A fuzzy stochastic approach for pre-positioning and ... · 020-0550 A fuzzy stochastic approach for pre-positioning and distribution of emergency supplies in disaster management Saeideh

020-0550

A fuzzy stochastic approach for pre-positioning and distribution

of emergency supplies in disaster management

Saeideh Tofighi1, S. Ali Torabi

1 and S. Afshin Mansouri

2*

1: Department of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran.

Emails: [email protected], [email protected]

2: Brunel Business School, Brunel University West London, UK.

Email: [email protected]

POMS 22nd Annual Conference,

Reno, Nevada, U.S.A.

April 29 to May 2, 2011

Abstract: Efficient management of humanitarian relief chains (HRC) is crucial due to the

increasing number and intensity of natural disasters around the globe. In this paper, a fuzzy

scenario-based stochastic programming (FSBSP) model is proposed for HRC-related

logistical problems which are characterized by inherent uncertainty in their input data. The

FSBSP model is an extension to an existing stochastic programming (SP) one. It determines

optimal policies to deal with pre and post-disaster events in two stages. The first stage

identifies the location of warehouse(s) and inventory levels of emergency supplies. The

second stage proposes a set of actions in response to a number of disaster scenarios.

Application of the proposed model is demonstrated through a case study adapted from the

literature. In addition, a number of randomly generated test problems are solved to validate its

performance. The results show promising performance of the proposed FSBSP model in

comparison with the extended SP one.

Key words: Humanitarian relief chains; Disaster management; Fuzzy stochastic programming

* Corresponding author, Tel: +44 (0)1895 265 361

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1. Introduction

In recent years, there has been an enormous increase in number and intensity of natural

disasters accompanied by massive global relief operations involving a large number of relief

organizations (Beamon and Kotleba, 2006). As a result, humanitarian relief chains (HRCs)

and logistics in the relief sector have received growing attention. Although, the impact of

disasters is unavoidable especially in emotional and financial aspects, it could be reduced by

a proactive approach through applying appropriate logistics planning techniques.

Response to a major disaster entails meticulous planning and firm decisions about the

deployment of people, equipment, and supplies. Timing and efficiency, rather than cost, are

the two essential characteristics of HRC operations which differentiates it from regular

commercial supply chains. HRC operations have to fulfill all human demands (e.g., medical

supplies and personnel, food and water) in a very short time, using the restricted resources to

minimize human suffering and death (Chern et al., 2010).

The quantitative methods and principles of HRCs have not been widely developed and

systematically implemented yet and consequently, the relief sector still suffers from the lack

of systematic approaches and tools to manage the relief chains. Regarding this fact, relief

agencies tend to manage the relief chains using ad-hoc methods, which may lead the disaster

response operations to be inefficient (high cost, resources' wastage, efforts' duplication) and

ineffective (slow response, unsatisfied demand) (Balcik and Beamon, 2008).

There are similarities in concepts between commercial supply chains and humanitarian

relief chains. As such, some of the tools and methods for commercial supply chains could be

used in HRCs (Aslanzadeh et al., 2009; Balcik and Beamon, 2008; Kovács and Spens, 2007).

However, commercial supply chains and HRCs have fundamental differences which bring

additional complexity and unique challenges to HRC management. The main characteristics

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of the relief chain design and management that are most relevant to our study are outlined by

Balcik and Beamon (2008) as follow:

unpredictability and uncertainty of demand, in terms of timing, location, type and size.

This leads to the application of stochastic/fuzzy approaches or combination of these two

(i.e., fuzzy stochastic);

sudden creation of demand in very large quantities for a wide variety of supplies to be

met in a short time;

high risks associated with adequate and on-time delivery;

lack of infrastructure and resources (supply, people, technology, transportation capacity,

money, etc).

As mentioned above, one of the main challenges of logistical planning in HRCs is due to

the uncertainty of demand where the available historical data are neither enough nor exact.

Moreover, most of the available data are of subjective and judgmental type extracted from the

experts' opinion. In this paper, we propose a hybrid fuzzy/stochastic approach in which the

imprecise nature of available data is dealt with fuzzy numbers into a scenario-based

stochastic programming framework.

The rest of the paper is organized as follow. The relevant literature is reviewed in Section

2. The logistics planning model in HRC and its corresponding fuzzy stochastic mathematical

model are demonstrated in Section 3. Section 4 provides a brief description of the

defuzzification method, followed by the equivalent auxiliary crisp model. In Section 5, the

proposed model is implemented on a case study adapted from the Mete and Zabinsky (2010).

Finally, conclusions are presented in Section 6.

