(PDF) Designing a capacitated multi-configuration logistics network under disturbances and parameter uncertainty: a real-world c

J Ind Eng Int (2018) 14:65–85 https://doi.org/10.1007/s40092-017-0206-x ORIGINAL RESEARCH Designing a capacitated multi-configuration logistics network under disturbances and parameter uncertainty: a real-world case of a drug supply chain Davood Shishebori1 • Abolghasem Yousefi Babadi2 Received: 27 January 2017 / Accepted: 13 May 2017 / Published online: 23 May 2017 The Author(s) 2017. This article is an open access publication Abstract This study investigates the reliable multi-con- Introduction figuration capacitated logistics network design problem (RMCLNDP) under system disturbances, which relates to Due to the competitive world of manufacturing in the locating facilities, establishing transportation links, and twenty-first century, nowadays, the supply chain network also allocating their limited capacities to the customers design (SCND) issue and its related topics have a special conducive to provide their demand on the minimum importance in the optimization research areas, because this expected total cost (including locating costs, link con- issue is one of the key factors that may arouse a key role in structing costs, and also expected costs in normal and the reduction of various costs (such as costs of location, disturbance conditions). In addition, two types of risks are construction, operation, production, transportation) as well considered; (I) uncertain environment, (II) system distur- as increase the efficiency of production and service bances. A two-level mathematical model is proposed for systems. formulating of the mentioned problem. Also, because of The past studies demonstrate that several mathematical the uncertain parameters of the model, an efficacious programming models have been developed to unravel a possibilistic robust optimization approach is utilized. To variegation of SCND problems. Some reviews have men- evaluate the model, a drug supply chain design (SCN) is tioned their models, algorithms and applications (Meixell studied. Finally, an extensive sensitivity analysis was done and Gargeya 2005; Daskin et al. 2005; Hatefi et al. 2015; on the critical parameters. The obtained results show that Cardona-Valdes et al. 2014). the efficiency of the proposed approach is suitable and is Since the SCND is known to be a long-term strategic worthwhile for analyzing the real practical problems. decision problem, considering various practical factors can help get more efficient solutions to the problem under Keywords Multi-product Multi-type transportation study. Risk is known to be a key factor in this context. links Multi-vehicles Multi-configuration Capacitated Recently, this topic in the SCND has been an incentive logistic network design (LND) Disturbances Two-stage context for most corporations in the trade growth world. decomposing heuristic LP relaxation Medical service On one view, there are two broad classes of risks that (drug) influence the design of the supply chain (SC): (I) the risk emanating from the adversities in moderating demand and supply and (II) the risk emanating from a fulmination of disruptions to usual processes, which includes the themes depending on indigenous catastrophes, stay-in strikes, & Davood Shishebori economical interruptions, and terroristic acts (Kleindorfer

and Saad 2005; Hatefi et al. 2015). 1 Department of Industrial Engineering, Yazd University, The first class of risks is studied in the major frame of Pajoohesh Square, Yazd 89195741, Iran the SC background. The existent uncertainties in the 2 School of Industrial and Systems Engineering, University of demand, lead times, transportation expenditures, and the Tehran, Tehran, Iran quality and quantity of returning commodities refer to this 123 66 J Ind Eng Int (2018) 14:65–85 class of risks. The stochastic programing and robust opti- definition and formulation’’ section is considered. In ‘‘So- mization (RO) are the most efficient tools used for con- lution approach’’, an efficacious solution approach is pro- sidering the uncertain parameters (Birge and Louveaux posed. A real practical case study is represented and 2011). analyzed in ‘‘Real life case study’’ section. In ‘‘Sensitivity Also, the second category of risks is considered in some analysis’’ section, a comprehensive sensitivity analysis at studies in the SC literature. Generally, this kind of risk is five subsections is represented. Finally, conclusions and known as system disturbances. In fact, SCs are subject to some future guidelines are discussed. several possible provenances of disturbances, where dis- turbances are unanticipated and unplanned incidents that interrupt the ordinary flow of materials and products Literature review through an SC. The disturbance at one part of an SC may cause a substantial influence over the whole chain. SCs are In the SCND and logistics literature, system disturbances subject to several possible provenances of disturbances. are known to be a special issue of supply uncertainty. Some of them are external sources (e.g., natural disaster, These disturbances are introduced as random incidents that terrorist attack, power outages, and supplier discontinu- lead some elements of the SC to lay off functioning, either ities), while others are internal sources (e.g., industrial partially or completely, for a (generally random) value of accidents, labor strikes). time. Several robust strategies and approaches have been Since most of the supply chain infrastructures are gen- proposed to relieve the efficacies of SC interruptions and erally widespread and their constructions are expensive and improve the efficiency of SC and its logistics at disturbance time consuming, it will be very costly and difficult to conditions. For more study, the scholar is referred to the rectify the design. Accordingly, it is most worthy to review by Snyder et al. (2006, 2016). scheme an SC that achieves efficiency and continuity under Conductive to place our contribution in the straight all kinds of risks from the beginning. In comparison to the panorama, four primary streams of the background may be existing literature, the contribution of this paper is as reviewed that can be of penchant for contrast: (I) the SCND follows: subject to system disturbances, (II) the capacitated SCND, (III) the SCND subject to parameter uncertainty, (IV) the • Investigation of multi-configuration (including multi- multi-configuration SCND. product, multi-type link, and multi-vehicle) structure in It is noted that in this study, the system disturbances are the network designing of the capacitated SC problem. defined as facility disturbances, transportation link distur- • Study of uncertain environment (containing uncertainty bances, and transportation vehicle disturbances. of transportation costs and demands) in the network designing of the capacitated SC problem. SCND with system disturbances • Investigation