Optimal Decisions for Prepositioning Emergency Supplies Problem with Type-2 Fuzzy Variables

the original


Introduction
Disasters caused massive casualties and tremendous economic and social damage.Much infrastructure lacking adequate structural stability collapses and human beings either die or need immediate help.These huge losses make emergency logistics management attract considerable research attention from both logistics academics and practitioners in recent years.More and more optimization models and algorithms are powerful tools to handle this kind of problems.Abounacer et al. [1] presented a three-objective locationtransportation model for disaster response and adopted an epsilon-constraint method for generating the exact Pareto frontier.Barzinpour and Esmaeili [2] developed a multiobjective relief chain location distribution model for preparation planning phase of disaster management which was applied to a real case study of an urban district in Iran.Extensive research has been documented on emergency logistics and disaster operations management.For more details, please refer to Altay and Green [3], Galindo and Batta [4], Anaya-Arenas et al. [5], and Özdamar and Ertem [6].
When creating an efficient and effective emergency plan, uncertainty is a paramount challenge.As pointed out by Bozorgi-Amiri et al. [7], information about demand, supply, and cost is unknown in advance due to the lack of related data.The existence of uncertainty has motivated some researchers to address the critical parameters in the form of stochastic approaches.For example, Salmerón and Apte [8] developed a two-stage stochastic optimization model for natural disaster asset prepositioning problem under the uncertainty of the event's location and severity.Rawls and Turnquist [9] formulated a two-stage stochastic mixed integer programming model that determined the facility location and amounts of various emergency supplies to be prepositioned and designed a heuristic solution approach called the Lagrangian L-shaped method to solve this problem.Balcik and Ak [10] studied the supplier selection problem in preparation for the sudden-onset disasters and developed a stochastic programming model by using a scenario-based method to represent demand uncertainty.In order to study the emergency medical services design problem under uncertainty, Zhang and Jiang [11] employed a robust counterpart approach and found model's Pareto-optimal solutions for costs and response times by the weighting method.Caunhye et al. [12] proposed a two-stage stochastic location-routing model that integrated preparedness and response schedule under uncertainties in demand and the state of the infrastructure.For a thorough coverage of stochastic disaster management problem, the interested reader may refer to Hoyos et al. [13] and references therein.
Fuzzy optimization techniques have been developed for emergency supplies management problem to model uncertainties mathematically.For example, Sheu [14] employed a hybrid fuzzy clustering optimization approach to a threelayer emergency logistics codistribution conceptual framework and demonstrated the applicability of the new method and potential advantages through numerical studies.Sun et al. [15] used the fuzzy set and rough set theory to build an emergency material demand prediction model and developed decision rules and computing methods based on the risk decision-making principle of classical operational research.Considering the multiple attributes of disaster relief supply chains, Zheng and Ling [16] proposed a multiobjective fuzzy emergency transportation planning model and designed a cooperative optimization method to solve efficiently the proposed problem.Ruan et al. [17] proposed a method for comparing triangular fuzzy numbers by combining a preference-based index and -cut idea and developed an approach for proportionately allocating limited relief supplies to emergency distribution points.Bai [18] proposed a two-stage emergency supplies allocation model to examine the impact of uncertain demands of the affected areas and the availability of path and supply amounts on the optimal allocation plan.Tofighi et al. [19] established two-stage possibilistic-stochastic programming that determined the locations of central warehouses and local distribution centers and designed a tailored differential evolution algorithm to find good enough feasible solution.