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2. Literature review

Beamon and Kotleba (2006) developed an inventory model for pre-positioned warehouses

to respond to the complex humanitarian emergencies. This model provides the optimum

reorder quantity based on reordering, holding and back-order costs. Their research is the first

step in developing strategic inventory management systems for HRCs. Balcik and Beamon

(2008) addressed the facility location problem for HRCs and developed an analytical

approach that enables relief authorities to make efficient and effective decisions regarding the

facility layout and stock pre-positioning. A maximal-covering type model was developed to

determine the number and locations of distribution centers in a relief network and the stocked

amount of relief supplies at each distribution center to fulfill the requirements of suffered

people. Facility location and inventory decisions were integrated as a model which considers

several types of emergency supplies, and includes budgetary constraints and capacity

restrictions. The effects of pre- and post-disaster relief funding, in terms of response time and

demand satisfaction, were demonstrated by their computational analysis on the relief system

performance.

The final stage of a HRC is the last mile distribution which refers to the delivery of relief

supplies from local distribution centers to the affected areas. Balcik et al. (2008) considered a

vehicle-based last mile distribution system, in which the emergency relief supplies are stored

in local distribution centers and afterwards, distributed among a number of allocated demand

locations. Moreover, to minimize the transportation costs and to maximize the benefits to aid

recipients, they proposed a mixed integer programming (MIP) model that determines delivery

schedules and routes for each vehicle throughout the planning horizon. The model properly

allocates resources based on supply, vehicle capacity, and delivery time restrictions.

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Balcik et al. (2010) reviewed the challenges in coordinating HRCs and summarized the

harmonized issues associated with the relief chain and logistics operations. Additionally,

some supply chain coordination mechanisms were examined and their adaptability to the

unique relief environments evaluated. To respond to the urgent relief demands in the critical

rescue period, Sheu (2007) presented a hybrid fuzzy clustering optimization approach to the

operation of emergency logistics co-distribution. A three-layer emergency logistics co-

distribution conceptual framework was resulted in a methodology involving two recursive

mechanisms: (1) disaster-affected area grouping, and (2) relief co-distribution.

MIP is a popular method to solve humanitarian supply chain planning problems. Although

MIP is often used to model such problems, these models are frequently unsolvable because of

the problems' complexity. To overcome this shortcoming, Chern et al. (2010) proposed a

heuristic algorithm, called the Emergency Relief Transportation Planning Algorithm

(ERTPA) to solve the aftermath demand planning problem for large-scale emergency

incidents. ERTPA sorts and categorizes demands according to the required products, defined

due dates, the possible shared capacities, and the distances between the depots and the

demand nodes. ERTPA is based on the shortest travelling-time tree and the minimum cost

production tree in order to plan the demands individually.

Mete and Zabinsky (2010) addressed a two-stage stochastic programming (SP) model for

storing and distributing problem of medical supplies under a wide variety of possible disaster

types and magnitudes. In the first stage, an optimal policy for warehouse(s) selection and

inventory levels is determined. In the second stage, a collection of all possible decisions on

transportation plans are defined for each disaster scenario to balance the preparedness and

risk in spite of the uncertainties of disaster events. Our model is inspired from this SP model.

The preparation phase of disaster relief has attracted remarkable attention in the academic

literature on humanitarian logistics. The main focus has been placed on the development of

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models for specific types of disaster scenarios while the relief agencies concentrate on the

immediate response phase after a disaster. Nevertheless, for dominating a disaster, all phases

of humanitarian logistics such as preparation, immediate response, and reconstruction are so

important. Kovács and Spens (2007) addressed this gap by reviewing the literature in the

field from academic and practical perspectives. Their work raised the knowledge of logistics

operations in disaster relief.

In spite of inherent uncertainty in most of the input data in HRC-related logistical problems

due to impreciseness and vagueness as well as the insufficiency of the available data which

makes the fuzzy set theory as a good candidate for modeling, our literature review reveals the

lack of using fuzzy approach in this field. Accordingly, we propose a two-stage fuzzy

stochastic model for a similar problem studied by Mete and Zabinsky (2010). The model

determines the optimal decisions in terms of both pre-disaster and post-disaster events. The

first stage determines the warehouse(s) locations and inventory levels of emergency supplies

and the next stage identifies a collection of possible decisions in response to each disaster

scenario.

It is worth noting that the concept of fuzzy scenario-based stochastic programming

(FSBSP) is in its infancy and the amount of relevant research is very scarce (Li et al., 2008;

Li et al., 2009; Li et al., 2010). This approach is significantly different from ordinary fuzzy

stochastic programming (FSP) concepts. In the common FSP, the concept of fuzzy random

variable (FRV) or random fuzzy variable (RFV) is meaningful. There are so many definitions

and descriptions about FRVs (e.g., Kwakernaak, 1978; Kruse and Meyer, 1987; Liu and Liu,

2003). Notably, the concept of RFVs was firstly introduced by Liu (2002).

