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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.19 No.1 pp.197-210
DOI : https://doi.org/10.7232/iems.2020.19.1.197

# The Multi-Objective Fuzzy Mathematical Programming Model for Humanitarian Relief Logistics

Chawis Boonmee*, Chompoonoot Kasemset
Department of Industrial Engineering, Faculty of Engineering, Chiang Mai University, Thailand
*Corresponding Author, E-mail: chawis.boonmee@cmu.ac.th
March 25, 2019 July 9, 2019 February 1, 2020

## ABSTRACT

The concern about humanitarian relief logistics has spurred an interest in disaster management. The use of warehouses for storing relief items is one of the most important parts that can help the victims immediately during a disaster occurrence. Effective warehouse location selection is quite difficult since there are several criteria such as budget, response time, and uncertainty of data. Thus, this study aims to propose a multi-objective fuzzy mathematical programming model for humanitarian relief logistics. This study presents an integrated facility location problem, inventory problem and distribution problem in humanitarian relief logistics under consideration of the inherent uncertainty of input parameters. The proposed model focuses on minimization of the response time and the planning budget that determines the location of warehouses, total inventory to be stored in each warehouse, distributed management of the warehouses, and the holding utilization of the warehouses. To solve the proposed multi-objective optimization model with fuzzy parameters, an equivalent auxiliary crisp model and an epsilon constraint approach are adopted in this study. A real case study in Pichit province, Thailand, is employed to validate the proposed model. Finally, the results are proposed and discussed that represent several alternatives to decision makers.

## 1. INTRODUCTION

Since the 1950s, the number of disasters has been greatly increasing (about 374 disasters per annum on average since the 2000s). In 2018, 394 natural disasters caused approximately 10,300 human fatalities and economic losses of USD 225 billion (Podlaha et al., 2019). Owing to the increasing number of disasters, concern about humanitarian relief logistics is becoming an issue of great interest in disaster management in order to support victims during disasters. Humanitarian relief logistics is defined as the process of evacuating people from disaster-stricken areas to safe places and planning, implementing and controlling the efficient, cost effective flow and storage of goods and materials, while collecting information from the point of origin to the point of consumption for the purpose of alleviating the suffering of vulnerable people (Boonmee et al., 2017).

During the pre-disaster phase and post-disaster phase, distribution of relief kits, food, and other supplies is a crucial task which must consider facility location, inventory, and routing problems simultaneously (Boon-mee et al., 2018). The location of a warehouse is the critical point that affects the response time in servicing vulnerable people. However, the location problem alone cannot be optimized to solve the problem because the selected location relates to assigning goods and distribu-tion planning. The disaster preparedness planning of warehouses in the pre-disaster phase essentially focuses on long-term planning decisions (Balcik et al., 2016;Torabi et al., 2018), which involve determining where to store and distribute emergency relief supplies at strategic locations before a disaster occurs. Optimizing location, inventory and distribution problems at the same time becomes a complex task during the preparedness stage; many research works have applied different techniques: simulation (Hu and Sheng, 2015), decision making (Kedchaikulrat and Lohatepanont, 2015), mathematical models (Abounacer et al., 2014), and queuing theory (He et al., 2013). Mathematical models with different objectives have been considered in many research works. Response time and planning budget are the most important objective functions (Manopiniwes and Irohara, 2014) separately optimized by many researchers. However, only few research works consider both objectives simultaneously (e.g. Edrissi et al., 2013).

The unique characteristic of humanitarian logistics problems is the uncertainty that leads to be more difficulty in deriving solutions. Deterministic models are considered when uncertainty parameters are assumed to be constant; however, to obtain more efficient solutions for real-world cases, the stochastic model is recommended (Habib et al., 2016).

As mention earlier, effective warehouse location selection is quite difficult and complex. As pointed out by several studies (e.g. Habib et al., 2016;Boonmee et al., 2017), the consideration of multi-objective function and the inherent uncertainty of input parameters for integrating decisions on warehouse location, inventory, and distribution in humanitarian relief logistics network should be simultaneously studied. Therefore, this study aims to propose a multi-objective optimization model with fuzzy parameters in humanitarian relief logistics management. The proposed model was developed based on Balcik and Beamon (2008) and Manopiniwes et al. (2014) because both studies addressed closely related issues. Balcik and Beamon (2008) proposed integrated decisions on location and inventory by considering unmet demand. On the other hand, Manopiniwes et al. (2014) did not allow for unmet demand. Nevertheless, neither article considered the inherent uncertainty of input parameters or multi-objective function. Hence, we aim to consider the inherent uncertainty of input parameters and multi-objective function in this study. Our mathematical model is able to (1) determine the optimal locations, distributions, and inventory of warehouses for pre-disaster preparedness; (2) consider both the response time aspect and planning budget aspect simultaneously in multi-objective function; (3) control the minimum holding utilization of warehouses; and (4) handle the inherent uncertainty of input parameters from un-availability or incompleteness and the imprecise nature of input data. In order to solve the proposed multi-objective optimization model with fuzzy parameters, an equivalent auxiliary crisp model and an epsilon constraint approach are proposed to apply in this study. Moreover, this study applies the proposed model to the real case of a flood in Pichit province, Thailand, in order to demonstrate the benefit of assisting humanitarian agencies in making more precise decisions.