In summary, the existing studies with regard to emergency supplies mainly depended on the deterministic optimization method.Although there were some references that employed the stochastic approaches or fuzzy techniques to deal with the uncertainty, they characterized the uncertain information with fixed probability or possibility distributions.In the case that the fixed distributions cannot be determined exactly in the real problem, then it is unsuitable to adopt the obtained solution to conduct the emergency management.In this paper, we propose a new parametric method to express uncertain parameters in the prepositioning emergency supplies problem by using variable possibility distributions instead of fixed possibility distributions.The variable possibility distributions are given by credibility value-at-risk (VaR) reduction method to the type-2 fuzzy variables.The type-2 fuzzy variable was introduced by Liu and Liu [20] as an extension of the type-2 fuzzy set [21] to depict the fuzzy phenomenon.Furthermore, four kinds of major reduction methods were refined for type-2 fuzzy variables to reduce the uncertainty embedded in the secondary possibility distributions.The mean value reduction method was given by Choquet fuzzy integrals of regular fuzzy variables [22].The critical value reduction method was defined by Sugeno fuzzy integrals of regular fuzzy variables [23].The equivalent value reduction method was introduced via classic Lebesgue-Stieltjes integrals of regular fuzzy variables [24] and the VaR reduction method was based on the possibility VaR or credibility VaR of regular fuzzy variables [25,26].For the recent development and other applications of type-2 fuzzy theory, we refer the reader to [27,28] for detailed discussion.The theoretical development above motivates us to study prepositioning emergency supplies problem from a new perspective.Bai [29] addressed a fuzzy prepositioning problem for emergency supplies on the basis of fuzzy possibility theory, where the transportation costs, the suppliers' supply amounts, the affected areas' demands, and the capacities of the roads were characterized by type-2 triangular fuzzy variables.This article differs from the mentioned work in exploring the decision maker's risk-averse attitude towards uncertainty in place of the risk-neutral criterion.
This paper aims to adopt fuzzy parametric optimization method to address the prepositioning emergency supplies problem.To the best of our knowledge, this kind of prepositioning problem with variable possibility distribution has not yet been studied in the existing literature.The main contributions of the paper are summarized as follows: (1) The postdisaster acquisition and transportation costs, the suppliers' supply, and affected areas' demand are unknown prior to the event and are characterized by type-2 fuzzy variables.This paper applies fuzzy optimization to examine the impact of the uncertain parameters on the optimal allocation plan.(2) This paper builds a fuzzy prepositioning emergency supplies model with risk-averse objective function and credibility constraints in a framework of type-2 fuzzy theory.(3) Our fuzzy prepositioning emergency supplies model itself is difficult to solve because it has a maximal credibility objective and several credibility constraints.We use the optimistic and pessimistic value formula to obtain an equivalent deterministic form without sacrificing optimality.This allows the original model to be turned to a tractable one computationally that can be solved by generic software.(4) We conduct numerical example in the context of emergency logistics to show the value of fuzzy model and the efficiency of the presented method.
The rest of this paper is organized as follows.Section 2 derives the pessimistic and optimistic value formula of the reduced fuzzy variables for common type-2 trapezoidal fuzzy variable.Section 3 formulates a new class of fuzzy prepositioning emergency supplies problem with maximal credibility criterion.In Section 4, making use of the results obtained, we convert the original prepositioning problem to its equivalent model.In Section 5, the proposed approach is applied to a practical fuzzy prepositioning example, and the computational results show the superiority of parametric method.Finally, Section 6 gives the conclusions.