of system disturbances, which subtends potential interruptions in facilities, the network trans- The literature system (including facility and link) distur- portation links, distribution centers, and vehicles. bances on SCND can be briefly reviewed as follows. Peng • Use of the production–distribution system which is a et al. (2011) considered the efficacy of studying unrelia- ‘‘customer to server’’ system. bility in LND problems in the presence of facility inter- • Applying the formulation in a real practical case study ruptions as several scenarios. They demonstrated that from the medical service (drug) supply chain. utilizing a reliable LND is frequently conceivable with These contributions have been somewhat indicated in insignificant increases in whole location and allocation the last row of Table 1 in terms of the formulation’s expenditures. Moreover, Jabbarzadeh et al. (2012) refor- characteristics and the background taxonomy. In this study, mulated a mixed-integer non-linear program (MINLP) for the mentioned problem is named robust and reliable an SC design quandary in which distribution centers can capacitated multi-configuration supply chain network have complete and partial interruptions. Aydin and Murat design problem (RR/CMc/SCNDP). To the best of our (2013) studied the capacitated reliable facility location knowledge, such a study has not been conducted till now. (CRFL) quandary in the presences of facility interruptions The rest of this article is structured as follows. Sec- as scenarios. They proposed an efficient algorithm as tion ‘‘Literature Review’’ purveys a relatively compre- hybridization of particle swarm intelligence (PSO) and hensive background in four main streams. The problem sample average approximation methods. (Shishebori et al. definitions and the proposed mathematical programming 2013), and Shishebori and Jabalameli (2013a, b) studied model are proposed in ‘‘Problem definition and formula- facility disturbances as a constraint for the maximum per- tion’’ section. Then, in ‘‘Mathematical modeling’’ section, missible disturbance expenditure of the system. They the reliable counterpart of the proposed model in ‘‘Problem proposed an MINLP formulation and considered it by a 123 Table 1 A comparable investigation of inquiry gap in the background Subsection Authors System Network System Single/multi-configuration Certain/uncertain problem space (year) capacity design disturbances Certain Uncertain Single Multi- Multi- Multi- Multi- Deterministic Intervals (as Probabilistic Fuzzy Etc. products type vehicles etc. scenario links based) (2.1) Peng et al. (2011) 4 4 4 4 J Ind Eng Int (2018) 14:65–85 Jabbarzadeh et al. 4 4 4 4 (2012) Aydin and Murat 4 4 4 4 (2013) Liberatore et al. 4 4 4 (2012) Shishebori et al. 4 4 4 4 (2013) Shishebori and 4 4 4 4 Jabalameli (2013a, b) Cardona et al. 4 4 4 4 (2014) Haldar et al. (2014) 4 4 4 Garcia-Herreros 4 4 4 4 et al. (2014) Ivanov et al. (2014) 4 4 4 4 4 Shishebori (2016) 4 4 4 4 4 4 (2.2) Mahajan et al. 4 4 4 4 (2002) Jemai et al. (2007) 4 4 4 4 Sitompul et al. 4 4 4 4 (2008) Jabalameli et al. 4 4 4 4 (2011) Toktas and Fusun 4 4 4 (2011) Nepal et al. (2012) 4 4 4 Duan and Liao 4 4 4 (2013) Shishebori et al. 4 4 4 4 4 (2013) Shishebori et al. 4 4 4 4 4 (2014) 67 123 68 Table 1 continued Subsection Authors System Network System Single/multi-configuration Certain/uncertain problem space (year) capacity design disturbances Certain Uncertain Single Multi- Multi- Multi- Multi- Deterministic Intervals (as Probabilistic Fuzzy Etc. 123 products type vehicles etc. scenario links based) Ashtab (2016) 4 4 4 4 (2.3) Tsiakas et al. 4 4 4 (2001) Jung et al. (2004) 4 4 Santoso et al. 4 4 4 (2005) You and 4 4 Grossmann (2008) Peidro et al. (2009) 4 4 Ben-Tal et al. 4 4 4 (2009) Bayati et al. (2013) 4 4 4 4 Karimi-nasab et al. 4 4 4 4 (2013) Gulpinar et al. 4 4 4 4 4 (2013) Singh et al. (2013) 4 4 4 Ayvaz and Bolat 4 4 4 4 (2014) Aqlan and Lam 4 4 4 (2016) Babazadeh et al. 4 4 4 (2016) (2.4) Chen and Lee 4 4 4 4 (2004) Park et al. (2007) 4 4 4 You and 4 4 4 Grossmann (2008) Ferrio and Wassick 4 4 4 4 (2008) El-Sayed et al. 4 4 4 (2010) J Ind Eng Int (2018) 14:65–85 Table 1 continued Subsection Authors System Network System Single/multi-configuration Certain/uncertain problem space (year) capacity design disturbances Certain Uncertain Single Multi- Multi- Multi- Multi- Deterministic Intervals (as Probabilistic Fuzzy Etc. products type vehicles etc. scenario links based) Mirzapour Al-E- 4 4 4 J Ind Eng Int (2018) 14:65–85 Hashem et al. (2011) Amrani et al. 4 4 4 4 (2011) Bashiri et al. (2012) 4 4 4 4 Jamshidi et al. 4 4 4 (2012) Badri et al. (2013) 4 4 4 4 Song et al. (2014) 4 4 4 4 Cardona-Valdes 4 4 4 et al. (2014) Pasandideh et al. 4 4 4 4 (2015) Jindal et al. (2015) 4 4 4 4 4 Sarrafha et al. 4 4 4 (2015) This 4 4 4 4 4 4 4 paper 69 123 70 J Ind Eng Int (2018) 14:65–85 case study. Garcia-Herreros et al. (2014) developed a two- frame conducive to specify the near-optimal SC replenish- stage stochastic program to plan resilient SCs in which the ment strategies in the presence of different demands and distribution centers are subject to the risk of interruptions at handle strategies for a capacitated SC. They examined a candidacy locations. Ivanov et al. (2014) formulated a capacitated single distributor–multi-retailer SC discipline in multi-commodity and multi-period distribution (re)plan- detail. Shishebori (2014) studied facility disturbances via a ning quandary for a multi-phase centralized network with constraint on the maximum permissive interruption expen- frame dynamics investigations held forth. Their approach diture of the system in the context of an FLNDP with permits investigating several implementation scenarios and interruptions. They proposed an MINLP model for the expanding motions on (re)planning in the case of pertur- quandary and illustrated it by a case study. Shishebori et al. bations. Shishebori (2016) studied the reliable multi-pro- (2014) considered a similar FLNDP and proposed an LP- duct multi-vehicle multi-type link logistics network design based heuristic to solve it. Ashtab (2016) studied a three- problem (RMLNDP) regarding the system disturbances. echelon capacitated SCND with customer zones, distribution They modeled the problem as an MIP model to provide a centers (DCs), and suppliers. The proposed formulation takes reliable sustainable multi-configuration logistic network into account the operative expenditures of an established DC system. Yousefi-Babadi et al. (2017) proposed a multi- due to its practical activity surface instead of the presumption objectives mixed-integer non-linear programming that an established DC functions at the uttermost acumen. (MINLP) model for a petrochemical supply chain under Some other related paper can be found in Taleizadeh et al. uncertainty environments, namely disruption risks and less (2008, 2009, 2010a, b, 2011, 2012, 2013a, b, 2014), Talei- knowledge of parameters. In