The parameters of the FSBSP are just random variables and are not related to FRVs or

RFVs. Although, the FSBSP is the same as scenario-based stochastic programming, which

has two/multiple stages but the equalities/inequalities are vague (not crisp) and follow the

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fuzzy concept (Li et al., 2008; Li et al., 2009; Li et al., 2010). In this paper, we present

another type of FSBSP which has been rarely considered before (Rommelfanger, 2007).

3. Problem description

A two-stage fuzzy stochastic model is proposed for pre-positioning and distribution of

emergency supplies in a HRC. The model covers both pre-disaster and post-disaster events.

In the first stage, the best location(s) for warehouse(s) among the potential candidates along

with the optimal inventory level for each relief supplies at each selected warehouse are

determined. Then, in the second stage, a collection of possible distribution plans in response

to different disaster scenarios are identified.

In the proposed FSBSP, the parameters in each scenario are considered to be ambiguous

and imprecise as fuzzy data. However, the method of converting the problem into the

equivalent crisp model may not be different in comparison with the current FSBSP structures.

Initially, the model is defuzzified for each scenario and afterwards, the related two-stage

stochastic program is converted into a crisp model according to the method presented by

Zhang (2001). Now, we elaborate our FSBSP model which is an extension to the stochastic

programming formulation presented by Mete and Zabinsky (2010).

Indices:

i Index of warehouses (i =1, 2, …, I)

j Index of destination (j =1, 2, …, J)

k Index of supplies (k =1, 2, …, K)

The disaster scenarios

Parameters:

Fuzzy operating cost of warehouse i

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Stage 1- warehouse(s) selection and inventory levels

The following MIP model determines which warehouses should be selected and how

much of each type of supply should be stored in the selected warehouses (sik).

(3.1)

s.t.

(3.2)

(3.3)

(3.4)

Fuzzy maximum available amount of kth type of supplies

Fuzzy storage capacity of the ith warehouse for kth type of supplies

Transportation time between warehouse i and destination j to reflect the road and

traffic conditions related to disaster scenario δ

Fuzzy penalty cost for each unit of unfulfilled demand of kth type of supplies at

destination j under scenario δ

Fuzzy demand for kth type of supplies at destination j under scenario δ

Fuzzy upper limit for penalty cost of unsatisfied demands for kth type of supplies at

destination j

(w1 , w2) Weight vector for the second stage's objective functions

Decision variable:

xi 1, if warehouse i is selected to operate; 0, otherwise

sik The inventory level of kth type of supplies in warehouse i

qijk(δ) The amount of kth type of supplies to be delivered from warehouse i to destination j

under disaster scenario δ

yjk(δ) The amount of unfulfilled demand for kth type of supplies in destination j

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The objective function (3.1) is to minimize the total operating cost of selected warehouses

in order to provide an executive preparation plan at the lowest possible cost as well as the

expected value of the second stage's objective function considering the possible disaster

scenarios. Constraints (3.2) enforce the restrictions on the availability of supplies and

constraints (3.3) consider limited capacity of warehouses. Finally, constraints (3.4) determine

the type of decision variables.

Stage 2- distribution plans and demand satisfaction decisions

In the second stage, the amount of distributed supplies between each warehouse and

destination (qijk(δ)) as well as unsatisfied demands (yjk(δ)) are determined for each disaster

scenario by solving the following weighted sum model:

(3.5)

s.t.

(3.6)

(3.7)

(3.8)

(3.9)

The objective function of the second stage (3.5) is to minimize the weighted sum of total

transportation time and the penalty cost for unfulfilled demand. Constraints (3.6) ensure that

the total distributed amount of each supply from each warehouse is limited to the

corresponding inventory level. Furthermore, constraints (3.7) maintain the balance between

inventory levels, distributed amounts and unfulfilled demands. Constraints (3.8) enforce the

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total penalty cost for each type of supply to be smaller than a threshold value, . Finally,

constraints (3.9) guarantee non-negativity of variables.

Consequently, the model provides the best locations for warehouses and their inventory

levels for different supplies as well as the required transportation amounts from warehouses

to affected areas and the corresponding unfulfilled demands at each scenario.

4. Solution procedure

We adopt an effective possibilistic approach proposed by Jimenez et al. (2007) along with

the decomposition approach of Zhang (2001) to transform the original FSBSP model into an

equivalent crisp one. Details of this process are provided in the following subsections.