Optimistic and Pessimistic Values of CVaR Reduced Variable
Let (Γ, A, Pos) be a fuzzy possibility space and ξ a type-2 fuzzy variable with secondary possibility distribution μξ ().
To reduce the uncertainty in μξ (), we employ the CVaR of Pos{ ξ = } as the representing value of μξ ().The method is referred to as the CVaR reduction [26].The reduced fuzzy variable obtained by CVaR reduction method is denoted by .
(i) If  ∈ (0, 0.5], the optimistic value of the CVaR reduced fuzzy variable  has the following parametric possibility distribution: (ii) If  ∈ (0.5, 1], the optimistic value of the CVaR reduced fuzzy variable  has the following parametric possibility distribution: Proof.We only prove the first assertion, and the second one can be proved similarly.
(i) If  ∈ (0,0.5], the pessimistic value of the CVaR reduced fuzzy variable  has the following parametric possibility distribution: (ii) If  ∈ (0.5,1], the pessimistic value of the CVaR reduced fuzzy variable  has the following parametric possibility distribution: Proof.It can be proved similarly as Theorem 1.

Assumptions and Notations.
To model our prepositioning emergency supplies problem, we make the following assumptions.
(i) The emergency supply chain is composed of some entities, mainly including suppliers, distribution centers, and affected areas.That is to say, the whole emergency network involves three levels of hierarchical structure, which is shown in Figure 1.
(ii) Each emergency distribution center can be opened in only one of three possible size categories: small, medium, and large, subject to the related fixed cost and maximal capacity.
(iii) There is more than one kind of emergency supplies considered, and each relief is associated with a fixed volume, procurement cost, and transportation cost.
(iv) The supply capacity of supplier, storage capacity of distribution center, demand amount of affected area, and unit procurement cost and transportation cost of emergency relief are uncertain prior to an emergency event.Different from the existing work [9], these uncertain parameters are characterized by variable possibility distributions.
In addition, we adopt the notations displayed in Notations to build our optimization model.

Maximal Chance Objective.
The prepositioning emergency supplies problem determines the location and size category of a distribution center as well as the quantities and allocations of the prepositioned emergency supplies with the aim of the least total cost.The total cost usually can be expressed as follows: where w = {  }, x = {  }, y = {  }, z = {  },  = { ξ },  = {η  }, and  = { ζ }; (⋅) is the CVaR reduced fuzzy variable of type-2 fuzzy variables' linear combination given in the parenthesis.In the right hand of ( 13), the first one is the fixed cost of opening the distribution centers, the second and third ones are acquisition cost and transportation cost for predisaster emergency supplies, and the last three ones are acquisition cost and transportation cost for postdisaster emergency supplies.
Our objective is to minimize the total cost incurred in the whole process.We adopt the following risk-averse criterion to construct the objective function.Evidently, (w, x, y, z; , , ) is a fuzzy variable.For a prescribed real number  0 , we denote Cr{(w, x, y, z; , , ) ≤  0 } as the credibility distribution of (w, x, y, z; , , ).In proposed prepositioning model, we consider maximizing the Cr of the total cost (w, x, y, z; , , ), that is, max Cr { (w, x, y, z; , , ) ≤  0 } .
By introducing an additional variable , the maximal chance objective is equivalent to the following representation: which implies that the total cost is less than or equal to  0 with credibility at least  (0 <  ≤ 1).

Credibility
Constraints.An efficient prepositioning strategy should react effectively in response to the demands of affected areas, in addition to cost-effective operations.Since the exact demands are not known in advance, we describe them by variable possibility distributions.If the demands of victims can be satisfied completely, then the requirement is represented as where   is the CVaR reduced fuzzy variable of type-2 fuzzy variable d .In practice, however, the assumption above is a challenging requirement for the decision makers due to a large number of complicated constraints.In this case, the demands of affected areas may be satisfied partly.The service quality constraint allows us to provide a predetermined service level   ∈ (0, 1) to meet the demands in the network [30].In fuzzy optimization theory, we can address the service quality by the following credibility constraint: On the other hand, since it is difficult to predict the magnitude of any disaster and subsequent impact on the transportation system, we treat the allocation of emergency supplies in a fuzzy manner.The uncertainty coming from the vulnerability of the transportation network leads to fuzzy capacities, which are characterized by variable possibility distributions.Without loss of generality, it is required that the capacity of supplier cannot be exceeded; that is, where   is the CVaR reduced fuzzy variable of type-2 fuzzy variable λ .Given a confidence level   , the capacity constraint of supplier can be expressed as follows: Similarly, it is required that the capacity of distribution center cannot be exceeded; that is, (20) where   is the CVaR reduced fuzzy variable of type-2 fuzzy variable δ .Given a confidence level   , the capacity constraint of distribution center can be expressed as follows: Cr The objective function ( 22) is to maximize the credibility  of the total cost in the sense of (23).Constraint (23) means that the total cost is less than or equal to the predetermined value  0 with credibility at least .Constraint (24) limits the number of open distribution centers at node  to no more than one.If a distribution center is made available at location , various reliefs can be stocked there.Constraints (25) and (26) require that each commodity prepositioned is assigned to the open facilities and the allocation does not exceed the facility capacity.Constraints ( 27) and (28) imply the capacity limitation of supplier and distribution center before disaster.Constraint (29) imposes the amount of commodity  for affected area  should meet the demand with a given service level   .Uncertainty resulting from the vulnerability of the transportation system leads to fuzzy capacities.So after disaster the acquisition and allocation cannot surpass the supplier's and distribution center's capacity with the respective predetermined credibility   and   , which can be written in constraints (30) and (31).Constraint (32) emphasizes that   equals 1 if there is a distribution center with size category  located at node , and 0 otherwise.Finally, constraint (33) ensures the nonnegativity of acquisition variables and allocation variables.

