A novel robust fuzzy stochastic programming for closed loop supply chain network design under hybrid uncertainty

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ScienceDirect Fuzzy Sets and Systems 341 (2018) 69–91 www.elsevier.com/locate/fss

A novel robust fuzzy stochastic programming for closed loop supply chain network design under hybrid uncertainty Mojtaba Farrokh a , Adel Azar b,∗ , Gholamreza Jandaghi a , Ehsan Ahmadi c a Department of Industrial Management, Farabi Campus, University of Tehran, Tehran, Iran b Department of Industrial Management, Faculty of Management and Economic, Tarbiat Modares University, Tehran, Iran c Department of Industrial and Systems Engineering, Russ College of Engineering and Technology, Ohio University, Athens, OH 45701, USA

Received 23 April 2016; received in revised form 29 March 2017; accepted 31 March 2017 Available online 4 April 2017

Abstract In today’s business environments, the high importance of economic benefits and environmental impacts of using scrapped products has caused most companies to move to designing the closed loop supply chain network. This paper considers the closed loop supply chain network design problem under hybrid uncertainty, while there are two sources of uncertainty for most parameters, thus require fortifying of the robustness of the decision. The first source is that some uncertain parameters may be based on the future scenarios which are considered according to the probability of their occurrence. The second source is that the values of these parameters in each scenario are usually imprecise and can be specified by possibilistic distributions. In this case, the best robust decision has some additional properties in terms of mean value and variability of the objective function. We introduced two types of the variability named scenario variability and possibilistic variability. Possibility theory is used to choose a solution in such a problem and a novel robust fuzzy stochastic programming approach is proposed that has significant advantages. The performance of the proposed model is also compared with that of other models in term of the mean cost and variability by simulation. © 2017 Elsevier B.V. All rights reserved. Keywords: Robust fuzzy stochastic programming; Possibilistic mean absolute deviation; Credibility measure; Closed-loop supply chain network design, hybrid uncertainty

1. Introduction A closed loop supply chain network (CLSCN) is an integrated system which includes both the forward and reverse supply chains, simultaneously. The CLSCN design is one of the infrastructure issues including decision making on the number, location, capacity, coordination of facilities, the flows through the network, purchasing and production values, and inventory holding in order to optimize the entire supply chain operations [6]. In the recent decade, the increasing importance of economic benefits and environmental impacts of using scrapped products has encouraged most companies to focus on the CLSCN design [52,39,47]. In fact, they have an interest in performing the CLSCN * Corresponding author.

E-mail address: [email protected] (A. Azar). http://dx.doi.org/10.1016/j.fss.2017.03.019 0165-0114/© 2017 Elsevier B.V. All rights reserved.

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activities such as recovering, recycling, remanufacturing and disposing operations [16]. Since the integrated design of forward and reverse supply chain is a critical factor in reducing the costs, improving service level, and responding to environmental issues, most researchers (e.g., [61,49,66,16,22]) have recently focused on the CLSCN design problem. In a practical decision-making environment, the CLSCN design problem is subject to many types of uncertainty including random and fuzzy (epistemic) uncertainties [43,46,6,56]. In the classical supply chain design problem, it is assumed that capacity, demand and costs are precisely known. It is obvious that this assumption is not realistic in practice. In most real environments we often try to describe a supply chain design problem whose parameters are not known a priori. While producing a solution we must take into account this uncertainty and consider that more than one possible realization may occur in the problem. We can rather define a range of possible values or sometimes probability or possibility distributions for handling the uncertainty. In uncertainty based programming approaches a set of possible realizations, called a scenario set, is provided. There are two approaches for defining this set. In the first one, called discrete uncertainty representation, we explicitly specify a set of all possible realizations with their corresponding probabilities for each parameter. In the second approach, called interval uncertainty representation, we define an interval of possible values for every parameter and limit it within a probability or possibility distribution function. Here scenario set is defined as the Cartesian product of all these intervals [29]. The discrete representation of uncertainty allows us to model a disruption (structural) uncertainty that is the one connected with some unpredictable events that have a global and main influence on the considered system. For instance, in a supply chain problem, several discrete scenarios for demand may be connected with the appearance of a new competitor, changes of customer usage patterns and economic crises and fluctuations. On the other hand, using the interval uncertainty representation we can handle an operational (local) uncertainty, corresponded with an imprecise nature of parameters in the supply chain problem. This type of risk is caused by uncertainties such as uncertain demands, supply, production and shipping costs, capacity, and lead time [54,6]. As the body of recent literature about CLSCN design problem shows, most developed mathematical programming models are the models under uncertainty. In the area of the supply chain network design, these programming models are ranged from stochastic (e.g., [1,50,66]), fuzzy (e.g., [55,45]) and robust (e.g., [46,22]) models to robust possibilistic (e.g., [48,7]) and robust stochastic (e.g., [40,21]) models. In the area of fuzzy mathematical programming, there are two different issues including flexibility or fuzziness in constraints and epistemic uncertainty in data, which are handled by flexible programming (e.g., [60,45]) and possibilistic programming (e.g., [34,55,47]), respectively. However, in some approaches, these two issues have been jointly considered in a coherent sense [8,43]. On the other hand, programming approaches under uncertainty involve a variety of modeling philosophy including minimization of expected (or mean) cost, minimization of cost deviations and minimization of maximum cost (worst case approach or minmax one). Some authors have used possibility theory to model the epistemic and fuzzy uncertainty including Dubois and Prade [11], Lai and Hwang [33], Liu and Iwamura [34], Inuiguchi and Ramik [26], Dubois et al. [15] and Jimenez et al. [27]. One of the main disadvantages of the possibilistic programming models either based on mean value [9] or expected value [27] is that they only address mean or expected values of the objective function in developing possibilistic programming models. In fact, despite of the wide investigation of risk issue in stochastic environments in different areas [40,21], risk and deviations control of objective function under the fuzzy conditions have been neglected and all decisions are made under average condition of realization of the uncertain parameters [64]. For addressing the risk issue in the stochastic programming, Mulvey et al. [40] proposed a flexible robust optimization approach for scenario-based stochastic programming models called robust stochastic programming. The programming approach, later developed by Yu and Li [58] and Leung et al. [38], has been widely applied in designing the supply chain networks (e.g., [6,44]). In the area of the possibilistic programming, Pishvaee et al. [48] developed a new approach called robust possibilistic programming. Also, Zhang et al. [64,63] produced the portfolio selection models based on the lower and upper possibilistic means and possibilistic variances of objective function. In formulation of Zhang and Zhang [65], absolute deviation is defined as a constraint and its upper bound is determined by decision maker’s (DM) preferences. Indeed, in their formulation risk of possibilistic objective function is not included in the objective function for optimization. Babazade et al. [7] also proposed a new formulation of possibilistic programming method which is able to minimize the deviation (risk) values beside the total mean of the problem with epistemic uncertainty. In the robust stochastic and robust possibilistic models, it has been tried to minimize the expected value of the objective function and the deviation over and under the expected optimal value under stochastic and possibilistic environments, respectively.

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Fig. 1. The CLSCN structure.