The remainder of this study is organized as fol-lows: Section 2 presents a literature review of related works. Section 3 presents the proposed mathematical model. Section 4 presents the solution method. Section 5 introduces a test case based on a flood disaster in Thailand. The results are reported and discussed in Section 6. Finally, a conclusion is given in Section 7.

## 2. LITERATURE REVIEW

This section provides a survey of the relevant literature. Comprehensive reviews about humanitarian relief logistics have been proposed by Galindo and Batta (2013), Habib et al. (2016), Boonmee et al. (2017), and Manopiniwes and Irohara (2014). There are many papers dealing with this work. Jia et al. (2007) proposed a maximal covering model along with multiple facility quality-of-coverage and quantity-of-coverage requirements for determining the facility locations of medical supplies during a large-scale situation. The goal was to maximize demand by ensuring a sufficient quantity of facilities at the stated quality level. Balcik and Beamon (2008) proposed a maximal covering location model that integrated the facility location problem and the inventory problem for humanitarian relief logistics under uncertain scenarios. The proposed model not only considered multiple item types but also considered planning budget constraints along with capacity constraints. Another covering model was proposed by Hale and Moberg (2005) for improving inventory management strategies under the consideration of the minimum number and possible locations for off-site storage facilities. Not only the covering problem but also the minisum problem are usually employed in this research area (Boonmee et al., 2017). Several scholars have focused on the budget or cost. Horner and Downs (2010) formulated a warehouse location model for locating disaster relief goods distribution; this formulation minimized the cost of the distribution of those relief goods. Rawls and Turnquist (2010) proposed an emergency response-planning approach for determining the location and quantities of various emergency supplies to be pre-positioned with uncertain conditions. The proposed model aims to minimize the expected cost over all scenarios under the uncertainty of demand. Khayal et al. (2015) developed a network flow model for selecting temporary distribution locations and allocating resources for response operations. The objective function aimed to minimize the deprivation and logistics costs of the relief distribution. Recently, response time has become an important issue for humanitarian logistics. Duran et al. (2011) proposed pre-positioning of emergency items for CARE International by determining a set of typical demand instances, given a specified upfront inventory, and sought the ideal configuration of the supply network while minimizing the average emergency response time over all the demand instances. Furthermore, this article determined the number and location of the warehouses and the quantity and type of items held in inventory in each warehouse for the supply network. Several scholars have also developed models mainly restricted to considering the number and location of facilities, in addition to the amount of supply to be stocked at the facilities chosen (Duran et al., 2011;Hong et al., 2012;Das and Hanaoka, 2013). Based on the above review, few articles have focused on response time in the network. Also, consideration of the utilization of each warehouse is lacking; therefore, we aim to consider response time and utilization of each warehouse in this study.

To consider multiple criteria, many papers pro-posed several objective functions simultaneously. Bozorgi-Amiri et al. (2013) presented a multi-criteria robust stochastic programming approach for disaster relief operations under uncertainty. The proposed model focused on demand, supplies, and the cost of procurement and transportation. It sought to locate the appropriate relief distribution centers so that the objective functions focused on the minimization of total cost and the maximization of demand coverage in the affected zone. Edrissi et al. (2013) developed a multi-criteria mathematical model to solve the planning problem for the recovery operations of damaged elements of the distribution network. Abounacer et al. (2014) proposed an exact solution approach for three-objective location–transportation model for disaster response with the aim of determining the number, position and mission of the required humanitarian aid distribution centers (HADC) within a disaster region. The objectives aimed to minimize total transportation duration, minimize the number of open agents and open distribution centers, and minimize the unmet demand for all demand points. Barzinpour and Esmaeili (2014) developed a multiple objectives mixed integer linear programming model for urban disaster management. This model was developed for the preparation stage that considered humanitarian and cost-based objectives by using a goal programming ap-proach. Ransikarbum and Mason (2016) proposed multiple objectives in an integrated network model for decision-making in the relief supply and network restoration stages during post-disaster management. Notably, few articles have focused on both planning budget and response time simultaneously in this area.