Solution Method
In order to solve our prepositioning model ( 22)-( 33), we require turning credibility constraints into their equivalent deterministic forms.In this section, we discuss this issue in detail.
In our prepositioning emergency supplies model, the parameter   is the transportation cost for emergency relief  from supplier  to distribution center .It can be expressed as   =     , where   is the unit transportation cost for commodity  before disaster and   represents the distance between supplier  and distribution center  in normal state.The parameter η is the type-2 fuzzy transportation cost for emergency relief  purchased by supplier  and transported to distribution center .It can be expressed as η =   η , where η is the unit transportation cost for commodity  after disaster.Similarly, the parameter ζ can be expressed as ζ =   ζ .In addition, the parameter λ is the type-2 fuzzy proportion of stocked material for emergency relief  at supplier  after disaster.It can be expressed as λ =   λ , where   and λ are the relative quality level of emergency relief  and the damage level of supplier , respectively.The parameter δ can be expressed as δ =   δ .
Theorem 3. Consider cost constraint (23) in prepositioning model ( 22)- (33).Let then cost constraint ( 23) is equivalent to where Proof.Since the acquisition cost ξ = ( that is, Based on Theorem 2, the pessimistic value of the CVaR reduced fuzzy variable has the following parametric possibility distribution: The proof of the assertion is complete. In the following, we deal with the analytical expression of service quality constraint (29); that is, Theorem 4. Consider service quality constraint (29) in prepositioning model ( 22)- (33).Let the demand d = (  1, ,   2, ,   3, ,   4, ;   , ,   , ) be a type-2 trapezoidal fuzzy variable.Suppose that   is its CVaR reduced fuzzy variables with    ∈ [0, 0.5]; then service quality constraint ( 29) is equivalent to where Based on Theorem 2, the pessimistic value of the CVaR reduced fuzzy variable has the following parametric possibility distribution: (50) The proof of the assertion is complete.

Deterministic Parametric Model.
On the basis of Theorems 3, 4, 5, and 6, we can transform the original model ( 22)-( 33) into its equivalent model as follows: where (⋅) inf (),  ,inf (  ),  ,sup (  ), and  ,sup (  ) are given by ( 39), ( 48), (53), and (57), respectively.The equivalent model ( 58) is parametric nonlinear programming.In addition, there are 0-1 integer decision variables, so it is a mixed-integer program, which can be solved by branch-and-bound method.The LINGO is a state-of-the-art optimization tool including the branch-and-bound IP code and can be used to solve model (58).

Illustrative Example
In this section, we conduct a collection of numerical experiments to show the effectiveness of our developed model.

Test Problem
. We now consider an example focusing on preparedness for earthquake threats.There are five affected areas that need to provide emergency responses.In order to give shelter and assistance to disaster victims as soon as possible, an emergency distribution center of any size can be opened at any one of four candidate locations in the network.Without loss of generality, three possible sizes of facilities are considered with fixed parameters as shown in Table 1.After disaster, since many resources are scarce, the related department is in charge of acquiring emergency supplies from suppliers and dispatching them to the disaster areas.The unit acquisition costs and the maximal supply capacity for suppliers are displayed in Table 2.The emergency resources stored in the distribution center contain two kinds of supplies: food and clothes.The costs and space estimates for acquiring, storing, and allocating these supplies are provided in Table 3.In the given network, the distances among different facilities, suppliers, distribution centers, and affected areas, are collected in Table 4.Because of the uncertainty about whether or not disasters will occur, and if they do, where and with what magnitude, the accurate values of cost, demand, and damage level are not known prior to disaster and characterized by type-2 trapezoidal fuzzy variables shown in Tables 5 and 6.Our goal is to find an optimal strategy for prepositioning and allocating the emergency supplies such that the total cost is no more than 105,000.