their model, two efficient zadeh and Pentico (2013), Taleizadeh (2014), Taleizadeh and queuing systems are applied in nylon plastic manufacturing Nematollahi (2014) and Taleizadeh and Pentico (2014). and recycling centers, in which a Jackson network is also used. The aims are to minimize the average tardiness to SCND with parameter uncertainty deliver products, total cost and transportation cost. Finally, they applied the Lagrangian relaxation based on a sub- In the background, different formulations for SC designing gradient approach to solve the proposed model. in the presence of uncertainty are described. Utmost are due to hybrid approaches (due to the accretion of simula- Capacitated SCND tion and analytic formulations), simulation approaches, or analytic approaches (i.e., stochastic formulations) (Peidro Capacity is known as another significant factor that plays a et al. 2009). In these approaches, the SC uncertainties are critical role in SCND. Several studies were done regarding presented with probability distributions which are com- SC limited capacity. Mahajan et al. (2002) dissected an SC monly forecasted from historical data. Howbeit, whenever consisting of uncapacitated/capacitated suppliers for diffus- statistical data are not reliable, or are not even existent, ing two autonomous commodities via multiple retailers and formulations due to the designation of these contingency studied the problem by means of the game theory. Jemai and distributions might not be the best selection (Wang and Shu Karaesmen (2007) considered a two-phase SC consisting of a 2005). In this setting, the possibility theory and the fuzzy retailer and a capacitated supplier in the structure of a Nash set theory could purvey a superseded approach to consider game. Sitompul et al. (2008) modeled the safety holding SC uncertainties. Tsiakis et al. (2001) dealt with a multi- assignment quandary for an n-phase capacitated consecutive stage, multi-product SC in the presence of demand uncer- SC and suggested a solution scheme with the objective of tainty due to scenario type. Jung et al. (2004) studied the maintaining the needed entire service surface at the lowest application of deterministic scheduling and planning for- expenditure. Jabalameli et al. (2011) formulated a budget- mulations which integrate safety holding surfaces as a constrained dynamic (multi-period) uncapacitated facility means of taking into account the demand uncertainties in location-network design problem (DUFLNDP). Their prob- the chemical process industry SC. Santoso et al. (2005) lem dealt with the determination of the optimal locations of dealt with an uncertain programming approach and disso- facilities and the design of the underlying network simulta- lution methodology for unraveling SCND quandaries of a neously. Toktas-Palut and Fusun (2011) formulated an M/M/ real practical scale under system uncertainty. You and 1 make-to-stock queuing network in a decentralized two- Grossmann (2008) proposed an optimizing formulation to phase SC including a manufacturer with limited production plan a multi-stage SC and the dependent inventory systems capacity and multiple autonomous suppliers. Nepal et al. regarding the demand uncertainty in the chemical mystery. (2012) indicated an investigation of a three-stage SC study- Peidro et al. (2009) modeled a fuzzy MIP model for SC ing capacity circumscription and step changes in SC uti- planning which deals with the process, demand, and supply lization rate according to life cycle demand stages. Duan and uncertainties in the presence of ill-known data. Moreover, Liao (2013) represented a simulation -based optimization Ben-Tal et al. (2009) modeled the multi-term inventory 123 J Ind Eng Int (2018) 14:65–85 71 control problem conducive to optimize the expected cost can include multi-product, multi-type link, and multi-ve- that includes the costs of flows, transfer links, capacities, hicle. In this paper, all of them are named (called) multi- and middle-stage facility locations. Singh et al. (2013) configuration. formulated a two-phase stochastic programming formula- Several studies were done at the SCND and logistics tion for capacitated network design of an SC with flexible with multi-configuration structure. Chen and Lee (2004) demands. Bayati et al. (2013) proposed an optimal pricing proposed a multi-period multi-product multi-stage for- and marketing planning where the primary objective mulation with multi-incompatible objectives of a multiple function is to maximize the total profit. Their mathematical stage SCN as an MINLP, where fuzzy sets were inves- model was considered with different input parameters and tigated to explain the uncertainties contained in product coefficients in the uncertain state. Also, they provided a prices and market demands. Park et al. (2007) developed suitable approach that calculates a lower and upper bounds a multi-product multi-period SC formulation, subtending for the objective function of the model. Karimi-Nasab et al. factory, supplier, and distribution region to optimize the (2013) proposed a multi-objective approach to determine total expenditure and represented a genetic algorithm the distribution policy for a wholesaler. In their approach, (GA) to unravel the problem. You and Grossmann (2008) the wholesaler distributes supplementary nutrition to a set proposed the optimization of a multi-echelon bi-criteria of local distribution centers positioned around the whole- SC under demand uncertainty with the objectives of saler, geographically. Their approach optimizes the selling minimizing the expected lead time and maximizing the price, carrying cost, batch size and services level of multi- net present value. Ferrio and Wassick (2008) proposed an items for each local distribution center in every planning MILP model for chemical multi-product supply network, period. Ayvaz and Bolat (2014) formulated a general two- containing production centers, an arbitrary number of phase probabilistic programming formulation to tackle DCs, and customers. The proposed model was applied for uncertainties in reverse logistics network design. Aqlan and optimizing and redesigning the network. Also, El-Sayed Lam (2016) presented a methodology and a software utility et al. (2010) considered a three-echelon forward–reverse for SC optimization in the presence of uncertainty and risk. multi-period logistics network design in the reverse Their methodology involves solving a deterministic multi- direction in the presence of deterministic customer objective model as well as using a simulation formulation demand and demand uncertainty in the forward direction to illustrate the stochastic elements of the SC. The two subject to maximizing the total expected profits. Mirza- formulations argue optimizing the lead time, profit, and pour Al-E-Hashem et al. (2011) modeled a multi-product, risk reduction via opting an integration of extenuation multi-site multi-period, three stage SC in the presence