On the other hand, in a real decision-making process, we often encounter a hybrid uncertainty including both local and structural uncertainty, simultaneously. For example, customer’s demand under different scenarios might have an epistemic uncertainty. This type of uncertainty arises when the value of the parameters under different scenarios have an epistemic uncertainty, concurrently. We name this type of the uncertainty as fuzzy scenarios. Usually, because of unrepeatability, incompleteness and other special features of each scenario, we do not have enough data for estimating the parameters of each scenario to find their probability distribution. In order to consider the parameters under each scenario, we can define the possibility distribution for the parameters in the form of fuzzy numbers. There is a wide range of complex problems which addressed these types of uncertainties such as Hwang [20], Luhandjula [37], Kasperski and Kulej [29] and Keyvanshokooh et al. [31]. The need to address both types of uncertainties in an integrated system has been confirmed recently by Melo et al. [41], Klibi and Martel [32] and Keyvanshokooh et al. [31]. However, a lot of papers devoted to the robust models with hybrid uncertainty in different problems, is scarce [57,31]. To the best of our knowledge, there is not any study using mathematical programming approach under hybrid uncertainty in a framework of robust optimization for supply chain network design. The aim of this paper is to extend a robust optimization method into fuzzy scenario-based stochastic programming model resulting in a novel approach called robust fuzzy stochastic programming (RFSP), that can cope with both the operational and disruption risks in a hybrid condition. To cope with the epistemic uncertainty in our RFSP model, among the current possibilistic programming approaches, we use the credibility constrained programming (CCP) approach (see [34]). This method is based on strong mathematical concepts and can support different kinds of fuzzy numbers. Moreover, this approach is computationally efficient to solve fuzzy linear problems as this method can preserve its linearity and do not increase the number of objective functions and inequality constraints [35,36]. Our method can be viewed as a generalization of the classical robust approach with both the discrete and interval uncertainties. 2. Problem description and formulation As illustrated in Fig. 1, we consider a single product, multi-echelon and multi-period CLSCN consisting of manufacturing, distribution, collection, recycling and disposal centers under demand, cost and capacity uncertainty. In the forward supply chain, the manufacturing centers produce goods by using different materials and components. Most of the materials are procured from different suppliers for new production and some materials, mainly costly ones, are provided by recycling facilities. Then the final products have to be distributed through the distributors to customer zones. The location of customer zones is fixed and there might be unsatisfied demands due to existence of operational and disruption risks. In the reverse supply chain, the customer leaves the used (scraped) products at stations called initial collection points where these products are replaced by new ones. Then the scraped products, which are collected at the collection centers, should be quickly transshipped to centralized return centers where returned products

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are inspected for quality failure and sorted as recyclable products or unrecyclable ones. Then, recyclable products are transported to disassembly/recycling plants where the products are crushed and separated into different components. Finally, the recycled materials are transported to market for selling to third parties or to the manufacturing centers where this secondary material is used along with the intact material for new production. Also, the unrecyclable products are transported to disposing centers. The considered CLSCN model has a general structure which is able to support recycling and disposing processes and thus can be used to different kind of industries such as battery (e.g., [30]), glass (e.g., [21]) and tire (e.g., [53]) industries. The main objective in the CLSCN design problem is to minimize the total supply chain costs by optimal determination of the number and location of facilities, their capacities, their type of used technology, and the flow quantities between the facilities along inventory levels under hybrid uncertainty. Since demand and costs could be affected by unexpected events such as the different economic situation, the appearance of a new competitor and changes of customer usage patterns, forecasting the precise distribution of future demands and costs of a new product is very difficult. Here, the stability of probability distributions cannot be guaranteed, especially in our multi-period planning horizon. Even if sufficient data are available to generate credible scenarios, we cannot generate a probability distribution for forecasting the future demands and costs of a new product in each scenario. Thus, we adopt a possibilistic approach of formulating the uncertainty for these quantities under each scenario. 2.1. Problem assumptions and notations The main characteristics and assumptions are described below. • The transportation cost per product from the supplier to the manufacturing plant is included in the raw material purchasing cost. • The supplier and customer locations are known and fixed. • The quantity of demands and some cost values (i.e., the material, manufacturing, distribution, collection and recycling costs) are as fuzzy scenario based variables. • There is no flow between the facilities of the same echelon. • Number of the scrapped products returned to the collection/inspection centers is a ratio of the customers’ demand. • The quantity of fixed cost for opening the facilities is uncertain and is described as fuzzy variables. • The inspection cost per item for the returned products are included in the collection cost. Indices: i c j k l h r m e t s

Index of fixed locations of suppliers Index of components (material) Index of potential locations for plants Index of potential locations for distribution centers Index of fixed locations of customers Index of potential locations for collection centers Index of potential locations for recycling centers Index of potential locations for disposing centers Index of different production technologies available for recycling centers Index of time periods Index of scenarios

We assume that there are I suppliers, J plants, C components, K distribution centers, L customers, H collection centers, R recycling centers, M disposing centers, T periods, S scenarios, and E different production technologies available for recycling centers. Parameters: Regarding the notation illustration, the symbol tilde (∼) is used whenever there is a fuzzy parameter or fuzzy quantity.

M. Farrokh et al. / Fuzzy Sets and Systems 341 (2018) 69–91

d˜lts cac f˜j f˜k f˜h f˜re f˜m t˜j kts t˜klts t˜hrts t˜rj ts t˜hmts p p˜ cits p o˜ lts mc˜j ts cc˜lhts r c˜crets s p˜ cts hc˜kts c˜ic c˜j c˜k c˜h c˜cre c˜m τl or our Wc Wh πl ps

73

demand of customer zone l at the time period t in scenario s amount of material c required in kilograms to produce one item of product fixed cost of opening plant j fixed cost of opening distribution center k fixed cost of opening collection center h fixed cost of opening recycling centers r with technology e fixed cost of opening disposing center m transportation cost per product unit from plant j to distribution center k at the time period t in scenario s transportation cost per product unit from distribution center k to customer zone l at the time period t in scenario s transportation cost per scrapped product unit from collection center h to recycling centers r at the time period t in scenario s transportation cost of materials in kilograms from distribution center k to customer zone l at the time period t in scenario s transportation cost per unrecyclable product unit from collection center h to disposing centers m at the time period t in scenario s purchasing cost of material c in kilograms from supplier i at the time period t in scenario s purchasing cost of scrapped product from customer zone l at the time period t in scenario s manufacturing cost per unit of product at plant j at the time period t in scenario s collection cost per scrapped product unit from customer zone l to collection center h at the time period t in scenario s recycling cost of one unit of material c in kilograms at recycling center j with production technology e at the time period t in scenario s sale price of material c in kilograms at the time period t in scenario s unit inventory holding cost at distribution center k at the time period t in scenario s maximum capacity of supplier i for component c maximum capacity of plant j maximum capacity of distribution center k maximum capacity of collection center h maximum capacity of recycling centers r with technology e for material c maximum capacity of disposing centers m ratio of return product from customer zone l percentage of recyclable product for the returned product percentage of unrecyclable product for the returned product amount of contribution of material c in kilograms for the recyclable product weight per unit of unrecyclable product in kilograms penalty cost per unit of unsatisfied demand of customer l probability of the occurrence of scenario s

Variables: Xcij ts Xj kts Xklts Ylhts Yhrets Ucrj ets Ucrets

quantity of material c transporting from supplier i to plant j at the time period t under scenario s quantity of products produced at plant j and shipped to distribution center k at the time period t under scenario s quantity of products shipped from distribution center k to customer l at the time period t under scenario s quantity of scrapped products shipped from customer zone l to collection center h at the time period t under scenario s quantity of scrapped products shipped from collection center h to recycling center r with technology e at the time period t under scenario s quantity of produced material c with technology e transporting from recycling center r to plant j at the time period t under scenario s sale quantity of produced material c by recycling center r with technology e at the time period t under scenario s