Optimal Prepositioning Plan.
We next employ LINGO software to solve our prepositioning emergency supplies model.For the sake of presentation, suppose that the parameters 's take their values as shown in Table 7.We set the parameters    =    =    = 0.35 for any index , , , , and  = 0.85.When the threshold values in credibility constraints are set to   =   =   = 0.95, we obtain that the corresponding optimal solution is reported in Figure 2 with the credibility more than 0.948.
From Figure 2, we first know that three distribution centers are opened, located in the candidate locations 1, 2, and 3.Moreover, the second distribution center is located in the medium category, and the first and third ones are located in the small category.Then, Figure 2 also tells us the quantities of various types of emergency supplies prepositioned as well as the amounts of every commodity replenished once an earthquake occurs.The predisaster acquisition amounts of emergency supplies are marked by 2-dimension vector in the thin lines between suppliers and distribution centers, while the postdisaster replenishment amounts of emergency supplies are labelled by 2-dimension vector in the thick lines.For example, the number pair (77,168)  the second supplier and the first distribution center indicates that 77 units of food and 168 units of clothes from the second supplier are prepositioned in the first distribution center before disaster.The number pair (220, 0) in thick link between the third supplier and the third distribution center implies that 220 units of food from the third supplier are replenished in the third distribution center after disaster and the other relief is abundant.Not only that, but also we can find that all the replenishment amounts for the second relief are zero.We now give some explanations about the computational results.Food belongs to perishable goods, so it is hard to maintain a high inventory for the distribution center.Usually, before the realized value of fuzzy variable demand is known, its conventional storage is low.Finally, Figure 2 also provides us the allocation mode of various types of emergency supplies.For example, the third distribution center is mainly responsible for the basic demands from the fourth and fifth affected areas.The number pair (74, 290) in thick line between the third distribution center and the fourth affected area indicates that 74 units of food and 290 units of clothes are transported from the third distribution center to the fourth affected area after disaster.Meanwhile, the number pair (276, 247) in thick line between the third distribution center and the fifth affected area indicates that the third distribution center provides 276 units of food and 247 units of clothes to the fourth affected area.

Parameter Analysis.
In order to identify the impact of model parameter  0 in the objective on solution quality, we find optimal solutions by adjusting the values of parameter  0 , changing it from 100000 to 106000 by 1000, and report the computational results in Figure 3. Figure 3 shows that there is much difference in terms of the optimal objective value when the parameter  0 changes.Furthermore, if the acceptable threshold turns large, the credibility of the total cost not exceeding  0 has the increasing trend.So it is concluded that there exist some positive correlations between the credibility and the threshold  0 .

Comparative Study.
In order to demonstrate the advantage of using type-2 fuzzy data, we compare the proposed model to the credibility constrained type-1 fuzzy prepositioning emergency supplies model.In a type-2 trapezoidal fuzzy variable ξ = (r 1 , r2 , r3 , r4 ;   ,   ), the parameters   and   are used to describe the degrees of uncertainty.When   =   = 0, the type-2 fuzzy variable degenerates to its corresponding type-1 trapezoidal fuzzy variable  = ( 1 ,  2 ,  3 ,  4 ).In this way, it is easy to obtain the related type-1 trapezoidal fuzzy data   ,   ,   ,   ,   , The uncertain parameters in model ( 22)-( 33) are CVaR reduced fuzzy variables which are not always normalized, while those data in model (59) are normalized type-1 trapezoidal fuzzy variables.So models (59) and ( 22)-( 33) are different.
For the type-1 numerical example, we use Theorems 3, 4, 5, and 6 to develop the equivalent transformation of model (59) in case of   =   = 0, denoted by model (60), which can correspond to the prepositioning problem in the type-1 fuzzy environment.For 0.5 <   ,   ,   ≤ 1, When we set the threshold values   =   =   = 0.95 in credibility constraints, the corresponding optimal credibility is more than 0.9804.From the above statement, it is clear that the result solved by the type-1 fuzzy prepositioning emergency supplies model is only a special case of the type-2 fuzzy model.The reason is stated as follows.On the one hand, in the type-1 fuzzy model, the uncertain data are assumed to have fixed possibility distribution, which usually depend on the decision maker's knowledge and experience.Under this environment, the optimal prepositioning strategy is related to the fixed possibility distributions.If the fixed possibility distribution cannot be determined accurately, then it is unmeaningful that we adopt such a solution.On the other hand, in our type-2 fuzzy prepositioning model, the uncertain parameters are described by variable possibility distribution.Once it is difficult to determine the fixed possibility distributions, the proposed parametric optimization method paves an effective way for decision maker.So, the decision makers can identify the informed emergency supplies schedule.In brief, the comparative study implies that the variable possibility distributions have some advantages over the fixed possibility distributions, such as alleviating the burden of the decision maker, avoiding the loss of information, and guaranteeing the accuracy of the prepositioning results.