In order to apply models in real-world cases, the uncertainty criteria become the main factor for formulating mathematical models. A few papers have addressed uncertainty criteria in this field. Barbarosoǧlu and Arda (2004) presented a two-stage stochastic programming framework for the distribution of relief products to disaster zones. A multi-commodity problem and multi-modal network flow problem were generated in this model. Moreover, this study considered the inventory problem for supporting the demand as well. Mete and Zabinsky (2010) proposed a stochastic optimization model for the distribution and storage problem of medical supplies to be used for disaster management related to the uncertainty of disaster type and magnitudes. The proposed model determined an optimal strategy of warehouse selections, the levels of inventory in each warehouse, and a collection of recourse decisions on transportation plans for each disaster scenario. In other related papers, Davis et al. (2013) and Paul and MacDonald (2016) proposed stochastic mathematical models in humanitarian relief logistics as well. Few papers have studied the inherent uncertainty of input parameters, though. Kamyabniya et al. (2018) proposed a multi-criteria robust optimization model for blood centers, hospitals, and temporary emergency shelters. The objective functions focused on the minimization of shortage function and cost function under the uncertainty of parameters for victims during a disaster situation. Also, Torabi et al. (2018) proposed a two-stage scenario-based mixed fuzzy-stochastic programming model for integrated relief pre-positioning and procurement planning based on a quantity flexibility contract under a mixture of uncertain data.

Based on the literature review, very few models have adopted a multi-objective function in the response time aspect and total planning budget aspect with fuzzy parameters in humanitarian logistics. Moreover, the consideration of multi-objective function in the response time aspect and total planning budget cost aspect with the inherent uncertainty of input parameters for integrating decisions on warehouse location, inventory, and distribution in humanitarian relief logistics in a case study in Thailand is lacking. Therefore, this research aims to propose a multi-objective fuzzy mathematical programming for humanitarian relief logistics in Thailand case study. The proposed mathematical model determines the locations of the warehouse, the total quantities of pre-positioned inventory for each warehouse, the distribution of multiple types of relief items from the original points to the destination points, and the holding utilization of warehouses while considering the minimum total response time and total planning budget as the objective functions.

## 3. PROPOSED MATHEMATICAL MODEL

The proposed mathematical model for pre-disaster preparedness was developed based on Balcik and Beamon (2008) and Manopiniwes et al. (2014). By considering both research studies, this study attempts to provide a comprehensive model in order to satisfy all relief demand and support the inherent uncertainty of input parameters under multi-objective function. The proposed mathematical model considers the integrated decisions related to the facility location problem, distribution problem and inventory problem under the inherent uncertainty of parameters. Also, this model aims to determine the distribution of multiple types of relief items from the original points to the destination points, the stock decisions at each warehouse, and the appropriate warehouse locations for relief distribution. We focus on time performance or response time between demand zones and warehouses and planning budget of warehouse-opening cost, holding cost, and shipping cost as objective functions in this formulation. In this research, the proposed model does not allow for unmet demand since the shortages and delays in a relief supply chain system can lead to more pain and suffering by survivors or other affected people. Further-more, the proposed model will control the holding utilization of each warehouse for efficient warehouse management. The objective functions and constraints are proposed as mixed integer programming (MIP) with fuzzy parameters. The assumptions of the problem are as follows: (1) The transport routes between demand zones and warehouses are the same as the normal situation. (2) The transport time taken is longer than normal situations by trucks due to obstructions caused by the disaster. (3) The warehouses have a limited capacity for storing relief items. (4) The fuzzy parameters are set as triangular fuzzy numbers (TFN). (5) All demand zones are serviced by some warehouses.

The notations and parameters:

• I : Set of demand zones (i = 1, 2, …, I)

• J : Set of candidate warehouse (j = 1, 2, …, J)

• K : Set of item types (k = 1, 2, …, K)

• $d ˜ i k$ : Expected demand of zone i in each item type k

• cj : Storage capacity at warehouse j

• vk : Unit volume of item type k

• $f ˜ j$ : Fixed cost of opening warehouse j

• $h ˜ j k$ : Holding cost of item type k at warehouse j

• $s ˜ k$ : Distributing cost of item type k per kilometer

• $r ˜ i j$ : Distance between warehouse j to demand zone i

• $t ˜ i j$ : Response time from warehouse j to demand zone i

• β : Threshold value for minimum storing utiliza-tion of warehouse

Decision variables: :

• $z i j k$ : Proportion of item type k that is served from warehouse j to demand zone i

• $q j k$ : Quantities of item type k to be stocked at warehouse j

• xj : 1, if warehouse j is selected for opening 0, otherwise;

• yij : 1, if warehouse j is assigned to serve demand zone i

• 0, otherwise;

The proposed multi-objective fuzzy mathematical model can be formulated as follows:

Objectives:

$Min Z 1 ∑ i ∈ I ∑ j ∈ J t ˜ i j y i j$
(1)

$Min Z 2 ∑ j ∈ J f ˜ j x j + ∑ j ∈ J ∑ k ∈ K h ˜ j k q j k + ∑ i ∈ I ∑ j ∈ J ∑ k ∈ K s ˜ k r ˜ i j d ˜ i k z i j k$
(2)