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Vhmts Invkst Mj Dk Ch Rrt Pm

quantity of unrecyclable returned products transported from the from customer zone l to the disposal site m at the time period t under scenario s ending inventory of product in distribution center k at the time period t under scenario s 1 if a plant is opened at location j, 0 otherwise 1 if a distribution center is opened at location k, 0 otherwise 1 if a collection center is opened at location h, 0 otherwise 1 if a recycling center is opened at location r, 0 otherwise 1 if a disposing center is opened at location m, 0 otherwise

2.2. Problem formulation The objective function in the formulation of CLSCN design problem minimizes the expected total cost including the fixed opening costs of facilities (FC), manufacturing and recycling costs (MRC), purchasing costs of raw material and scrapped products minus material sales (PC), collection costs (CC), transportation costs (TC) and holding cost (HC). These components are fuzzy quantities and can be formulated as follows:      F C˜ = f˜j Mj + f˜k Dk + f˜h Ch + f˜rt Rrt + f˜m Pm j

k

˜ s= M RC

 j

P C˜ s = C C˜ s =

c

k

 c

i

j

l

h

t

+

p p˜ cits Xcij ts +

t



j

cc˜lhts Ylhts



e

j

j

e



h

r c˜crets Ucrj ets

t

p o˜ lts Ylhts −

t

h



∀s

 c

k

t

l

t˜rj ts Ucrj ets +

t˜klts Xklts +



t

hc˜kts Invkts

t

r

e

s p˜ cts Ucrets

∀s

t

∀s

t˜j kts Xj kts +

 r

r

l

t

k

c

H C˜ s =

mc˜j ts Xj kts +

r



t



T C˜ s =

h

m

h

 h

r

t˜hrts Yhrts

t

t˜hmts Vhmts

∀s

t

∀s

t

k

In terms of the above notation and the cost terms, the CLSCN problem can be formulated as follows:  ˜ s + P C˜ s + C C˜ s + T C˜ s + H C˜ s ) Min F C˜ + ps (M RC s.t.



s

cac Xj kts =

k

Invks(t−1) + 





Xcij ts +

 r

i

e

Xj kts = Invkts +

j



Ucrj ets

∀c, j, t, s

(1) (2)

t

Xklts

∀k, t, s

(3)

l

Xklts ≥ d˜lts ,

∀l, t, s

(4)

k



Ylhts ≤ τl d˜lts ,

∀l, t, s

h

 h



r

e

Wc Yhrets ≥

h

 m

t

Vhmts =

Yhrets = 

 k

(5)

 l

∀t, s

(6)

∀c, r, e, t, s

(7)

or Ylhts ,

h

Ucrj ets + Ucrets ,

j

W h.our .Ylhts

∀m, t, s

(8)

M. Farrokh et al. / Fuzzy Sets and Systems 341 (2018) 69–91



Xcij ts ≤ c˜ic

∀i, t, c, s

75

(9)

j



Xj kts ≤ c˜j Mj ,

∀j, t, s

(10)

∀k, t, s

(11)

k



Xklts ≤ c˜k Dk ,

l



Ylhts ≤ c˜h Ch

∀h, t, s

(12)

l



Ucrj ets + Ucrets ≤ c˜cre Rre ,

∀c, r, e, t, s

(13)

j



Vhmts ≤ c˜m Pm

∀m, t, s

(14)

h



Rre ≤ 1,

∀r

(15)

e

Xj ks , Xkls , Ylhs , Yhrs , Invkts ≥ 0,

integer ∀j, k, l, h, r, t, s

(16)

Xij ts , Ucrj ets , Ucrets , Vhmts ,

∀c, i, j, r, e, t, s

(17)

Mj , Dk , Ch , Rre , Sm ∈ {0, 1},

∀j, k, h, r, e

(18)

Constraint (2) ensures the material flow balance at each plant center. Constraint (3) ensures that, in each period, the sum of the flow entering each distribution center from all plants and its residual inventory from the previous periods is equal to the sum of the flow exiting from each distributor and the residual inventory of the existing period. Constraint (4) guarantees that demands of customers can be satisfied. Constraint (5) describes the relationship of customer demands with the flow of the scrapped products transferred from customer zones to collection centers. Constraints (6) ensure sum of the scrapped products entering the recycling centers is equal to the sum of the recyclable products entering the collection centers. Constraint (7) ensures that, for each material and in each period, the material transferred from each recycling center to market or the manufacturing centers does not exceed the material obtained from recyclable products in the recycling center. Constraints (8) ensure sum of the scrapped products entering the disposing centers is equal to the sum of the unrecyclable products entering the collection centers. Constraints (9)–(14) are capacity constraints on supply, production, distribution, collection, recycling and disposing centers, respectively. Constraint (15) ensures that at most one technology can be assigned to each recycling center at each potential location. Finally, constraints (16)–(18) enforce the integer, non-negativity and binary restrictions on the corresponding decision variables. Because of inherent variability and unavailability of data in real environments, most of the parameters included in such supply chain network design problem have a nature of hybrid uncertainty. Similarly, most of the parameters of CLSCN design problem in this paper, including demand of customers and different costs, have a high degree of the uncertainty because of dynamic and turbulent nature of the supply chain. So, in order to model the hybrid uncertainty of these ill-known parameters we use appropriate fuzzy and stochastic approaches and propose a novel RFSP approach that its details and advantages against common programming approaches are considered in the next sections. 3. Preliminaries Some definitions and results, which are needed in the following section, will be introduced herein. ¯ δ, )LR be two LR-fuzzy numbers (see [11]) and let λ ∈ R be a real Lemma 1. Let A˜ = (a, a, ¯ α, β)LR and B˜ = (b, b, number. Then the addition of fuzzy numbers and multiplication of a fuzzy number by a real number is defined by the extension principle [62]. (1) (2)

¯ α + δ, β + )LR A˜ + B˜ = (a + b, a¯ + b, ˜ ˜ ¯ α + δ, β + )LR A − B = (a − b, a¯ − b,

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 (3) λA˜ =

(λa, λa, ¯ λα, λβ)LR (λa, ¯ λa, |λ|α, |λ|β)LR

λ>0 λ<0

Definition 1. The lower and upper possibilistic mean values of an LR-fuzzy number A˜ as ([9]) ˜ =2 M∗ (A)

1

ρ(inf A˜ ρ )dρ

0

˜ =2 M (A) ∗

1

ρ(sup A˜ ρ )dρ

0

where inf A˜ ρ and sup A˜ ρ show the left and right extreme points of the ρ-level cut of A˜ for ρ ∈ [0, 1], respectively. By the definition, the lower and upper possibilistic means are respectively expressed as ˜ = a − α/3, M∗ (A) ˜ = a¯ + β/3. M ∗ (A)

(19)

Its interval-valued possibilistic mean is stated by Carlsson and Fullér [9] as the following interval:   ˜ = M∗ (A), ˜ M ∗ (A) ˜ M(A)

(20)