of policies and allocating inventory and orders. Babazadeh uncertainties of demand fluctuations and cost parameters. et al. (2016) modeled a multiple objective stochastic pro- Amrani et al. (2011) formulated an MIP model for a gramming formulation for designing a second-generation multiple product production–distribution network such biodiesel SC network regarding the risk investigating total that the considered network has the property of alterna- cost and environmental impact minimization. They pre- tive facility configuration. Bashiri et al. (2012) studied a sented a new formulation of stochastic programming multi-echelon network considering several time resolu- method which is able to minimize the total mean and risk tion decisions and tactical and strategic planning and values of stochastic -based uncertain problems. Keyvan- developed a new multi-product mathematical formula- shokooh et al. (2016) developed a novel hybrid robust- tion. Jamshidi et al. (2012) modeled a bi-objective multi- stochastic programming (HRSP) approach to simultane- echelon SCN design problem in which multifold trans- ously model two different types of uncertainties by portation selections at each level of the SC are consid- including stochastic scenarios for transportation costs and ered with a capacity constraint and several costs. polyhedral uncertainty sets for demands and returns in the Keyvanshokooh et al. (2013) addressed the problem of closed-loop supply chain network (CLSCN) design prob- designing and planning a multi-period, multi-commodity lem. Transportation cost scenarios are generated using a and capacitated integrated forward/reverse logistics net- sampling method and scenario reduction is applied to work/closed-loop supply chain network. Badri et al. consolidate them. (2013) proposed a new multiple commodity SCND for making strategic and tactical decisions. Maximizing the SCND with multi-configuration total net income thorough the time is the obvious objective function of the proposed problem. Moreover, Considering several possible aspects of the proposed Sarrafha et al. (2015) studied an SCND containing fac- problem can help to find more practical solutions. This can tories, suppliers, retailers, and DCs. They proposed a help decision makers to have several alternatives for the multi-period structure such that a flow-shop scheduling proposed SCND and logistics. The several possible aspects formulation in the manufacturing portion of the SC and 123 72 J Ind Eng Int (2018) 14:65–85 also the shortage in the framework of backorder are 4. Which vehicle between which facilities should be combined in each period. Jindal et al. (2015) studied the established? optimization of a multi-echelon, multi-time multi-product 5. How many products should be transported between capacitated closed-loop SCNDP in an uncertain circum- facilities? ference. The uncertainties were relevant to several 6. Which vehicles transport these products? parameters such as return volume, product demand, inventory cost, transportation cost, and processing cost. Fattahi et al. (2015) developed different mixed-integer Problem definition and formulation linear programming (MILP) models for designing cen- tralized and decentralized supply chains using two-stage The working conditions of the RR/CMc/SCNDP can be stochastic programming. They investigated a multiple described as follows. Let GS, GT, and GD signify the sets of period replenishment problem based on (s, S) policy for supply, transportation, and demand sites, respectively. these supply chain models. Also, the G0 is defined as the set of all supply and trans- portation sites (G0 = GS [ GT), the sites for which close/ Research gap and our contribution open decisions are necessary. The set G0 can be called ‘‘facilities’’. All of the facilities and transportation links are Some of the previous works have studied the SCND and capacitated and have a maximum level (capacity) in order logistics with facility disturbances and regarding link dis- to service to the demand sites. Suppose S be the set of turbances separately. Moreover, most of them did not scenarios, each of which distinguishes a set of facilities, consider the capacity (including facilities, vehicles, and transportation links, and transportation vehicles that are transportation links). However, in some manufacturing simultaneously disrupted. Suppose s = 0 as the nominal industries, there are some capacitated SC and logistics scenario in which no disturbances happen. The set P illus- systems in which a variety of disturbances (failures) may trates several types of products that should be produced and occur. The most obvious examples are SC of different transported to the demand sites. The set L presents different spare parts, food product manufacturing, petrochemicals, kinds of transportation links. For example, for each link, it etc. is assumed that three various quality levels (i.e., |L| = 3) In contrast to previous works in this constitution, this may be considered; each of which is defined as follows: the paper investigates the problem of robust and reliable paved road with the standard quality (type 1), the paved designing of a capacitated SC network (SCN), which road with low quality (type 2), and the dirt road (type 3). consists of suppliers, DCs, transportation vehicles and As seen, if a link with type 1 quality is constructed, its demand sites as well as some transportation links. They establishment cost will be more than the other types, while are potential and it should be decided that which its capacity is more and its transportation cost is lower than potential sites and links should be built. Moreover, the the other quality types of transportation links. Here, links SCN has a multi-configuration structure; i.e., there are (roads) with three quality types are defined; however, the multi-product, multi-type link, and multi-vehicle several quality types can be defined for the problem. Also, states in the considered SCN. Also, two types of risks the set V shows several types of transportation vehicles. are considered: (I) uncertain environment, (II) system The best type of the vehicle has the highest cost of disturbances. It is obvious that modifying this SCN and investment, but the lowest cost of transportation and the its related logistics will be very difficult and costly. worst type is vice versa. Several vehicles, which can be Therefore, it is important to design a reliable and robust used and categorized as different types of vehicles are SCN that reaches suitable stability and performance airplane, helicopter, train, refrigerated pickups/truck, etc. under several kinds of risks from the beginning. A two- These vehicles are selected with regard to the required level mathematical model is proposed for modeling the investment cost, the necessity of the product (e.g., drug or mentioned problem. Also, because of uncertain param- medical service), probability of system risk, or other eters of the model, an