi Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We develop a robust optimization model for designing a three-echelon supply chain network that consists of manufacturers, distribution centers, and retailers under both demand uncertainty and supply disruptions. The market demands are assumed to be random variables with known distribution and the supply disruptions caused by some of the facilities faults or connection links interruptions are formulated by several scenarios with unknown occurrence probabilities. In particular, we assume the probabilities that the disruption scenarios happen belong to the two predefined uncertainty sets, named box and ellipsoid uncertainty sets, respectively. Through mathematical deductions, the proposed robust SCN design models can be transformed into the tractable linear program for box uncertainty and into second-order cone program for ellipsoid uncertainty. We further offer propositions with proof to show the equivalence of the transformed problems with the original ones. The applications of the proposed models together with solution approaches are investigated in a real case to design a tea supply chain network and validate their effectiveness. Numerical results obtained from model implementation and sensitivity analysis arrive at important practical insights.

#### 1. Introduction

Designing and managing a supply chain network have become crucial due to the increasing market competition, variable customer demands, and the fast development of the economic and technological globalization. An efficient supply chain network will contribute to quickly responding to the customers’ demands and achieving the success of the supply chain which depends on the cooperation and coordination among all members. Supply chain network (SCN) design incorporates both strategic and tactical decisions on the number, location, capacity, and mission of the supply, production, and distribution facilities required to provide goods to a customer base [1]. In recent years, lots of mathematical models have been developed to solve various supply chain network design problems. A. Nagurney and L. S. Nagurney [2] consider a firm that is engaged in determining the capacities of its various supply chain activities and develop a rigorous modelling and analytical framework for the design of sustainable supply chain networks. Bashiri et al. [3] present a mathematical model for strategic and tactical planning in a multiple-echelon, multiple-commodity production-distribution network. Mahdi et al. [4] indicate the SCN design can significantly affect the economic viability of a biofuel technology and develop a mixed-integer linear programming to determine the optimal supply chain design and operation under uncertain environment. Jeihoonian et al. [5] adopt a mixed-integer programming model to design a closed-loop supply chain network for durable goods and coordinate forward flow and reverse flow to determine the position of each type of facilities in the entire network. For comprehensive reviews of SCN design models, see Klibi et al. [6], Tokman and Beitelspacher [7], and Farahani et al. [8].

By investigating the supply chain models, Melo et al. [9] find that both strategic and tactical decisions can be highly affected by various sources of uncertainty such as demand and supply interruptions, lead time variability, exchange rate volatility, and capacity variations [10]. From the perspective of risk management, Tang [11] defines two categories of risks in supply chains. One is operational risks arising from business-as-usual incidents, such as machine breakdowns and power outages which lead to uncertainties in matching supply and demand. The second category is disruption risks, which arise from natural and man-made disasters such as earthquakes, floods, hurricanes, and terrorist attacks. Operational risks are usually captured through incorporating such inherent uncertainties in the input data such as uncertain customer demand, uncertain supply capacity, and uncertain costs due to dynamic and fluctuating nature of these parameters. Disruption risks are usually captured by incorporating disruption scenarios in the model formulation of underlying decision problem since they are unlikely to occur but have a high impact when they do occur [12]. To cope with the disadvantage effect of the uncertainties on supply chain operations, Klibi et al. [6] suggest that stochastic programming can be used as a powerful technique to tackle uncertainties. Furthermore, a so-called scenario-based stochastic programming has been paid more attention in SCN design problems. Scenario-based stochastic programming optimizes the expected value of the objective functions by considering a set of discrete scenarios and their occurrence probabilities for random variables. Following such logic, Mulvey et al. [13] initially propose a scenario-based robust stochastic optimization for large-scale systems and define two measures of robustness, namely, model robustness and solution robustness, respectively. A solution to an optimization model is defined as robust solution if it remains close to optimal for all scenarios of the input data and robust model if it remains almost feasible for all data scenarios. By using a parameter reflecting the decision-maker’s preference between the solution and model robustness, it incorporates the conflicting objectives of the two. The scenario-based robust optimization can be regarded as a two-stage stochastic programming in essence and has been adopted as an effective tool for designing and managing the supply chain operating in uncertain environments.

Using the robust optimization approach proposed by Mulvey et al. [13], Babazadeh and Razmi [14] develop an efficient mixed-integer linear programming to handle both operational and disruption risks of the agile supply chain network. Baghalian et al. [15] consider both demand-side and supply-side uncertainties and develop a stochastic mathematical formulation for designing a network of multiproduct supply chains. Jabbarzadeh et al. [16] present a dynamic supply chain network design model for the supply of blood during and after disasters which can assist in blood facility location and allocation decisions for multiple postdisaster periods. Jeong et al. [17] provide an integrated framework to design emergency logistics networks comprising distribution warehouses, disaster recovery centers, and neighbourhood locations based on efficiency, risk, and robustness metrics, where the efficiency and risk are determined using transportation cost and relief item loss cost, while the robustness is analyzed in terms of perturbed transportation cost through diverse damage scenarios to major facilities. Fattahi and Govindan [18] apply a two-stage stochastic program to address a multistage and multiperiod supply chain network design problem under stochastic and highly variable demands.