The lower and upper possibilistic mean values of a fuzzy number are interpreted as optimistic and pessimistic mean values of a fuzzy variable, respectively. It can be expressed that its interval-valued possibilistic mean is an interval in which is expected the variable occurs. Lemma 2. Now, assume that A˜ and B˜ be two LR-fuzzy numbers, and let λ, γ ∈  be two real numbers. Then the following results can be found as following ([9]). ˜ = M∗ (A) ˜ + M∗ (B), ˜ M∗ (A˜ + B)  ˜ λ≥0 λM∗ (A) ˜ = M∗ (λA) ˜ λ<0 λM∗ (A)  ˜ λ≥0 λM ∗ (A) ˜ = M ∗ (λA) ˜ λ<0 λM ∗ (A)

˜ = M ∗ (A) ˜ + M ∗ (B) ˜ M ∗ (A˜ + B)

˜ = λM(A) ˜ + γ M(B) ˜ M(λA˜ + λB) Definition 2. Let A˜ be a LR-fuzzy number. Then, the crisp possibilistic mean is defined by ([9]):   ¯ ˜ + M∗ (A) ˜ /2 = M(A) = M ∗ (A)

1

ρ(inf A˜ ρ + sup A˜ ρ )dρ

(21)

0

By the definition, the possibilistic mean value are stated as ¯ A) ˜ = a + a¯ + β − α M( 2 6 Accordingly, we can confirm the following relations: ¯ A˜ + B) ˜ = M( ¯ A) ˜ + M( ¯ B) ˜ M( ¯ A˜ + λB) ˜ = λM( ¯ A) ˜ + γ M( ¯ B) ˜ M(λ

(22)

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77

˜ the possibilistic absolute deviation between them is Definition 3. For any two given LR-fuzzy numbers A˜ and B, defined by ([65]): ˜ B) ˜ = v(A,

 1  ¯ ˜ ¯ A) ˜ − M( ¯ B) ˜ M A + B˜ − M( 2

(23)

The possibilistic absolute deviation can be defined for the given LR-fuzzy number A˜ as ˜ = a¯ − a + v(A)

α+β 3

(24)

Definition 4. Let A˜ be a LR-fuzzy number with membership function μ(x), and let r be a real number. The credibility measure is defined as follows ([35]):  1 Cr{A˜ ∗ r} = Pos{A˜ ∗ r} + Nec{A˜ ∗ r} 2 ∗ is any of the relations ≤, =, ≥. Then, the corresponding credibility measures are as follows: ⎧ 0 −∞ ≤ r ≤ a − α ⎪ ⎪ ⎪ ⎪ ⎪ r−a+α a − α ≤ r ≤ a ⎪ ⎨ 2α Cr(A˜ ≤ r) = 12 a ≤ r ≤ a¯ ⎪ ⎪ r−a+β ¯ ⎪ ⎪ a¯ ≤ r ≤ a¯ + β ⎪ 2β ⎪ ⎩ 1 a¯ + β ≤ r ≤ +∞ ⎧ 1 −∞ ≤ r ≤ a − α ⎪ ⎪ ⎪ ⎪ ⎪ a+α−r a − α ≤ r ≤ a ⎪ ⎨ 2α Cr(A˜ ≥ r) = 12 a ≤ r ≤ a¯ ⎪ ⎪ a+β−r ¯ ⎪ ⎪ a¯ ≤ r ≤ a¯ + β ⎪ 2β ⎪ ⎩ 0 a¯ + β ≤ r ≤ +∞

(25)

(26)

Based on Eqs. (25) and (26), for ρ-critical values greater than 0.5, the following substitutions could be used: Cr{A˜ ≤ r} ≥ ρ Cr{A˜ ≥ r} ≥ ρ

⇔

r ≥ (2 − 2ρ)a¯ + (2ρ − 1)(a¯ + β)

⇔

r ≤ (2ρ − 1)(a − α) + (2 − 2ρ)a

(27)

The above two relations can be used directly to convert fuzzy constraints into their equivalent crisp ones [36]. 4. The proposed RFSP approach How to address the objective function and constraints of CLSCN design problem under hybrid uncertainty will be introduced in the following subsections. Costs and risk will be measured by the possibilistic mean value and deviation on the fuzzy scenarios based costs, respectively. Also for coping with the possibilistic constraints, among the common possibilistic programming approaches, we have applied the credibility constrained programming approach. The proposed RFSP model is similar to the flexible robust programming model proposed by Mulvey et al. [40] and Pishvaee et al. [48] except that our approach has been developed for hybrid uncertainty. 4.1. Robust programming It is obvious that the modeling philosophy of minimizing pure expected or mean values of the objective function in mathematical programming models under uncertainty does not guarantee achievement of robustness. In other words, these models are risk-neutral. Instead, the robust models lead the DMs to address risk-averse aspects in decision making besides considering the average value under uncertainty. Here, before introducing the proposed RFSP model, somewhat based on the previous efforts (see [40,63,48]), we provide a clear and comprehensive description and classification for the types of robustness and robust programming (RP) approaches.

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According to the literature in robust programming methodologies, robustness is mainly grouped into optimality robustness (solution robustness) and feasibility robustness (model robustness). A solution is defined as optimality robustness if it remains close to optimal for (almost) all scenario set of the input data. In other word, the optimality robustness refers to the insensitivity of the model performance such as total cost and other criteria in the presence of uncertainty [2]. In literature of the RP, for taking into account this type of robustness, the variance of objective function can be applied as a measure of optimality robustness besides the expected or mean value [40,48,21,7] or considered as a constraint where its upper bound is determined by DMs preferences [59,63]. However, using the variance measure in objective function produces a non-linear programming model and thus needs computational effort for solving the problem. To escape the obstacle in the robust stochastic programming, absolute deviation along with a simple change of variables has been used instead of the quadratic form of deviations by Yu and Li [58] and Leung et al. [38]. In the robust possibilistic programming, the difference between the two values of extreme possible cost [48] or the possibilistic absolute deviation of fuzzy cost [7] has been used instead of the non-linear form of deviations. In other hands, a solution is defined as feasibility robustness if it remains almost feasible for (almost) all possible scenario set of the input data. Using confidence levels in stochastic and possibilistic programming or infeasibility penalty in the objective function to penalize violation of the constraints under scenario set [40,48] are common methods for addressing the type of robustness in the literature. Generally, RP approaches can be divided into two categories. In the first approach called bounded uncertainty sets based robust approach, input data in form of bounded uncertainty sets is used to capture the uncertainty data. When the approach can be applied we cannot identify any probability or possibility distribution or even define uncertainty as discrete scenarios for the uncertain data. In this approach, the feasibility robustness is only addressed. As the pioneer research in this area, Soyster [51] proposed a RP method for inexact linear programming problems. However, the resulting model produces solutions that are too conservative in the sense that we give up too much of optimality robustness in order to verify feasibility robustness (worst case logic). To address the issue of over-conservatism, a significant step forward for developing robust optimization models was taken by Ben-Tal and Nemirovski [3,4], Ghaoui et al. [18,19] and Bertsimas and Sim [5] by considering uncertain linear problems with different convex uncertainty sets. This research works proposed less conservative models in a flexible framework. Unlike the first approach, the second one is applied when we can identify a scenario set as discrete uncertainty set or interval uncertainty set in form of probability or possibility distributions. We name the approach as scenario set based robust approach. In the area of possibilistic programming, the first efforts towards RP were performed by Inuiguchi and Sakawa [24,25] using the min–max and min–max regret approach. This research work has been later followed by Nie et al. [42] and Kasperski and Kulej [29]. Also, in the area of stochastic programming, the min–max regret and min–max relative regret approach firstly introduced by Kouvelis and Yu [28], all uncertain parameters in RP model can take their values from a finite set of possible scenarios with unknown joint probability distribution. In the approach, the number of scenarios is part of the input data. Because of the lack of complete knowledge about the joint probability distribution of uncertain parameters, criteria for the first-stage decisions can be to minimize the maximum regret or the relative regret between the optimal objective function value under perfect information and the resulting objective function value under the robust decisions over all possible realizations of the uncertain parameters (scenarios) in the model. In the min–max regret and min–max relative regret approaches it is assumed that a DM wants to minimize the cost of a decision in the worst case. It is obvious that the approaches also give up too much of optimality robustness and only address the feasibility robustness. To cope with the issue, in some approaches, it has been tried for developing models that include both the measures of optimality robustness and feasibility robustness along with establishing a trade-off between robustness and the cost of robustness. Unlike the worst case approach, in these approaches the violation of constraints is allowed and the method tries to find a relative robust solution based on the value of total interest or DM’s preferences (e.g., [40,48]). 4.2. RFSP model In our approach, the uncertainty of the parameters is modeled by means of fuzzy scenarios and fuzzy variables; hence different definitions of the average of a fuzzy number has been stated in different papers including probabilistic mean or expectation [12,27] and possibilistic mean [9] of a fuzzy number. These definitions can be used to evaluate both the expected cost and the risk of a CLSCN. Dubois and Prade [13,14] introduced the interval-valued expectation of a fuzzy number as a closed interval bounded by the expectations obtained from its lower and upper probability mean