efficient robust optimization practical factors. approach is applied. To avoid infeasible situations, a penalty fee is ordained In other words, the following questions are answered in for the demands of sites that cannot feasibly be met. It can this paper: be denoted that these demands are fulfilled from some 1. Which facility should be located in which site? outside suppliers as emergency facilities, but with high 2. Which links between which facilities should be transportation costs. Also, it can be interpreted that the established? demand of a site can be not to provide if the cost of pro- 3. Which kind of link quality should be established? viding the demand of the site is greater than the penalty of 123 J Ind Eng Int (2018) 14:65–85 73 it. In this paper, this contingency is considered by sup- M = a large number, posing that NS subtends an ‘‘emergency facility’’ that does BC = upper limit of investment budget constraint, not have any establishment cost and is never laid on risky PR = desired robustness level, situations; also, it has infinite capacity. Obviously, it is ever ns = the optimal cost of unreliable scenario s, s 2 S, more open in the optimum solution and does not have any Nsj = disturbance parameter of facility at site j, (j 2 NS) risky situations. For each transportation link, from the in unreliable scenario s, (s 2 S) Nsj 2 ½0; 1 , emergency facility to other sites, the unit transportation cost is equal to the unmet demand penalty fee. fsij = disturbance parameter of link (i, j) 2 A in The sets and parameters of the problem are as follows: unreliable scenario s, (s 2 S) fsij 2 ½0; 1 , ‘lvs ij = disturbance parameter of vehicle v (v 2 V) at link Sets (i, j) 2 A with quality type l (l 2 L) in unreliable scenario G: set of sites (G = GS [ GT [ GD), ‘ lvs P: set of products (p = 1, …, NP), s, (s 2 S) ij 2 ½ 0; 1 , L: set of several quality types of transportation links Usj = 1 if facility at site j 2 N0 is failed in scenario s 2 S, (l = 1, …, NL), 0 under other conditions, V: set of several types of transportation vehicles (v = 1, Xsij = 1 if link (i, j) 2 A is failed in scenario s 2 S, 0 …, NV), under other conditions, A: set of potential transportation links (i = 1, …, NI and Dlvs ij = 1 if vehicle v (v 2 V) at link (i, j) 2 A with quality j = 1, …, NJ), type l (l 2 L) is failed in scenario s 2 S, 0 under other S: set of scenarios. In all possible scenarios, each conditions. scenario illustrates the facilities and also link distur- Although Usj , Xsij, and Dlvs ij are defined as binary bances (s = 1, …, NS). parameters, the proposed mathematical programming model has good functionality if these parameters can be fractional, presenting partial disturbances. Parameters fj = fixed cost of locating the facility j 2 G0, ps = probability of happening of unreliable scenario s 2 Decision variables S; ps 2 [0, 1], Zj = 1 if a facility is opened at site j 2 G0, but is 0 under clij = construction cost of link (i, j) 2 A with quality type other conditions, l (l 2 L), Xlij = 1 if link (i, j) 2 A is established with quality type l tplv ij = unit transportation cost of commodity p (p 2 P) on (l 2 L), but is 0 under other conditions, link (i, j) 2 A with quality type l (l 2 L) by vehicle v (v 2 Wvij = 1 if vehicle v (v 2 V) is established at link (i, j) 2 V), A, but is 0 under other conditions, cvij = investment cost of vehicle v (v 2 V) at link (i, j) 2 dj = 1 if no facility is opened at site j and it remains as A, demand site, but is 0 under other conditions kvij = capacity of vehicle v (v 2 V) at link (i, j) 2 Yplvs ij = amount of flow of product p (p 2 P) on link (i, j) A (according to kilogram criteria), 2 A with quality type l (l 2 L) by vehicle v (v 2 V) in Cj = capacity of facility at site j 2 NS (according to scenario s 2 S. processing time criteria), Therefore, we propose the following MIP model for the upj = processing time of commodity p (p 2 P) at site j 2 RR/CMc/SCNDP. GS, Hpj = capacity of facility at site j 2 NT (according to kilogram criteria), Mathematical modeling Plij = capacity of link (i, j) 2 A with quality type l (l 2 L) (according to kilogram criteria), Reliable counterpart stage 1 wp = weight of product p (p 2 P), Umax ij = maximum number of vehicle types that may be The reliable counterpart stage 1 of RR/CMc/SCNDP, as used at link (i, j) 2 A, model (I), can be represented as follows: dpj = demand of product p (p 2 P) at site j, (j 2 GD), Model (I): 123 74 J Ind Eng Int (2018) 14:65–85 • Objective function: logical flow out and flow in. Constraints (6) vouch that each site can be specified as either supply site or demand X X X X X site; i.e., each site cannot be simultaneously known as min ETC ¼ fj Zj þ clij Xijl þ cvij Wijv j 2 G0 l2L ði;jÞ 2 A v2V ði;jÞ 2 A demand site and supply site; in other words, each site can XXX X be just a demand site or supply site, but not both of them, þ tijplv :p0 Yijplv0 : X X X X X p2P v2V l2L ði;jÞ 2 A fj Zj þ clij Xijl þ cvij Wijv BC: ð7Þ ð1Þ j2G0 l2L ði;jÞ2A v2V ði;jÞ2A Constraint (7) represents the investment constraint. The objective Eq. (1) optimizes the expected total costs X X X X p plvs (ETC), containing constant location costs, link establish- uj Yij 1 Usj 1 Nsj Cj Zj ment costs, vehicle establishment costs, as well as the p2P v2V l2L ði;jÞ2A ð8Þ expected transportation costs for all possible scenarios with 8j 2 GS ; s 2 S: respect to their probabilities. Constraints (8) ensure that the summation of the • Constraints: processing time of products, produced by supply site j 2 GS, is definitely less than its total processing time 2X X X X X 3 capacity Cj when it is opened (i.e., Zj = 1) and prevents fj Zj þ clij Xijl þ cvij Wijv 6 j2G0 l2L ði;jÞ2A v2V ði;jÞ2A 7 any flow and is fully functional in scenario s 2 S, when 6 XXX X 7 ð1 PR Þn it is disrupted or closed. Accordingly, the supply site j 2 4 þ tijplv :ps Yijplvs 5 s GS can have some partial disturbances; therefore, it can p2P v2V l2L ði;jÞ2A 8s 2 S=f0g: work with 1 Nsj % of the nominal capacity. It is ð2Þ presented as Nsj 2 ½0; 1: XXX X Constraints (2) persuade the PR-robust criterion, entail- wp Yijplvs 1 Usj 1 Nsj Hpj Zj p2P v2V l2L ði;jÞ2A ð9Þ ing that the unreliable scenario costs have to be less than 8j 2 GT ; s 2 S: 100 (1 ? PR) % of the optimum scenario costs n*s . If PR = 1, the formulation will be equivalent to a determin- Constraints (9) guarantee that the summation of weights istic SCNDP and the PR -robustness constraints will be of the flow of products, transported by DC j 2 GT, is inactive. definitely less than its capacity Hj, when it is opened (i.e., X X X plvs X X X plvs Yij Yji Zj = 1), and prevents any flow and is fully functional in v2V l2L ðj;iÞ2A scenario s 2 S, when it is disrupted or closed. Also, the v2V l2L ðj;iÞ2A ð3Þ distribution center j 2 GS can have some partial distur- þ djp 1 Nsj Hpj þ MUsj þ M 1 zj bances; therefore, it can work with 1 Nsj % of the 8j 2 G0 ; s 2 S; p 2 P; X X X plvs X X X plvs nominal capacity. It is presented as