Applications of robust stochastic programming techniques for SCN design models are limited because of the shortage of historical data for fitting probability distributions for uncertain parameters [19]. According to Klibi et al. [6], stochastic programming techniques usually require the perfect information of probability distributions of random variables, such as the likelihood of an interruption occurrence and its magnitude of impact. Such historical data, especially for those rare events, is limited or nonexistent making it difficult or impossible to estimate the actual distribution of uncertain parameters [20]. A robust optimization approach which depends on the complete distributions of the random variables on no condition could be adopted to handle uncertain parameters. Soyster [21] was the first to introduce a robust optimization formulation with interval data uncertainty, which will lead to the overconservative solutions because the probability at which uncertain parameters reach their worst values is as low as that at which they reach their normal values. Subsequently, Ben-Tal and Nemirovski [22], Bertsimas and Sim [23], and Ben-Tal et al. [24, 25] reduce the conservation of Soyster’s approach by well defining the uncertainty sets to which the uncertain parameters belong and can thus be applied to design the robust supply chain network. Pishvaee et al. [26] assume the demand and the returns of used products vary in a specified closed bounded box and propose a robust optimization model for handling the inherent uncertainty of input data in a closed-loop supply chain network design problem. Hasani et al. [19] propose a general comprehensive model for strategic closed-loop supply chain network design, in which the uncertainties of parameters such as demand and purchasing cost in the proposed model are handled via an interval robust optimization technique. Hatefi and Jolai [27] propose a robust and reliable model for an integrated forward-reverse logistics network design, which simultaneously takes uncertain parameters and facility disruptions into account. More recently, Hasani et al. [28] propose a robust optimization model based on the uncertainty budget concept to consider uncertain parameters in global supply chain network design. Akbari and Karimi [29] use the similar robust optimization approach to design a multiechelon, multiproduct, multiperiod supply chain network under process uncertainty. Hasani and Khosrojerdi [30] develop a mixed-integer, nonlinear model and consider six flexible and resilience strategies simultaneously in designing robust global supply chain networks under disruptions and uncertainties. In addition, they present an efficient parallel Taguchi-based memetic algorithm to solve the proposed model.

From the literatures mentioned above, it can be seen that the disruption issues have drawn attention in SCN design problems. However, both demand uncertainty and supply disruption are rarely considered simultaneously [15, 31], especially for the unknown disruption occurrence probability. In this paper, we develop a path-based three-echelon robust SCN design model under both demand and supply uncertainties. For the uncertain market demand, we assume, without the loss of generality, it is a random variable with a known distribution such as normal distribution, which may lead to a nonlinear model. To improve the solving efficiency, a piecewise linearization method is applied to transform the nonlinear terms into the linear ones. For the supply uncertainty, we consider some supply disruption scenarios caused by the manufacturers’ facilities faults or the path interruption between the supply chain nodes. In particular, we use two predefined uncertainty sets, named box and ellipsoid, respectively, to describe the unknown probabilities that the disruption scenarios happen, which motivates us to adopt the robust optimization to build the SCN design models. Furthermore, through mathematical operations, all formulated robust optimization models are transformed into tractable convex programs and can thus be solved efficiently.

The remainder of this paper is organized as follows. We briefly introduce the SCN design problem under demand uncertainty and supply disruption in Section 2 and then propose a SCN design model in Section 3. In Section 4, we proposed two robust SCN design models when the disruption scenarios probabilities are bounded in the box uncertainty set and ellipsoid uncertainty set, respectively. We further transform them into the tractable linear and second-order cone programs to solve. A case study is given in Section 5 to illustrate the proposed models and solutions and to confirm their effectiveness. In the last section, we summarize the paper and discuss future research directions.

#### 2. Problem Description

In this paper, we focus on the SCN design of a three-echelon supply chain under both the demand uncertainty and supply disruptions. The supply chain involves some potential manufactures, distribution centers, and retailers located in some candidate locations of the markets. As the market terminals, the retailers are in charge of the selling and face uncertain market demands that are considered as random variables with known distribution. Before observing the market demands, retailers need to place orders from the upstream. The distribution centers integrate the orders from the retailers and pass them to the manufacturers. To achieve economies of scale, the manufactures are inclined to integrate the orders and then organize production or purchase from the outer suppliers. Once the goods are produced and ready for shipment, they will be transported to distribution centers by consolidation and then arranged for delivery to multiple destinations. To meet the requirement of each market, a cross-dock operation can be adopted in the distribution centers to replenish fast-moving store inventories.

Figure 1 depicts the basic network structure. We also use the concept of a path or route instead of defining flow variables among the nodes of the supply chain [15]. Thus, each link starting from a manufacturer, passing through a distribution center, and ending at a retailer can be regarded as a potential transportation route. We further consider disruption risks in supply chain which impose great impacts on the companies’ performance. According to Azad et al. [32], sources of disruption risks can be segmented into two categories: random disruption risks which may occur at any physical point of supply chain network, for example, natural hazards and earthquakes, and premeditated disruption risks which are deliberately planned to inflict the supply chain with maximum damages.