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values, viewing them as consonant random sets. They also explained that this expectation is additive in the sense of addition of fuzzy numbers. Also, Carlsson and Fullér [10] introduced a lower and upper possibilistic mean values of a fuzzy number, viewing them as possibility distributions, their definition being consistent with the extension principle and also based on the set of level-cuts. The possibilistic mean of a fuzzy number is the half point of its interval-valued mean. It has been proved that interval-valued possibilistic mean is a proper subset of its interval-valued probabilistic mean. Also, it has been proved that possibilistic mean of a symmetric fuzzy number is equal of its probabilistic mean [9]. Also, Fullér and Majlender [17] defined weighted mean values as a generalization of the possibilistic mean with a possibility for incorporating the importance of ρ-level sets. 4.2.1. Possibilistic mean formulation In this section, we use the compact form of the CLSCN model stated as follows: Min f˜x + c˜s ys s.t. Ays ≥ d˜s Bx = 0

(28)

˜ Dys ≤ Ex x ∈ {0, 1}, ys ≥ 0 x and y represent the binary and other decision variables, respectively. A, B, D, E are parameter matrices, while f, c, d are parameter vectors. A, B, D are known deterministically, while f , and E are fuzzy variables related to opening costs of the facilities and the capacities for each facility, respectively. Also, c and d are fuzzy scenario based of a fuzzy scenario based variables related to some costs and demand of customers, respectively. A specific realization  parameter is called a scenario, which is denoted by s and associated probability ps ( s ps = 1). Note that a scenario is a series of realizations over the planning horizon for considered parameters. We use S to denote the set of scenarios. Fuzzy coefficients f and E can be denoted as f˜ and E˜ and the fuzzy scenario based variable d and c can be expressed as c˜s and d˜s for each uncertain scenario. Also, variable y, which is subject to adjustment when one scenario is realized, can be denoted as ys for scenario s. In the proposed model, to form the original possibilistic chance constrained programming model, we have used the possibilistic mean value operator to model the objective function. The credibility measure is also employed to convert the possibilistic constraints to its equivalent crisp ones. According to the above-mentioned descriptions, the original fuzzy stochastic programming (OFSP) model can be formulated as follows: ¯ f˜]x + Min M[



¯ c˜s ]ys ) ps (M[

s

s.t.

Cr(Ays ≥ d˜s ) ≥ ρs

∀s (29)

Bx = 0 ˜ ≥ ϕs Cr(Dys ≤ Ex)

∀s

x ∈ {0, 1}, ys ≥ 0 Now, F˜s (x, ys ) = f˜x + c˜s ys is a fuzzy cost of solution (x, ys ) under scenario s. We will assume that the costs of the CLSCN design problem, f˜i = (f i , f¯i , αi , βi )LR and c˜j s = (cis , c¯j s , αj s , βj s )LR (i = 1, 2, ..., n; j = 1, 2, ..., J ; s = 1, 2, . . . , S), are described by trapezoidal fuzzy numbers. The addition of positive linear combinations of LR-fuzzy numbers has been extensively considered when their reference functions are linear or all of them have the same shape for L and R [23]. We will assume LR-fuzzy numbers with the same reference functions L and R, thus we can use the well-known Zadeh extension principle for addition and multiplication. The Eq. (30) states the total fuzzy cost on a scenario, denoted as F˜s (x, ys ), if all costs are fuzzy intervals. After this, to be more convenient, we denote F˜s (x, ys ) as F˜s (also more, F˜ , F s , F¯s , Ms , Ns , F , F¯ , M and N ). We can apply Lemma 1, which yields:

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F˜s =



f i xi +

i



c j yj s ,



j

f¯i xi +

i



c¯j yj s ,

j



αi xi +

i



αj s yj s ,

j



βi xi +

i

= (F s , F¯s , Ms , Ns )LR

 j

 βj s yj s LR

(30)

So, F˜ s is a trapezoidal fuzzy interval. Also, it is not difficult to compute F˜      ˜ f i xi + ps c j s y j s , ps c¯j s yj s , αi xi F= f¯i xi + i

+

j

 j

s

ps αj s yj s ,

s

i



βi xi +

j



i

j

s

i



ps βj s yj s

s

LR

= (F , F¯ , M, N )LR

(31)

Derived from Definition 2, the possibilistic mean value of the costs under each scenario (Fs ) and the weighted possibilistic mean value of the objective function (F ) can be stated as follows:   ¯ F˜s ) = ¯ f˜i )xi + ¯ c˜j s )yj s Fs = M( M( M( i

=

j

 f + f¯i i i

¯ F˜ ) = F = M(

2



+

  cj s + c¯j s βj s − αj s βi − αi xi + + yj s , 6 2 6



¯ f˜i )xi + M(

i

=

 f + f¯i i 2

i

s = 1, ..., S

(32)

j

 j

+

¯ c˜j s )yj s ps M(

s

     cj s + c¯j s βj s − αj s βi − αi ps xi + + yj s 6 2 6 s

(33)

j

¯ α, β)LR , the original According to the above-mentioned descriptions and assuming d˜s = (d s , d s , αs , βs )LR , E˜ = (E, E, crisp fuzzy stochastic programming model can be formulated as follows: Min

F

(34)

s.t.

Ays ≥ (2 − 2ρs )d s + (2ρs − 1)(d s + βs )

(35)

Bx = 0   Dys ≤ (2ϕs − 1)(E − α) + (2 − 2ϕs )E x

(36)

x ∈ {0, 1}, ys ≥ 0

(38)

(37)

where ρs and ϕs denote the DM’s minimum confidence level for satisfaction of possibilistic chance constraints under each scenario. In the above formulation, it is realistic that the chance constraints should be satisfied with a confidence level greater than 0.5 (i.e., ρs , ϕs ≥ 0.5). In this approach, DM should determine the minimum confidence level of chance constraints. It is obvious that in this approach the final value is selected as subjective and we cannot recognize that the selected value for confidence level is the best possible choice. As a result, this method is difficult in practice to adjust level of the feasibility robustness, especially in the scenario based problem which number of chance constraints is very high. Additionally, the deviation of objective function cannot be under control in this formulation to achieve the optimality robustness. 4.2.2. Robust formulation In this section, the proposed novel RFSP model is formulated to eliminate the difficulty related to determining the minimum confidence level and address the deviations of objective function under hybrid uncertainty. The proposed model, based on mean and absolute deviation of fuzzy numbers, is formulated as follows:

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      Min F + λv(F˜ ) + γ ps M¯ M(F˜ ) − F˜s + ω ps (d s + βs ) − (2 − 2ρs )d s − (2ρs − 1)(d s + βs ) s   s  (39) ps (2ϕs − 1)(E − α) + (2 − 2ϕs )E − (E − α) x +φ s

s.t.