Subject to:

$∑ j ∈ J z i j k = 1 ∀ i , k$
(3)

$z i j k ≤ y i j ∀ i , j , k$
(4)

$∑ k ∈ K v k q j k ≤ c j x j ∀ j$
(5)

$∑ k ∈ K v k q j k c j ≤ β x j ∀ j$
(6)

$∑ i ∈ I d ˜ i k z i j k ≤ q j k ∀ j , k$
(7)

$y i j ≤ x j ∀ i , j$
(8)

$0 ≤ z i j k ≤ 1 ∀ i , j , k$
(9)

$x j , y i j ∈ { 0 , 1 } ∀ i , j$
(10)

The first objective function (1) is to minimize the total response time in order to satisfy the relief logis-tics system demand. The total response time is obtained from the response time between the ware house to the demand zone, which can be estimated by aid agency experts based on disaster-related experience. The second objective function (2) is to minimize the total planning budget, defined as warehouse-opening cost, holding cost, and distribution cost. Equation (3) confirms that all demands are fully serviced. Equation (4) stipulates that a demand zone can only be served by the warehouse assigned to that particular demand point. Equation (5) provides the capacity of each warehouse, in which an amount of inventory stored in any warehouse cannot exceed its capacity. Equation (6) controls the minimum holding utilization of a selected warehouse; that is, when a warehouse is opened, the utilization of that warehouse needs to exceed the pre-determined threshold value. Equation (7) ensures that the inventory level at a warehouse is no smaller than the demand at each demand zone. Equation (8) confirms that the warehouse can service each demand zone when it is selected. Finally, the non-negativity and binary equations are defined as equation (9) and (10), respectively.

The solution of the proposed mathematical model including the service of the warehouses, the number of selected warehouses, total response time, total planning budget, maximum response time, amount of items to be stored in warehouses and holding utilization of warehouses can be calculated. These calculations can be analyzed related to the inherent uncertainty of the input data. Our results can serve emergency management purposes. The first utility of this information is to help in the preparation stage by recommending the spatial distribution of warehouses, assignment of warehouses, amount of inventory, and planning budget. The second use is to help in the response stage in order to provide relief distribution flow and service from warehouses at the minimum response time.

## 4. SOLUTION METHOD

As mentioned earlier, the proposed model is a mixed-integer programming with fuzzy parameters. To solve the proposed model, the approach of Jiménez et al. (2007) and Pishvaee and Razmi (2012) is applied in this study. This approach is described as follows:

### 4.1 The Equivalent Auxiliary Crisp Model

This approach used by Jiménez et al. (2007) is based on the definition of the “expected interval” and “expected value” of a fuzzy number. Suppose d is a triangular fuzzy number. The definition of the membership function can be provided as follows:

$μ d ˜ ( x ) = { f d ( x ) = x − d p d m − d p , i f d p ≤ x ≤ d m 1 , i f x = d m g d ( x ) = d o − x d o − d m , i f d m ≤ x ≤ d o 0 , i f x ≤ d p o r x ≥ d o$
(11)

Notice that the triangular phase number is de-fined as three prominent points (the most likely value (m), the most pessimistic value (p), and the most optimistic value (o)). Accordingly, Jiménez et al. (2007) researched (investigations) the expected interval (EI) and the ex-pected value (EV) of the triangular fuzzy number $d ˜$ that is defined as equation (12) and (13):

$E I ( d ˜ ) = [ E 1 d , E 2 d ] = [ ∫ 0 1 f c − 1 ( x ) d x , ∫ 0 1 g c − 1 ( x ) d x , ] = [ 1 2 ( d p + d m ) , 1 2 ( d m + d o ) ]$
(12)

$E V ( d ˜ ) = E 1 d + E 2 c 2 = d p + 2 d m + d o 4$
(13)

Based on the ranking method of Jiménez et al. (2007), for each pair of fuzzy numbers $a ˜$ and $b ˜$, in which the degree of $a ˜$ is greater than $b ˜$ can be defined as:

$μ M ( a ˜ , b ˜ ) = { 0 , i f E 2 a − E 1 b < 0 E 2 a − E 1 b E 2 a − E 1 b − ( E 1 a − E 2 b ) , i f 0 ∈ [ E 1 a − E 2 b , E 2 a − E 1 b ] 1 , i f E 1 a − E 2 b > 0$
(14)

when $μ M ( a ˜ , b ˜ ) ≥ α$ it can be said that $a ˜$ is greater than or equal to $b ˜$ at least in the degree of α, it will be represented as $a ˜ ≥ α b ˜$. Note that α is the value of the minimum acceptable feasibility degree, which means that value of vagueness degree. Based on our proposed model, a is set as the crisp value and $b ˜$ is set as the fuzzy number. Hence, each pair of a and $b ˜$ can be reformulated as follows:

$μ M ( a , b ˜ ) = { 0 , i f a − E 1 b < 0 a − E 1 b E 2 b − E 1 b i f 0 ∈ [ a − E 2 b , a − E 1 b ] 1 , i f a − E 2 b > 0$
(15)

Now, this considers following the fuzzy mathe-matical model in which all inherent uncertain input pa-rameters are defined as triangular fuzzy numbers.