Nsj 2 ½0 ; 1. Yji Yij djp þ Mdj " # XXX X v2V l2L ðj;iÞ2A v2V l2L ðj;iÞ2A ð4Þ Hpj Yijplv 1 Xsij 1 fsij Plij Xijl þ Xjil 8j 2 G0 ; s 2 S; p 2 P; p2P v2V l2L l2L X X X plvs X X X plvs Yji Yij djp þ M 1 dj 8s 2 S; ði; jÞ 2 A: v2V l2L ðj;iÞ2A v2V l2L ðj;iÞ2A ð10Þ 8j 2 GD ; s 2 S; p 2 P; Constraints (10) emphasize that the total flow through ð5Þ the link (i, j) 2 A does not overstep its capacity Plij, when it is constructed (i.e., Xlij ? Xlji = 1) and any flow prevented, dj þ 1 Usj zj ¼ 1 8j 2 G; s 2 S: ð6Þ and is fully functional in scenario s 2 S when it is disrupted Constraints (3)–(5) are known as the flow conservation or closed. Also, the link (i, j) 2 A can have some partial constraints. Constraints (3) monitor the supply sites and disturbances (e.g., bad weather conditions, avalanche, control the supply that flows. Constraints (4)–(5) screen the earthquake, etc.); therefore, it can used with 1 fsij % demand sites and control the demand to be equal in the of the nominal capacity; i.e., its capacity is reduced to flow in and flow out of the site. Each supply/demand site may be employed as a transportation site and may have a fsij %. It is presented as fsij 2 ½0; 1: 123 J Ind Eng Int (2018) 14:65–85 75 " ! # X X X X X X a lvs min ns ¼ fj Zj þ clij Xijl þ cvij Wijv wp Yijplvs 1 Dlvs ij 1 Wijv kvij 8ði; jÞ 2 A; j 2 G0 l2L ði;jÞ 2 A v2V ði;jÞ 2 A p2P ij XXX X s 2 S; v 2 V; l 2 L: þ tijplv :ps Yijplvs ; p2P v2V l2L ði;jÞ 2 A ð11Þ ð20Þ Constraints (11) guarantee that the summation of XX X XX X weights of the flow, transported by vehicle v through the s:t: Yijplvs Yjiplvs þ djp 1 Nsj Hpj v2V l2L ðj;iÞ2A v2V l2L ðj;iÞ2A link (i, j) 2 A with type l, does not exceed its capacity kvij when it is established (i.e., Wvij = 1) and any flow pre- þ MUsj þ M 1 zj 8j 2 G0 ; p 2 P; vented, and is fully functional in scenario s 2 S, when it ð21Þ is disrupted or closed. Similarly, also, the vehicle (i, j) XX X XX X v 2 V can have some partial disturbances and can work Yjiplvs Yijplvs djp þ Mdj ‘ v2V l2L ðj;iÞ2A ð22Þ v2V l2L ðj;iÞ2A with 1 lvs ij % of the nominal capacity. It is presented 8j 2 G0 ; p 2 P; ‘lvs X X X plvs X X X plvs as ij 2 ½0; 1: Yji Yij djp þ M 1 dj X v2V l2L ðj;iÞ2A v2V l2L ðj;iÞ2A Xijl þ Xjil 1 8ði; jÞ 2 A: ð12Þ l2L 8j 2 GD ; p 2 P; ð23Þ Constraints (12) prevent links from being opened in both directions with several quality types at once. dj þ 1 Usj zj ¼ 1 8j 2 G; s 2 S; ð24Þ X Wijv Uijmax 8ði; jÞ 2 A: ð13Þ X X X X p plvs v2V uj Yij 1 Usj 1 Nsj Cj Zj p2P v2V l2L ði;jÞ2A ð25Þ Constraints (13) restrict the maximum number of vehi- 8j 2 GS ; cle type that may be used through the link (i, j) 2 A. X XXX X Wijv Xijl þ Xjil 8v 2 V; ði; jÞ 2 A: ð14Þ wp Yijplvs 1 Usj 1 Nsj Hpj Zj l2L p2P v2V l2L ði;jÞ2A ð26Þ 8j 2 GT ; Constraints (14) guarantee that the vehicle establishing " # XXX X at link (i, j) 2 A can be happed when the link, with at least Hpj Yijplvs 1 Xsij 1 fsij Plij Xijl þ Xjil one of the quality type, is opened. p2P v2V l2L l2L Zj 2 f0; 1g 8j 2 G0 ; ð15Þ 8ði; jÞ 2 A; Wijv 2 f0; 1g 8v 2 V; ði; jÞ 2 A; ð16Þ ð27Þ " ! # X a lvs Xijl 2 f0; 1g 8l 2 L; ði; jÞ 2 A; ð17Þ wp Yijplvs 1 Dlvs 1 Wijv kvij 8ði; jÞ 2 A; ij p2P ij dj 2 f0; 1g 8j 2 G0 ; ð18Þ v 2 V; l 2 L; Yijplvs 0 8ði; jÞ 2 A; p 2 P; v 2 V; l 2 L; s 2 S: ð28Þ ð19Þ Yijplvs 0 8ði; jÞ 2 A; p 2 P; l 2 L; v 2 V; ð29Þ Constraints (15)–(18) declare the binary variables and, finally, constraints (19) guarantee that the variable Yplvs and constraints (8) and (12–18). ij will be non-negative. Reliable counterpart stage 2 (for each reliable Solution approach scenario) A hint The reliable counterpart stage 2 of RR/CMc/SCNDPs, as a model (II), can be presented as follows: Basically, most of the SCs and logistic systems can be affected Model (II): from two broad categories of risk. The first category is a 123 76 J Ind Eng Int (2018) 14:65–85 XX X XX X parameter (e.g., demand, fixed prices, operational prices, lead Yjiplvs Yijplvs djp qdd Gpj þ Mdj times) uncertainty and the second category is system distur- v2V l2L ðj;iÞ2A v2V l2L ðj;iÞ2A bances (e.g., economic disturbances, strikes, natural disasters, 8j 2 G0 ; s 2 S; p 2 P; and terrorist attacks). One of the most efficient approaches to ð34Þ deal with the first category is robust optimization (RO). Since XX X XX X the late 1990s, this approach has been applied broadly in the Yjiplvs Yijplvs djp þ qd Gpdj area of uncertain control and optimization. Several RO v2V l2L ðj;iÞ2A v2V l2L ðj;iÞ2A approaches are presented in the literature review (e.g., Mulvey þ M 1 dj et al. 1995; Ben-Tal and Nemirovski 1998, 1999, 2000; Yu and 8j 2 GD ; s 2 S; p 2 P; Li 2000), (Bertsimas and Sim 2003, 2004; Leung et al. 2007), ð35Þ (Bozorgi-Amiri et al. 2011; Pishvaee et al. 2011; Shishebori and X X X X p plvs Babadi 2015). According to the efficient beneficial properties of j Yij þ gplvs u s uij 1 Uj 1 Nsj Cj Zj the Pishvaee et al. (2011) approach for our problem, in this p2P v2V l2L ði;jÞ2A study, this robust approach is considered which is conducive to 8j 2 GS ; s 2 S; cope with the related parameter uncertainty. For more detailed ð36Þ reading about this robust optimization approach, we refer and constraints (6, 7) and (9–19). researchers to Pishvaee et al. (2011). In the following, the application of the mentioned robust Robust reliable counterpart stage 2 (for each approach for our problem is represented. reliable scenario) Robust reliable counterpart stage 1 Solving the model (IV) can determine the optimal scenario costs (ns ) for the RR/CMc/SCNDPs for each of the sce- Due to previous explanations, the robust counterpart of the nario s which is presented as follows: suggested market to market closed-loop SC network Model (IV): (Model (III) for the RR/CMc/SCNDP and Model (IV) for the RR/CMc/SCNDPs) design problem with stochastic min Rns ¼ ; ð37Þ demands, transportation costs specified by box sets are s:t: equipollent to the following MIPs: 2X X X X X 3 Model (III): fj Zj þ clij Xijl þ cvij Wijv 6 j2G0 l2L ði;jÞ2A v2V ði;jÞ2A 7 min RETC ¼ 9; ð30Þ 6 7 6 X X X X 7; 2X X X X X 3 4 þ plv plvs tij :ps Yij þt gijplvs 5 fj Zj þ clij Xijl þ cvij Wijv p2P l2L v2V ði;jÞ2A 6 j2G0 l2L ði;jÞ2A v2V ði;jÞ2A 7 6 7 6 X X X X plv 7 9; ð31Þ ð38Þ 4 þ t :p0 Y plv0 þt gplv0 5 ij ij ij p2P l2L v2V ði;jÞ2A qt Gplvs plvs tij gtij 8ði; jÞ 2 A; p 2 P; l 2 L; v 2 V; s:t: ð39Þ 2 X X X X v3 fj Z j þ clij Xijl þ cvij Wij qt Gplvs plvs tij gtij 8ði; jÞ 2 A; p 2 P; l 2 L; v 2 V; 6 j2G0 ði;jÞ2A v2V ði;jÞ2A 7 6 7 ð40Þ 6 XXX X 7 ð1 PR Þns 4þ tplv :p Y plvs þt gplvs 5 2XX X XX X 3 p2P v2V l2L ði;jÞ2A ij s ij ij Yijplvs Yji plvs 6 v2V 7 p 4 l2L ðj;iÞ2A v2V l2L ðj;iÞ2A 5 1 Nsj Hj 8s 2 S=f0g; þdjp qd Gpdj ð32Þ 2XX X XX X 3 þ MUsj þ M 1 zj Yijplvs Yjiplvs 8j 2 G0 ; p 2 P; 6 v2V 7 p 4 l2L ðj;iÞ2A v2V l2L ðj;iÞ2A 5 1 Nsj Hj ð41Þ þdjp qd Gpdj XX X XX X Yjiplvs Yijplvs djp qdd Gpj þ Mdj þ MUsj þ M 1 zj v2V l2L ðj;iÞ2A v2V l2L ðj;iÞ2A 8j 2 G0 ; s 2 S; p 2 P; 8j 2 G0 ; p 2 P; ð33Þ ð42Þ 123 J Ind Eng Int (2018) 14:65–85 77 Suppliers Distribution centers Demand