Constraints (35)–(38)

The first term in the objective function denotes the weighted possibilistic mean value of the objective function. The second and third terms address the optimality robustness under operational and disruption uncertainties, respectively. A high variability for objective function means that the solution is a high-risk one. In other words, a small change of the values of uncertainty can cause a big variability in the value of the objective function. Indeed, the operational and disruption uncertainties can cause variability in the objective function, where the two types of risks are defined by the possibilistic variables and different scenarios, respectively, to handle the uncertainties. We name the second and third terms as possibilistic variability and scenario variability of objective function, respectively. A decrease in value of the variability, done by increasing the coefficients of them, can increase the optimality robustness. Here, λ and γ are defined as the importance (weight) coefficients of the possibilistic variability and scenario variability, respectively. When they are small, it will be given small importance to the variability of the objective function. But when these coefficients are big, then it will be given excessive importance to the variability of the objective function. Therefore, it is necessary to test the RFSP model with various λ and γ on a real network design problem. Finally, the fourth and fifth terms state weighted penalty for possible violation of each constraint (i.e., unsatisfied demand and the shortage of capacity) in the objective function determined by adjusting the values of ω and φ. Based on the above descriptions it can be concluded that the RFSP model is to achieve a reasonable trade-off between the three parts of objective function: (1) mean cost (first term), (2) optimality robustness (second and third terms) and (3) feasibility robustness (fourth and fifth terms). DMs can decide which values for the four coefficients are suitable for their problems according to risk and cost importance. Obviously, risk-averse DMs prefer higher values for the coefficients. In the following, we consider the second to fifth terms with its details, respectively. To cope with the variability, we have used the second and third terms. The second term states sum of the weighted possibilistic absolute deviation of F˜ . Therefore, the existence of the second term results in minimization of deviation over and under the mean value of F˜ . The coefficient λ represents the importance of this term against the other terms in the objective function. Thus, the term measured the possibilistic variability of proposed model can be adjusted by the coefficient λ. From Definition 3, the possibilistic absolute weighted deviation of the objective function (F˜ ) of the CLSCN can be stated as:   ¯ F˜ ) = F¯ − F + M + N v(F˜ ) = M¯ F˜ − M( 3       αj s + βj s αi + βi ¯ = ps c¯j s − cj s + xi + yj s fi − f i + 3 3 s i

(40)

j

Also, the third term expressed the absolute deviation of the weighted possibilistic mean of the difference between the M(F˜ ) and F˜s under all scenario, where γ denotes the weight placed on the term. This term can control the scenario variability. Now, we consider the term: v(F˜s , F˜ ) =



  ps M¯ M(F˜ ) − F˜s

(41)

s

It is widely accepted that the interval-valued expectation M(F˜ ) remains additive in the sense of the addition of fuzzy ¯ numbers [12,9], which is needed to calculate the mean difference on M(M( F˜ ) − F˜s ). Theorem 1. The third term is a possibilistic absolute value; it can be easily converted into an equivalent linear form. One additional variable θ s along with two constraints can be introduced to linearize it. We propose below technique to transform the model (39) to the following linear programming model:

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Min

F + λv(F˜ ) + γ +φ







ps θ s + ω



s

  ps (d s + βs ) − (2 − 2ρs )d s − (2ρs − 1)(d s + βs )

s

 ps (2ϕs − 1)(E − α) + (2 − 2ϕs )E − (E − α) x

(42)

s

s.t.

  M¯ M(F˜ ) − F˜s + θs ≥ 0, ∀s ¯ F˜s − M(F˜ )) + θs ≥ 0, ∀s M( Constraints (35)–(38)

(43) (44)

¯ ¯ ¯ F˜s − M(F˜ )) = −θs . On the Proof. It can be verified that if M(M( F˜ ) − F˜s ) ≥ 0, then θs = M(M( F˜ ) − F˜s ) and M( ¯ ˜ ˜ ¯ ˜ ˜ ¯ ˜ ˜ other hand, if M(Fs − M(F )) ≥ 0, then θs = M(Fs − M(F )) and M(M(F ) − Fs ) = −θs . As can be seen, the solution from the model (39) is identical to that from model (42)–(44). 2 Theorem 2. The above model can be transformed to following form:     Min F + λv(F˜ ) + γ ps θ s + ω ps (d s + βs ) − (2 − 2ρs )d s − (2ρs − 1)(d s + βs ) s   s  ps (2ϕs − 1)(E − α) + (2 − 2ϕs )E − (E − α) x +φ s  (F − 13 M − F¯s ) + (F¯ + 13 N − F s ) Ns − Ms + + θs ≥ 0, 2 6   (F s − F¯ − 13 N ) + (F¯s − F + 13 M) Ms − Ns + + θs ≥ 0, 2 6 Constraints (35)–(38)



s.t.

∀s

(45)

∀s

Proof. First of all, we should evaluate the difference between a crisp interval-valued mean and an LR-fuzzy number. In order to do that, let us assume that M(F˜ ) is a trapezoidal fuzzy number with right and left spread zero, that is M(F˜ ) = (F − 13 M, F¯ + 13 N, 0, 0). Then for Eqs. (43) and (44) under each scenario using Lemma 1 we obtain:   1 1 ˜ ˜ ¯ M(F ) − Fs = F − M, F + N, 0, 0 − (F s , F¯s , Ms , Ns )LR 3 3 LR   1 1 = F − M − F¯s , F¯ + N − F s , Ns , Ms (46) 3 3 LR   1 1 ˜ ˜ ¯ ¯ Fs − M(F ) = (F s , Fs , Ms , Ns )LR − F − M, F + N, 0, 0 3 3 LR   1 1 = F s − F¯ − N, F¯s − F + M, Ms , Ns 3 3 LR which are also trapezoidal fuzzy numbers. Then we obtain its crisp possibilistic mean:     (F − 13 M − F¯s ) + (F¯ + 13 N − F s ) Ns − Ms M¯ M(F˜ ) − F˜s = + 2 6   1 1   (F s − F¯ − 3 N ) + (F¯s − F + 3 M) Ms − Ns + M¯ F˜s − M(F˜ ) = 2 6

(47)

(48) (49)

Indeed, these two terms can control the optimality robustness under hybrid uncertainty. The fourth and fifth terms determine the confidence level of each chance constraint in which ω and φ are the penalty unit of the possible violation of each constraint including the imprecise parameter(s). In these terms, the terms inside brackets state the difference between the worst case value of imprecise parameter and the value that is used in chance constraints [48]. Indeed, these terms control the feasibility robustness of the solution vector. In the CLSCN model

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the value of ω and φ can be considered as the penalty of non-satisfied demand and shortage of capacity, respectively. Moreover, when the technological coefficient matrix E is tainted with epistemic uncertainty, the RFSP model is resulted in a non-linear mathematical programming model. However, the non-linear term can be converted into the linear one by defining a new variable and adding several constraints to this model. To escape from the complexity of the non-linear model, let us be a new variable which is defined as follows: us = ϕs x

(50)

Then, the above-mentioned non-linear model can be converted to an equivalent linear one as follows:     Min F + λv(F˜ ) + γ ps θ s + ω ps (d s + βs ) − (2 − 2ρ)d s − (2ρ − 1)(d s + βs ) s   s  ps (2us − x)(E − α) + (2x − 2us )E − (E − α)x +φ s  (F − 13 M − F¯s ) + (F¯ + 13 N − F s ) Ns − Ms + + θs ≥ 0, 2 6   (F s − F¯ − 13 N ) + (F¯s − F + 13 M) Ms − Ns + + θs ≥ 0, 2 6 Ays ≥ (2 − 2ρs )d s + (2ρs − 1)(d s + βs )



s.t.