$Min z = d ˜ t x Subject to a i x ≥ b ˜ i y , i = 1 , ... , I$
(16)

Based on Jiménez et al. (2007), a decision vector in this study is feasible in the degree of α if $min { μ M ( a i x , b ˜ i y ) } = α .$ According to equation (15), the equation $a i x ≥ b ˜ i y$ is equivalent to the following equation. Note that x and y are the crisp decision vector.

$a i x − E 1 b i y E 2 b i y − E 1 b i y ≥ α i = 1 , ... , I$
(17)

Equation (17) can be rewritten as follows:

$a i x ≥ α E 2 b i y − ( 1 − α ) E 1 b i y i = 1 , ... , I$
(18)

According to the above description about the phase number and the definition of expected interval (EI) and expected value (EV) of the fuzzy number, the equivalent crisp α-parameter metric model of equation (16) can be represented as follows:

$M i n E V ( d ˜ ) x a i x ≥ α E 2 b i y − ( 1 − α ) E 1 b i y i = 1 , ... , I$
(19)

Thus, the proposed mathematical model can be reformulated as follows:

Objective:

$Min Z 1 ∑ i ∈ I ∑ j ∈ J ( t i j p + 2 t i j m + t i j o 4 ) y i j$
(20)

$Min Z ∑ j ∈ J ( f i p + 2 f i m + f i o 4 ) x j + ∑ j ∈ J ∑ k ∈ K ( h j k p + 2 h j k m + h j k o 4 ) q j k + ∑ i ∈ I ∑ j ∈ J ∑ k ∈ K { ( s k p + 2 s k m + s k o 4 ) ( r i j p + 2 r i j m + r i j o 4 ) ( d i k p + 2 d i k p + d i k p 4 ) } z i j k$
(21)

Subject to:

$∑ j ∈ J z i j k = 1 ∀ i , k$
(22)

$z i j k ≤ y i j ∀ i , j , k$
(23)

$∑ k ∈ K v k q j k ≤ c j x j ∀ j$
(24)

$∑ k ∈ K v k q j k c j ≤ β x j ∀ j$
(25)

$∑ i ∈ I ( α ( d i k m + d i k o 2 ) + ( 1 − α ) ( d i k p + d i k m 2 ) ) z i j k ≤ q j k ∀ j , k$
(26)

$y i j ≤ x j ∀ i , j$
(27)

$0 ≤ z i j k ≤ 1 ∀ i , j , k$
(28)

$x j , y i j ∈ { 0 , 1 } ∀ i , j$
(29)

### 4.2 Fuzzy Solution Approach

We now consider a multi-objective problem with fuzzy parameters, which is more complex than single-objective optimization problems. There are many tech-niques to overcome this challenge, such as the weight sum method, LP-matrix, weighted-norm method, epsilon-constraint method, and non-preemptive goal programming. The most popular method is the weight sum method, but it is not appropriate for this problem because this solution space is mixed integer programming and it is known that when the solution space is not convex, this method cannot find all solutions (Chanta et al., 2014). However, the weighted-norm method, epsilon-constraint method and non-preemptive goal programming can find all of the solutions of integer problems and it is suitable for solving a model with multiple and conflicting objectives. Here, we selected the epsilon-constraint method that was produced by Haimes et al. (1971) for solving our problem. The concept of the epsilon constraint approach is to maximize or minimize one objective function while the other objectives are bounded at acceptable fixed values. The fuzzy solution method based on the epsilon constraint method has been proposed and shown to be successful by Pishvaee and Razmi (2012), Kamyabniya et al. (2018), and Mohammed and Wang (2017). Hence, we will propose the fuzzy solution method of Pishvaee and Razmi (2012) based on the epsilon constraint method in this study. The algorithm is presented as follows:

• Step 1: Reformulate the MOFMP model into the equivalent auxiliary crisp model by using the approach proposed in Section 4.1 and determine the minimum acceptable feasibility degree of decision vector (α).