centers Fig. 1 Generic pattern for SC configuration (e.g., drug SC) XX X XX X Yjiplvs Yijplvs djp supply/distribution chain network by delivering of demands v2V l2L ðj;iÞ2A v2V l2L ðj;iÞ2A ð43Þ to customers on time and with also minimum total costs (e.g., drug stores, clinics, and hospitals). According to the degree þ qd Gpdj þ M 1 dj ; 8j 2 GD ; p 2 P of the demand emergency, different transportation vehicles, X X X X p plvs j Yij þ gplvs s 1 Nsj Cj Zj including road transportation (e.g., refrigerated trucks, etc.), u uij 1 Uj p2P v2V l2L ði;jÞ2A rail transportation, and air transportation (e.g., helicopter, 8j 2 GS ; etc.), can be used. At this time, there are two supply and eight distribution centers. This company wants to develop the ð44Þ extent of its supply/distribution network in the country via and constraints (8), (12–18), (24) and (26–29). increasing the number of the supply and distribution centers and improving the transportation network. The case study can be modeled by the RR/CMc/SCNDP Real-life case study as a real practical problem. Considering the security of financial information of the company, this information is Here, a real-life case study represents the application of the not reported here. In the following, other complementary RR/CMc/SCNDP. In this case study, the goal is to access details of the case study are explained; then, the problem is improvement of the urban residence centers (towns) to MS solved and the parameter sensitivity is done (Fig. 1). centers. The case is related to one of the biggest drug supply/ Figure 2 presents the existing supply/distribution centers distribution chain networks named Daru Pakhsh. This and also potential sites for constructing new facilities (new company is one of the biggest drug supply/distribution chain supply/distribution centers) and new transportation links. networks in the Middle East and has several essential drugs As a reminder point, because of better presentation, the and also numerous cosmetic accessories. Daru Pakhsh is demand sites are not shown in Fig. 2; however, all drug trying to improve the customer satisfaction index in drug stores, clinics, and hospitals in the villages and cities are 123 78 J Ind Eng Int (2018) 14:65–85 Fig. 2 Existing facilities (including suppliers and distribution centers) and the potential sites for establishing new facilities the demand sites. The policy of the company is to have a Figure 3 demonstrates the opening of new facilities at distribution center at each province for easier and faster sites shown as green stars. The optimum objective transportation of the demands of the sites. Moreover, there function value is 2,581,423,932 MU (including the preferably is a supply center at each region of the country optimum value of the fixed cost 41,932,960 MU, and the to reduce the transportation costs. Since a supply center can optimum value of operational cost 2,623,356,892 MU). serve as a distribution center too, if a supply center is Likewise, with respect to Fig. 3, it is evident that the constructed in a province, then a distribution center will not provinces Zanjan, Mazandaran, North khorasan, South be constructed there and the supply center will work as a khorasan, Kermanshah, Kerman, Sistan Balochestan and distribution center too. There are three types of road, rail, Abomosa Island are sites that have the minimum loca- and air transportation methods. tion costs; hence, these provinces were selected for Due to the aforementioned statuses, it is clear that the locating new facility centers (new distribution centers). mentioned case can be literally considered as an RR/CMc/ Also, new roads (including air, earth and rail lines) are SCNDP. As an outcome, the RR/CMc/SCNDP and the RR/ presented in Fig. 3. It is noted that all of the supply/ CMc/SCNDPs formulations are adequate mathematical distribution centers and demand sites are connected via formulations for the case study. The models were coded in road lines. Therefore, only new applied air and rail lines the GAMS 24.1 software and executed by CPLEX solver. are presented. All of the demand sites and centers can be Figure 3 presents the optimum solution state of the case connected to each other directly/indirectly via a kind of study. line (road, air, or rail). 123 J Ind Eng Int (2018) 14:65–85 79 Fig. 3 The optimum solution state of the case study Fig. 4 Changing of the BC regarding fixed and operational costs 123 80 J Ind Eng Int (2018) 14:65–85 Fig. 5 Changing of the BC regarding the objective function value Table 2 A comparative Objective value Fixed costs Operational costs analysis of general disturbance and fiddling disturbance General disturbance 3,058,166,039 30,947,927 3,089,113,966 Fiddling disturbance 2,581,423,932 41,932,960 2,623,356,892 Fig. 6 Changing PR according to the fixed and operational costs Table 3 A comparative analysis of changing PR operational costs are increased and decreased, respectively, due to growth of the parameter BC. The reason for this No. PR Objective value Fixed costs Operational costs behavior is improved affordability of the new facility TP8 0 4,803,573,965 24,724,118 4,778,849,847 location and new link establishment. Accordingly, the TP9 0.1 5,243,131,789 24,894,734 5,218,237,055 operational costs can be considerably reduced. Regarding TP10 0.2 6,499,778,660 27,005,654 6,472,773,005 Fig. 5, it is evident that the total costs, including the fixed TP11 0.3 5,383,295,009 22,544,225 5,360,750,784 costs and operational costs, are reduced due to growth of TP12 0.4 4,691,692,071 22,028,219 4,669,663,852 the parameter BC. Regarding Fig. 4, as a suitable conclu- TP13 0.6 8,363,086,205 26,583,794 8,336,502,411 sive remark, if the junction of the fixed costs and opera- TP14 0.8 10,458,346,768 28,885,271 10,429,461,496 tional costs is brought up as the optimal value of the BC, TP15 1 13,107,819,927 31,399,030 13,076,420,896 then this optimal value can be considered in [9,500,000, 11,000,000]. Sensitivity analysis Comparing general disturbance and fiddling partial Changes in BC disturbance Figures 4 and 5 demonstrate that the changing of the It is noted that the efficiency of the facility with general investment budget (BC) parameter affects the objective disturbance is 0%; however, the efficiency of the facility in function. It is observed that the fixed costs and the fiddling partial disturbance is determined as (1 - k) %, in 123 J Ind Eng Int (2018) 14:65–85 81 Fig. 7 Changing of PR according to the objective value Fig. 8 The sensitivity analysis of changing probability of disturbance by objective value Fig. 9 The sensitivity analysis of changing probability of disturbance by the objective value which k is the failure probability. Accordingly, the related facility can service with a part of the nominal capacity. comparative analysis of the case study parameters is Therefore, the growing of the total cost of the fiddling obtained as Table 2. partial disturbance happening is not similar to the growing