∀s ∀s (51)

Bx = 0 Dys ≤ (2us − x)(E − α) + (2x − 2us )E us ≤ Mx us ≥ M(x − 1) + ϕs us ≤ ϕs x ∈ {0, 1}, ys , us ≥ 0, 0.5 ≤ ρs , ϕs ≤ 1 where M is a sufficient large number. Also, the three added constraints ensure that the new variable is equal to zero if x = 0; and is equal to ϕs if x = 1. 5. Computational experiments and evaluation To show the performance and practicality of the RFSP model, a test instance for a CLSCN design problem is considered and the related results are presented in this section. In this study, the CLSCN design problem has seven locations for establishing new plants identified as candidate ones. There are eight potential locations for distribution centers, seven potential locations for collection centers, six alternatives for recycling facilities and four alternatives for disposing centers. We consider one type of component with fourteen customer zones having uniformly distributed yearly demands. Scenarios are selected for four possible future economy conditions, i.e., either boom, good, fair and poor, with unequal probabilities. Then, for each scenario, fuzzy parameters are generated. To do so, without loss of generality and just to simplify the generation of fuzzy parameters, we applied LR-fuzzy numbers for our numerical test. To generate the fuzzy parameters, the four prominent values (i.e., the most lower and upper likely values a and a, ¯ the right spread and the left spread values α and β) are estimated for each imprecise parameter with an appropriate probability distribution. For this purpose, one of the most likely values (here, a) of each parameter is first produced randomly by using the uniform distributions specified in Table 1 (i.e., a = uniform(a, b)). Then, three random numbers (r1 , r2 , r3 ) are generated between 0 and 0.4, 0 and 0.2, and 0 and 0.2, respectively, by using uniform distribution. Finally, the upper likely values, the right spread and the left spread values are calculated as follows: a¯ = (1 + r1 )a α = r2 a

(52)

β = r3 a Similarly, other parameters are generated randomly according to the uniform distributions specified in Table 2. The problem was solved using GAMS 23.5/CPLEX 12.2 on a Pentium dual-core 2.10 GHz computer with 3 GB RAM.

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Table 1 The Random values of fuzzy scenarios based parameters in the test instance. Scenarios (s)

The lower likely values (a) Poor

Fair

Good

Boom

Scenario probability t˜j kts t˜klts t˜hrts t˜rj ts t˜hmts mc˜j ts cc˜lhts r c˜crets pp˜ cits p o˜ lts s p˜ cts hc˜kts d˜lts

0.20 U(60–80) U(50–70) U(40–60) U(20–30) U(40–60) U(450–500) U(100–120) U(30–35) U(55–60) U(200–240) U(500–550) U(40–60) U(9,070–10,085)

0.25 U(70–90) U(60–80) U(50–70) U(30–40) U(50–70) U(500–550) U(120–140) U(33–38) U(60–66) U(220–260) U(550–600) U(60–80) U(10,085–11,025)

0.25 U(80–100) U(70–90) U(60–80) U(40–50) U(60–80) U(550–600) U(140–160) U(36–41) U(66–72) U(240–280) U(600–660) U(80–100) U(11,045–12,465)

0.30 U(90–110) U(80–100) U(70–90) U(50–60) U(70–90) U(600–650) U(160–180) U(39–44) U(71–78) U(260–300) U(660–720) U(100–120) U(12,480–15,500)

Table 2 Random data for fuzzy parameters in the test instance. Parameter

The lower likely values (a)

Parameters

The lower likely values (a)

f˜j f˜k f˜h f˜re f˜m c˜ic c˜j c˜k c˜h

Uniform (225,000–240,000) Uniform (12,000–13,000) Uniform (10,000–12,000) Uniform (100,000–120,000) Uniform (10,000–13,000) Uniform (180,000–250,000) Uniform (20,000–32,000) Uniform (18,000–27,000) Uniform (22,500–28,700)

c˜cre c˜m πl τl Wc Wh cac or our

Uniform (200,000–320,000) Uniform (250,000–380,000) 10 Uniform (0.6–0.9) Uniform (5–7) Uniform (20–22) Uniform (5–10) 90% 10%

5.1. Robustness analysis In this step, we perform sensitivity analysis on both the possibilistic variability and scenario variability, which represents optimality robustness and feasibility robustness, to show the behavior of the proposed model. To assess the performance of the proposed RFSP model, this model is separately solved for different values of the coefficients of optimality robustness and feasibility robustness (i.e. λ, γ , ω and φ). Fig. 2 illustrates the results in a curve for mean cost and possibilistic variability of the objective function. It is seen that increase of the value of λ (risk factor) in the objective function cause increasing of total mean cost value and decreasing of the possibilistic variability. Also, Fig. 3 illustrates that by reducing risk factor via increasing values of γ , the total mean cost value is increased and in contrast, scenario variability is decreased. Indeed, by determining the value of two coefficients (i.e., λ and γ ), we can appropriately control the optimality robustness of the solutions according to DM preferences. Figs. 2 and 3 show when using the mean approach (i.e., λ = 0 and γ = 0), the risk of decision making is very high, because the possibilistic variability and scenario variability has the highest values. Fig. 4 indicates that, with the increase of ω, the un-fulfillment cost of demands decreases, eventually to zero. For a given value of λ and γ , a larger value of ω results in a larger value for mean cost and a high optimality robustness. It is clear that the performance of the shortage of capacity, which is adjusted with the coefficient φ, is the same as the performance of the un-fulfillment cost of demands. In reality, by increasing the ω and φ, the feasibility robustness is increased.

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Fig. 2. Possibilistic mean value and possibilistic variability for different values of λ.

Fig. 3. Possibilistic mean value and scenario variability for different values of γ .

Fig. 4. Possibilistic mean value and un-fulfillment cost for different values of ω.

5.2. Assessing the performance of the RFSP model 5.2.1. Comparison with Mulvey et al. model In this section, the RFSP model is compared with robust stochastic programming model (RSP) developed by Mulvey et al. [40]. It should be noted that the objective of RSP is to minimize the mean cost and the scenario variability. This model is as follows:

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Fig. 5. Comparing the mean cost of RSP and RFSP for different levels of stochastic risk (γ ).

Fig. 6. Comparing the scenario variability of RSP and RFSP for different levels of stochastic risk (γ ).