• Step 2: Consider the range of each objective function (Eq. (20) and (21)) in feasibility degree α by solving each objective function separately. The optimistic (o) and pessimistic (p) values are able to be calculated as follows:

$min { Z 2 | Z 1 ≤ Z 1 α o } = Z 2 α p$
(30)

$min { Z 1 | Z 2 ≤ Z 2 α o } = Z 1 α p$
(31)

• Step 3: Determine the membership function in each objective function as follows:

$μ Z 1 ( x ) = { 0 , i f Z 1 < Z 1 α o Z 1 α p − Z 1 Z 1 α p − Z 1 α o i f Z 1 α o ≤ Z 1 < Z 1 α p 1 , i f Z 1 > Z 1 α p$
(32)

$μ Z 2 ( x ) = { 0 , i f Z 2 < Z 2 α o Z 2 α p − Z 2 Z 2 α p − Z 2 α o i f Z 2 α o ≤ Z 2 < Z 2 α p 1 , i f Z 2 > Z 2 α p$
(33)

where: $μ Z i ( x )$ is the satisfaction degree of ith objective function (Z) that is ranged in the interval [0, 1]. (The lowest satisfaction degree of ith objective function (Z) is equal to 0, while the highest satisfaction degree of ith objective function (Z) is equal to 1)

• Step 4: Reformulate the multi-objective model into a single objective model based on the epsilon-constraint method. The reformulated mathematical model is repre-sented as follows:

• $ε Z 1$ = the acceptable bound of the satisfaction de-gree of Z1

Objective function:

$Max μ Z 2 ( x )$
(34)

Subject to:

$μ Z 1 ( x ) ≥ ε Z 1$
(35)

$∑ i ∈ I ∑ j ∈ J ( t i j p + 2 t i j m + t i j o 4 ) y i j = Z 1$
(36)

$∑ j ∈ J ( f i p + 2 f i m + f i o 4 ) x j + ∑ j ∈ J ∑ k ∈ K ( h j k p + 2 h j k m + h j k o 4 ) q j k + ∑ i ∈ I ∑ j ∈ J ∑ k ∈ K { ( s k p + 2 s k m + s k o 4 ) ( r i j p + 2 r i j m + r i j o 4 ) ( d i k p + 2 d i k p + d i k p 4 ) } z i j k = Z 2$
(37)

The reformulated model is presented as equation (34) subject to equations (22) – (29), (32) – (33) and (35) – (37); the satisfaction degree of the total planning budget is set as the main objective function while the satisfaction degree of the total response time in order to satisfy the relief logistics system demand is bounded at an acceptable value. The proposed model varies the value of $ε Z 1$ from 1 to 0 for generating the Pareto-optimal solutions. The non-dominated solutions are identified in each sub-solution.

• Step 5: Observe and seek the preferred solution. If the decision maker is satisfied with the generated solution in step 4, stop and go to step 6; otherwise, select a new feasibility degree (α) and go back to step 2.

• Step 6: Make a second iteration for investigating more precisely in the interested feasibility degree. The new grid is provided by reducing the range to obtain more Pareto-optimal solutions.

• Step 7: Select the preferred non-dominated solution as the final solution.

To validation our proposed model, we applied a real case study in Thailand to demonstrate our proposed model; the information from this case study is described in Section 5.

## 5. CASE STUDY

This section presents a case study in which we applied our approach—a real case study in Pichit prov-ince in Thailand. Pichit province is vulnerable to flooding every year since there are two main rivers that flow through the province from the north to the south—the Yom River and the Nan River. Floods usually occur late in July – November, which is during the rainy season and the water overflows from northern Thailand. Pichit province was faced with a costly flood disaster in 2011; more than 2,300 million square meters were hit by the flood. The geographical map of Pichit province and the affected zone of the flood disaster in 2011 are shown in Figure 1 (a). The parameters and data for this case study were provided by the Department of Disaster Prevention and Mitigation (DDPM) of Pichit province. All imprecise parameters were estimated by determining the three prominent values (i.e., the most likely, the most pessimistic and the most optimistic values) of the triangular fuzzy parameters according to experts’ knowledge and available data. The operation cost of opening facilities is shown in Table 1. The warehouses in this system are categorized into two types: large warehouses with high costs and small warehouses with low costs. This also affects the carrying cost for relief items as staff members have to be employed for that purpose.

Large warehouses with high cost may include a local government office buildings, in which those ware-houses have higher operating costs for opening and carrying. Conversely, small warehouses with low cost may include temples or schools, which have lower operating costs for opening and carrying. For the capacity of the two types of warehouses, we assumed that the capacity of the large warehouse is 200,000 units and the capacity of the small warehouse is 100,000 units. All warehouses distribute and store multiple types of relief items. The formulated system involved 12 communities (Amphur), 90 demand zones (Tambon), and 42 candidate warehouses (16 small warhouses and 26 large warehouses). The popula-tion density in each demand zone and the location of candidate warehouses are shown in Figure 1 (b). The proposed mathematical model considers the minimum holding utilization of a selected warehouse, which was assumed to be 20 percent. Finally, two relief items were considered in this study. The first type satisfies individual demands while the second type satisfies the demands of a number of affected households.