Regarding to the Table 2, it is obvious that both of the of the total costs of the general disturbance happening. fixed and operational costs grow with varying of the fid- dling partial disturbance to the general disturbance. It is Changes of robustness level logical; because when a general disturbance happens, the related facility completely fails and cannot service any Figure 6 and fourth and fifth columns of Table 3 present demand; therefore, extra alternative facilities must be the changes of the fixed and operational costs of the opened conducive to the service of the unmet demands. objective function vs. the increases of the PR. Also, Fig. 7 However, if the fiddling partial disturbance occurs the and the third column of Table 3 present the changes of the 123 82 J Ind Eng Int (2018) 14:65–85 Fig. 10 The sensitivity analysis of changing probability of disturbance by the objective value Fig. 11 The sensitivity analysis of changing qd by the objective value Fig. 12 The sensitivity analysis of changing qt by the objective value objective function vs. the increases of PR. It can be seen changes of the partial disturbance parameters in facilities ‘ that the right hand side value of the constraint and (Nsj ), links (fsij ), and transportation vehicles ( lvsij ). It is accordingly the objective value decrease with increase of obvious that the total system costs grow with increase of PR. This decrease event continues to PR & 0.4. In the ‘ each of the (Nsj ), (fsij ), and ( lvs ij ). As a more comprehen- following, the value of objective function considerably increases for PR [ 0.4. As a concluding remark, it is clear sive concluding remarks, when the (Nsj ) increases up to that the best value of PR can be 0.4. 60%, the total costs considerably increase; because the strategic (fixed) costs grow in this state. Moreover, when ‘ Changes of fiddling disturbance parameters (Nsj , fsij , (fsij ) and ( lvs ij ) increase up to 70 and 80%, respectively, ‘lvs the total costs significantly increase. This analysis ij ) emphasizes that the importance of (Nsj ) is much more than ‘ Figures 8, 9, and 10 present the changing procedure of the that of (fsij ) and ( lvsij ), conducive to ameliorate the relia- fixed costs, operational costs, and total costs regarding the bility and accessibility of the system. 123 J Ind Eng Int (2018) 14:65–85 83 Furthermore, Figs. 8, 9 and 10 corroborate that the (II) system disturbances. It is obvious that modifying this increase of operational costs, regarding to the growing of SCN and its related logistics will be very difficult and the (Nsj ), is more than this increase, regarding to the costly. Therefore, it is important to design a reliable and ‘ robust SCN that reaches suitable stability and efficiency growing of the (fsij ) and ( lvs ij ). However, the increase of ‘ under several kinds of risks from the start. A two-stage strategic costs, regarding the growing of the ( lvs ij ), is less mathematical formulation was proposed for modeling of s than this increase, regarding to the (fij ). the mentioned problem. Also, because of uncertain parameters of the model, an efficacious possibilistic robust Changes of the uncertainty level in robust approach optimization approach was applied. To validate the model, (qt, qd) and comparison by a certain model a drug SCN was studied and the results of the solution model were described and analyzed. Finally, an extensive Figure 11 presents the variable behavior of the total costs sensitivity analysis was done on the critical parameters regarding the variation of parameter qd in the certain and such as the investment budget, robustness level, probability uncertain states of the suggested model. It is observed that of disturbance in the facilities (including DCs, vehicles and the total costs increase when the uncertainty at the value of lines), and uncertainty level. The sensitivity analysis indi- demand (qd) grows. This motif will be more obvious when cates the effect of several changes in the key parameters of the value of uncertainty grows up to 50%. Moreover, the model. Fig. 11 emphasizes that whatever the uncertainty decrea- Throughout this study, we dealt with the questions that ses, the total costs at the uncertain state come closer to the can be proposed as future research for scholars. First, we total costs at certain states. On the other hand, whatever the considered the possibilistic robust aspect of the presented uncertainty increases, the gap between the total costs of the model; however, other aspects of uncertainties (e.g., sev- uncertain state and the total costs of the certain state will eral probability distributions, intervals, fuzzy sets …) can amplify. be considered. Second, some solution algorithms can be Figure 12 presents the changes of the total costs with applied to gain the optimal value of the model in the large- respect to the variation of parameter qd in the certain and scale problem. Third, in this study a scenario-based uncertain states of the proposed formulation. It is evident approach of the system disturbance was applied to consider that the total costs flourish when the uncertainty at the the reliability. Fourth, the study of other uncertainty con- value of unit transportation costs (qd) grows. While this sideration approaches, e.g., several probability distribu- uncertainty grows up to 60%, the total costs will be sig- tions, are suggested to formulate the considered problem nificantly increased. This behavior is related to the con- and compare these approach behaviors with each other. siderable growth of the total transportation costs. Accordingly, the proposed model can be more applicable Moreover, Fig. 12 emphasizes that whatever the increases when the agility and resiliency concept are applied to that. in the value of uncertainty at transportation costs, the dif- ference in the total costs, in the certain state and uncertain Open Access This article is distributed under the terms of the state of transportation cost, will increase. Creative Commons Attribution 4.0 International License (http://crea tivecommons.org/licenses/by/4.0/), which permits unrestricted use, As another concluding remark, comparing Figs. 11 and distribution, and reproduction in any medium, provided you give 12, it is clear that the growth of the total system costs, due appropriate credit to the original author(s) and the source, provide a to the uncertainty of demands, is less than the growth of the link to the Creative Commons license, and indicate if changes were total system costs, due to uncertainty of transportation made. costs. References Conclusion Amrani H, Martel A, Zufferey N, Makeeva P (2011) A variable neighborhood search heuristic for the design of multi commodity This paper investigated the problem of robust and reliable production-distribution networks with alternative facility con- designing of a capacitated SCN, which consists of suppli- figurations. 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