Min s.t.













 ps ξ s + γ ps ξ s − ps  ξs  + 2θs + ω ps η s s∈  s∈ s∈ s  ∈ ξs − ps  ξ s  + θ s ≥ 0 s∈

θs ≥ 0

ηs indicates the violation of constraints under some of the scenarios. Other constraints of the RSP model are similar to the constraints of the proposed model. In order to do the comparison, both models must be solved at a predetermined confidence level, e.g., 90 percent. Also, the possibilistic mean cost under each scenario is used for computing the scenario variability in the RSP model. Moreover, because the possibilistic variability in the RSP model is not considered, the possibilistic risk factor (λ) in the RFSP model will be considered zero. The mean cost and the scenario variability for different levels of stochastic risk (γ ) have been shown in Figs. 5 and 6. In Figs. 5 and 6, by increasing the stochastic risk factor (γ ), the mean cost increase and the scenario variability decrease. As it can be seen in the figures, the value of the mean cost and scenario variability under both models do not have a significant difference. Therefore, the RFSP model is able to control the scenarios variability as accurate as the RSP model.

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Fig. 7. Comparing the mean cost of RPP and RFSP models for different levels of possibilistic risk (λ).

Fig. 8. Comparing the possibilistic variability of RPP and RFSP models for different levels of possibilistic risk (λ).

5.2.2. Comparison with Pishvaee et al. model In this section, the RFSP model is compared with the robust possibilistic programming model (RPP) developed by Pishvaee et al. [48]. It should be noted that the objective of RPP is to minimize the mean cost and the possibilistic variability. The objective function of the RPP model is as follows:    Min F + λ(Fmax − Fmin ) + ω ps πl (d s + βs ) − (2 − 2ρs )d s − (2ρs − 1)(d s + βs ) +φ



s



 ps (2ϕs − 1)(E − α) + (2 − 2ϕs )E − (E − α) x

s

where Fmax and Fmin are determined as follows   Fmax = (f¯ + βi )xi + ps (c¯j s + βj s )yj s i

j

s

Fmin =

  (f − αi )xi + ps (cj s − αj s )yj s i

j

s

The constraints of the RPP model are similar to the constraints of the RFSP model. Because the RPP model does not involve the scenario variability, the scenario risk factor in the RFSP model is determined to be zero. The value of mean cost and possibilistic variability of the RPP and RFSP models for different levels of the possibilistic risk factor (λ) has been shown in Figs. 7 and 8, respectively. In Figs. 7 and 8, by increasing the possibilistic risk factor (λ), the mean cost increase and the possibilistic variability decrease. As it can be seen in the figures, in some values of the risk factor, the value of mean cost and possibilistic variability under the RFSP model has a better performance. Indeed, the RFSP model can obtain lower possibilistic variability with a lower mean cost compared with the RSP model. However, it is essential to validate these results by simulation. 5.2.3. Simulation of the results To validate the RFSP model, the obtained solutions in different models have been evaluated by using the simulation. To do so, 100 random realizations are generated uniformly and then the performance of the solutions

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Table 3 Comparing the performance of the different models with other in terms of mean cost and total variability under simulation. Model

Risk factor

RSP RPP RFSP RFSP RFSP

Simulated values

γ

λ

Scenario variability

Possibilistic variability

Total variability

Mean cost

3 – 3 0 3

– 1.5 0 1.5 1.5

1.068E+08 2.629E+8 1.057E+08 2.335E+8 1.065E+08

9.97E+08 8.095E+08 9.36E+08 7.960E+08 7.990E+08

1.10E+09 1.07E+09 1.04E+09 1.03E+09 9.06E+08

2.861E+09 2.846E+09 2.849E+09 2.851E+09 2.941E+09

obtained by the RSP, RPP and RFSP models are tested under each realization. For example, for imprecise parameter ¯ α, β)LR , each realization is obtained by generating a random number uniformly between the two extreme A˜ = (a, a, points of the corresponding possibility distribution function (i.e., a real ∼ [a − α, a¯ + β]). Afterward, the solutions obtained by the models under nominal data (x ∗ , ys∗ ) will be replaced in a linear programming model that its compact form is as follows: Min f real x ∗ + csreal ys∗ + ωSsdem + φSs

cap

s.t.

Ays∗ + Ssdem ≥ dsreal Bx ∗ = 0

(53)

Dys∗ − Ss

cap

cap

Ssdem , Ss

≤ Esreal x ∗

≥0 cap

In the programming model (53), Ssdem and Ss are the only decision variables that determine the violation of chance constraints of the demand and the capacity under random realization, respectively. For assessing the optimality robustness of the RFSP model, the results of this model are compared with the results of the other models. The results of different models under 100 times simulation in terms of the mean cost and the total variability including the scenario variability and the possibilistic variability have been shown in Table 3. In this table, the results indicate the superiority of the proposed RFSP model to other models in controlling the total variability, where the amount of the stochastic and possibilistic risk factors have been considered 3 and 1.5, respectively. Indeed, the RFSP model is able to reduce the total variability more than two other methods. It should be noted, because of the significant differences in the total variability of the RFSP model with the other two models, the mean cost of the RFSP model would also be more than the other two models. The value of the difference in costs is called excess robustness cost of the developed model. 5.3. The managerial implications The RFSP model has been developed for most today’s complex situations to cope with both the disruption risks such as the different economic situation, appearance of a new competitor and changes of customer usage patterns, and the operational risks such as uncertain demands, supply, production and shipping costs, and capacity. Most managers tend to minimize the disruption risks for the long term and conversely, minimize the operational risks for the short term. To do so, they can control both risks in the CLSCN design problem by using the proposed model with considering a positive value for the risk factors (λ and γ ) related to terms of optimality robustness. Considering suitable values for these risk factors are challenging. Since the DMs preferences and the required degree of conservatism in the interest problem are different, they should be considered as the main factor to determine the most suitable value for the coefficients in the RFSP model. Thus, the DMs should determine the appropriate λ and γ values according to the degree of their risk-averse and the nature of the problem led to a trade-off between mean cost and optimality robustness (variability measure). It is clear that the RFSP is a more risk-averse model and it is expected that its average performance has a lower quality when compared to the other two models. This model can be suitable for some complex situations as it assures that the objective function under hybrid uncertainty will has a low value of the both variabilities including the scenario variability and possibilistic one.

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6. Conclusions To achieve the benefits of using scrapped products in today’s competitive supply chains, this paper addresses the issue of designing a CLSCN model, including both the recycling and disposing processes, under hybrid uncertain conditions. To this aim, a mixed integer programming model was developed that optimizes the configuration of supply chain network with respect to both disruption and operational risks. To cope with fuzzy scenarios based uncertain parameters, a novel programming approach, called robust fuzzy stochastic programming, is proposed. The approach is applicable in a situation that the value of parameters in each scenario is unknown and we should define the possibility distribution for the parameters. The robust formulation has four terms in the objective function including mean values, scenario variability and possibilistic variability, and penalty for the unmet demand and shortage of capacity. To illustrate the behavior of the proposed model, a test instance for a CLSCN design problem was considered. Furthermore, by using the simulation method, we compared the proposed robust model in terms of the mean costs and the total variability with the Mulvey et al. [40] and Pishvaee et al. [48] models. These two models can control either the scenario variability or the possibilistic variability, while the proposed model can control both variabilities. The results indicate the superiority of the proposed model over two other models in decreasing the total variability as a measure of the optimality robustness. As a result, it is suitable for most managers to control both disruption and operational risks by considering the scenario variability and the possibilistic variability, simultaneously, with an excess cost. 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