We coded and solved the mathematical model in LINGO 16.0 optimization software. All tests were carried out on a personal computer with an Intel (R) Core (TM) i7-6700 CPU (3.40GHz) and 16 GB of RAM. In this study, the feasibility degree (α) is generated from 0-1, with an increment of 0.2 for performance testing (α-levels: 0, 0.2, 0.4, 0.6, 0.8 and 1). Because the planning budget is the main criteria that should be considered for proposing to decision makers or the government, the satisfaction degree of the second objective function was set as the main objective function in this study, while the satisfaction degree of the first objective function objective was set at an acceptable fixed value. The objective was maximized while the acceptable fixed value or value of epsilon ranged from the upper bound to lower bound, between 0-1. In each feasibility degree, six non-dominated solutions were generated. The bounds consisted of 1, 0.8, 0.6, 0.4, 0.2, and 0, respectively. The solutions are presented in Table 2 and Figure 2.

The results reported in Table 2 show that when the decision maker focused on each satisfaction degree of Z1 with a higher α-level, the total planning budget and the number of warehouses opened are also increased. For example, at $μ Z 1 ( x )$ = 0, when α-level = 0.6, the total planning budget is only spent as 6.64E+06 THB and only 5 warehouses are opened, whereas when α-level = 0.8, the total planning budget is spent as 6.98E+06 THB and 6 warehouses are opened. Also, the results in Table 2 confirm that two criteria are conflicting objectives, meaning that no solution simultaneously achieves both criteria since a decrease in total planning budget leads to an increase of total response time in order to satisfy the relief logistics system demand. The second objective function (Z2) has a trend towards the concentrated network to minimize total planning budget cost that consists of opening cost, holding cost, and shipping cost. This objective function trends to be concentrated network because when lower satisfaction degrees of the first objective function are compared to the higher ones, the number of opened warehouses at lower satisfaction degrees is less than the higher satisfaction degrees. For example, at α-level 0, when $μ Z 1 ( x )$ = 0.2, only five warehouses are opened, whereas six warehouses are opened when $μ Z 1 ( x )$ = 0.4. Moreover, we can see that the first objective function (Z1) has a trend towards scatter network to minimize the total response time in order to satisfy the relief logistics system demand. This objective function is scatter network since more warehouses are opened in this case compared to a concentrated network; this therefore leads to shorter response time for transporting relief items from warehouses to demand zones. As the results in Table 2 show, more warehouses are opened, and the formulated system gives more importance to the first objective function.

Table 2 also shows the related data of response time such as average, standard deviation, and the maxi-mum response time between warehouse and demand zone. We can see that the optimal solution of the first objective function is reached at $μ Z 1 ( x )$ = 1 in every acceptable feasibility degree; there are average, standard deviation, and maximum response times as 22.84 minutes, 12.81 minutes, 55.63 minutes, respectively. When the satisfaction degree of the first objective function decreases, these three values simultaneously increase.

Figure 2 depicts a graphical representation of each acceptable feasibility degree that represents the total planning budget cost, opening cost, holding cost, and shipping cost of each solution with respect to total response time. As can be seen from Figure 2, the value of total response time at the final solution of satisfaction degree is reduced when the acceptable feasibility degree increases. The opening cost is the biggest contributor to total planning budget cost in which the trend of the opening cost is the same as the total planning budget cost. When the total response time extends, the opening cost decreases. For the holding cost, this cost is a minor contributor to total planning budget cost that has a lower value than the opening cost. The tendency of holding cost is reduced when the total response time increases. On the other hand, the tendency of shipping cost is an inverse function with the other costs. When the total response time is increased, the shipping cost is also augmented in direct variation.

As was mentioned in the solution approach at step 6, the new Pareto-optimal solution is generated. In this study, the decision maker selected the acceptable feasibility degree at 0.8. To investigate more precisely, the decrement of $ε Z 1$ was provided as 0.05. Thus, the range of epsilon was adjusted as 20 segments and 21 grid points. The results are represented in Table 3 and Figure 3. Figure 3 demonstrates the average utilization of warehouses, which shows the efficiency of the warehouse for holding relief items. The tendency of average utilization of warehouses increases when the total response time increases. The average holding utilization of warehouse started at 20 percent in the satisfaction degree of the first objective function at 1, and then the average holding utilization of warehouse increases significantly until it closes to 80 percent in the satisfaction degree of the first objective function at 0. In other words, when the number of selected warehouses declines, this leads to higher holding utili-zation of the selected warehouse. In order to more easily understand, three example schemes of warehouse location and distribution area of each warehouse are illustrated in Figure 4.