Conclusions
In this paper, we studied a new fuzzy prepositioning emergency supplies problem, including model formulation, property analysis, and numerical experiments.We next summarized the main results of the paper.
(i) Theorems 1 and 2 present the optimistic value and pessimistic value formula for the CVaR reduced fuzzy variables of type-2 trapezoidal fuzzy variable.Using the formulas, we can lessen the complexity of computing the credibility constraints so that much time can be saved when solving the proposed prepositioning model.
(ii) We established a new fuzzy maximal credibility prepositioning emergency supplies model, in which the postdisaster acquisition and transportation costs, the damage level, and affected areas' demand were uncertain and assumed to be type-2 fuzzy variables.
(iii) On the basis of the CVaR reduction method and obtained formula, Theorem 3 discussed the equivalent expression of chance objective, while Theorems 4-6 gave the parametric forms of credibility constraints.Thus, we converted the original optimization problem into its equivalent parametric programming model and solved the corresponding optimal solutions through LINGO software.
(iv) We provided a numerical example to demonstrate the effectiveness of the proposed model.The computational results showed that the variable possibility distributions have some advantages over the fixed possibility distributions for prepositioning problem.
There are several directions for our future research.On the one hand, we did not consider the availability of path after disaster.Once some links were unusable, the transportation of emergency supplies would use more circuitous routes.The case of adding other transportation modes such as air and sea transportation is an important issue for further research.On the other hand, we assumed the possibility distributions of the type-2 fuzzy variables were trapezoidal.If not, it is unsuitable for our proposed model to employ the general-purpose software.Under this condition, it is of special value to develop a more efficient customized evolutionary algorithm [32,33], like GA, PSO, and BBO, by exploiting the structure of the prepositioning model.

Figure 1 :
Figure 1: An example for emergency network structure.

Table 1 :
in thin link between Parameters related to different distribution centers.

Table 2 :
Parameters related to different suppliers.

Table 3 :
Parameters related to different emergency supplies.

Table 4 :
Parameters related to distances of different network nodes.

Table 5 :
Parameters related to acquisition costs and proportions of different emergency supplies.

Table 6 :
Parameters related to demands of different affected areas.

Table 7 :
Values of parameter .
Indices : Set of suppliers, indexed by  : Set of distribution centers, indexed by  : Set of affected areas, indexed by  : Set of size categories for each distribution center, indexed by  : Set of emergency supplies, indexed by .  : Unit acquisition cost for emergency relief  obtained from supplier    : Unit transportation cost for emergency relief  from supplier  to distribution center    : Unit space volume for emergency relief    : Fixed cost of opening a distribution center  with size category  V  : Maximal allocation capacity of distribution center with size category    : Maximal supply capacity of emergency relief  from supplier  : A big real number.Type-2 fuzzy acquisition cost for emergency relief  obtained from supplier  after disaster d : Type-2 fuzzy demand for emergency relief  at affected area  η : Type-2 fuzzy transportation cost for emergency relief  purchased by supplier  and transported to distribution center  ζ : Type-2 fuzzy transportation cost for emergency relief  shipped from distribution center  to affected area  λ : Type-2 fuzzy proportion of stocked material for emergency relief  at supplier  after disaster δ : Type-2 fuzzy proportion of stocked material for emergency relief  at distribution center  after disaster.  : Binary variable indicating whether a distribution center with size category  is located at node  or not   : Amount of emergency relief  purchased by supplier  and prepositioned at distribution center    : Amount of emergency relief  purchased by supplier  and transported to distribution center  after disaster