Based on these results, some recommendations can be made for the decision makers. If they are more focused on response time for satisfying the relief logistics system demand, they should choose the non-dominated solution in the satisfaction degree of the first objective function at 1, in which the total response time is 2,055.18 minutes. For this option, the total planning budget cost is 24,492,221.82 THB. On the other hand, if the decision makers are more concerned about the planning budget, they should choose the non-dominated solution in the satisfaction degree of the first objective function at 0, in which the total planning budget cost is 6,975,572.01 THB. For this option, though, the total response time is 3,984.75 minutes. If the decision makers want to balance between the two criteria, they should select the non-dominated in the satisfaction degree of the first objective function at 0.5, in which the total response is 3,147.71 minutes and the total planning budget cost is 8,330,912.57 THB. To en-hance relief logistics planning and warehouse selection efficiently, the decision maker should be interested in the Pareto set since it represents the solution to a problem with several choices. Ultimately, the final point depends on the decision maker’s preference.

As mentioned earlier, the area in this case study experiences a flood disaster virtually every year. This situation usually faces several problems such as ineffi-cient performance and errors including unsuitable opened warehouse sites, inadequate relief items, excess relief item inventory, long response time for satisfying the relief logistics system demand, and a miss assignment. In this study, we confirmed that our proposed conceptual model could help to overcome those problems. Specifically, this model could consider the financial perspective, the response time in order to satisfy the relief logistics system demand, amount of items to be stored in warehouses, holding utilization of warehouses, and the inherent uncertainty of the input data simultaneously. Nevertheless, if the fuzzy parameters are not considered in this proposed model, the proposed model is not able to determine the inherent uncertainty of the input data. The proposed model will become the deterministic models, in which all un-certainty parameters are assumed to be constant over time. Hence, the solutions of the proposed model without the fuzzy parameters cannot satisfy issues in real-world cases. On the other hand, when fuzzy parameters are considered in this proposed model, the proposed model can determine the inherent uncertainty of input data and can obtain more efficient solutions for real-world cases. Thus, consideration of the fuzzy parameters is quite essential for obtaining the solution in real-world cases (Habib et al., 2016).

This study still has some limitations. The pro-posed case study is a small-scale problem, so the exact algorithm could be employed and could find the solution easily. However, if the case study is a large-scale and complicated problem, we recommend applying an advanced algorithm or heuristic algorithm for obtaining the solution since it requires less time and can solve more complicated problems. To improve pre-disaster preparedness, this proposed model should consider road closures or traffic congestion, difference in travel speed depending on the mode selection, accessibility of demand zones, and uncertainty of disaster. Although this proposed model was developed for the pre-disaster phase, the proposed model could be adjusted to the post-disaster phase.

## 7. CONCLUSIONS

This research proposed a multi-objective math-ematical programming model with fuzzy parameters for humanitarian relief logistics network in order to optimize integrated decisions on facility location, inventory and distribution problems under the uncertainty of parameters. The proposed model determined the distribution of multiple types of relief items from the original points to the destination points, stock decisions of relief items at each warehouse, appropriate warehouse locations for relief distribution, and the minimum holding utilization of warehouse. Two criteria were simultaneously considered as the objective functions: response time and planning budget cost. As the problem was formulated, an equivalent auxiliary crisp model and an epsilon constraint approach were applied in this study. A real case study in Pichit province, Thailand, was used to demonstrate the proposed model. This study presented several solutions for decision makers on selecting an efficient solution that shows the service of the warehouses, the number of selected warehouses, total response time, total planning budget, maximum response time, amount of items to be stored in warehouses, and holding utilization of the ware-houses. This research will be of great significance in helping decision makers consider the strategic placement of warehouses, inventory management, the satisfaction of the relief logistics system demand and financial perspective under the inherent uncertainty of input parameters. In future research, the model should consider road closures or traffic congestion, as well as mode of shipping that may affect the efficiency of relief distribution. Moreover, this model should concentrate on other objective functions simultaneously (maximum response time, penalty cost, etc.) because it will be an advantage for the decision makers. However, our proposed mathematical model can be applied to any other city in several disaster situations as well.

## ACKNOWLEDGMENT

This work was supported by the Department of Disaster Prevention and Mitigation (DDPM) of Pichit province. The authors would like to thank them for their valuable collaboration and support data.

## Figure

Geographical maps of Pichit province in Thailand: (a) flood area in 2011; (b) the location of candidate warehouses.

The graphical representation of each acceptable feasibility degree.

The graphical solution of the second iteration.

The geographical solution of three example schemes for warehouse location and distribution area at acceptable feasibility degree at 0.8: (a) Scheme 1 with the maximum satisfaction degree of the first objective function ( ); (b) Scheme 11 with the median satisfaction degree of the first objective function and second objective function ( ); (c) Scheme 21 with the minimum satisfaction degree of the first objective function ( ).

## Table

The categorization data of the warehouses in the case study

The computational results of each objective function in the different α-levels

The computational results of the second iteration

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