4. 1. Defuzzification method

In the proposed FSBSP, the parameters are fuzzy in each scenario. Therefore, we could

assume that the model for each scenario is a fuzzy programming problem which has to be

converted to a crisp model. Subsequently, the stochastic part could also be changed to a crisp

model by applying appropriate strategies which will be presented in the next subsection. In

this subsection, initially we describe the Jimenez et al. (2007) method which is used for

defuzzifying the linear programming problem.

As a basis, assume that we have a fuzzy number ( ) with the following membership

function ( ):

(4.1)

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Now, we use two definitions made by Heilpern (1992) as the expected interval and the

expected value of a fuzzy number , denoted by and respectively as follow:

(4.2)

(4.3)

To compare two fuzzy numbers, the fuzzy ranking method proposed by Jimenez (1996) is

used due to its efficiency, strong mathematical foundation and generality (for more details

see Pishvaee and Torabi, 2010). In this regards, the following definitions are useful:

Definition 1. For any pair of fuzzy numbers and , the degree in which is bigger than is

as follows (Jimenez, 1996):

(4.4)

where

and

are the expected intervals of and , respectively. It can be

said that is bigger than, or equal to, at least in a degree when and it is

represented by .

Definition 2. A decision vector is feasible in degree (or -feasible) if (Jimenez et

al., 2007):

(4.5)

where .

In other words, could be resulted by the equation (4.4)

obviously:

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(4.6)

or in fact:

(4.7)

For equivalent constraints, we could say if:

and (4.8)

Similarity, they could be rewritten as:

(4.9)

Similarly, for a symmetric triangular fuzzy number (TFN) as

: instead of the equations (4.7) and (4.9), we could have the following ones:

(4.10)

(4.11)

4. 2. Decomposition approach for two-stage stochastic programming

The model which is known as two-stage stochastic programming, involves decisions at

two steps: first the decisions for the whole model considering all of scenarios and their

possibilities, and then the decisions are made based on the occurrence of each specific

scenario and the specified first stage decisions (Zhang, 2001). For more explanation of this

method, initially we consider the simple model of two-stage stochastic programming:

Stage 1- (4.12)

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s.t.

Stage 2- (4.13)

s.t.

where w shows the random nature of variables.

To solve this model; the expected value of the second stage objective function is first

considered with the first stage's objective function and it is solved as a common linear

programming (LP) as follows:

s.t.

(4.14)

Afterwards, as the decision vector in the first step (X) is determined, the second stage

problem (4.13) is solved with the given X and specified scenario (w) to determine the

corresponding .

5. Case study

5. 1. Implementation

In order to demonstrate the proposed approach, a case study of an earthquake in the Seattle

area (Fig. 1) is provided. This case study which is adapted from Mete and Zabinsky (2010), is

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based on discussions with an Emergency Management Coordinator of a large Seattle medical

center. For simplicity, only a single type of medical supplies is considered here.

The Seattle fault and the Cascadia fault are the two main factors which cause the

earthquakes in the southern part and the northern part of Seattle area, respectively. The

medical demands were estimated in different parts of the city and at different times of the day

based on the impact of earthquakes and population around hospitals. Three periods of time

were considered in a day: working hours (W), rush hours (R), and non-working hours (N).

Consequently, six disaster scenarios were produced as given in Table 1. Table 2 provides

estimated demand to ten hospitals (destinations) in each scenario. There are five possible

warehouses whose capacities and operating costs are represented in Table 3. Table 4 provides

the transportation times between each pair of warehouse and destination respecting to each

disaster scenario. It should be mentioned that the numbers in the above tables represent the

center of respected fuzzy numbers with symmetrical parts being equivalent to ten percent of

the central values.

Fig. 1. Schematic locations of hospitals and possible warehouses in the Seattle area

1

2

3

4

5

6

7

8

9

10

3

1

5

2

4

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Table 1. Probabilities of scenarios (Mete and Zabinsky, 2010)

Scenario Seattle fault Cascadia fault

W R N W R N

Probability 0.11 0.07 0.22 0.17 0.11 0.32

Table 2. The center of symmetric fuzzy demands (adapted from Mete and

Zabinsky, 2010)

Hospital Seattle fault Cascadia fault

W R N W R N

1 6313 6042 9491 9234 8306 13,624

2 3409 3857 3994 5296 3958 7149

3 4969 3732 6466 5922 5147 9357

4 1532 3454 4254 5422 7114 7507

5 2293 3487 4836 7185 8750 10,258

6 3129 2508 2913 3801 1814 2112

7 10,021 5932 3869 12,410 6830 7639

8 7342 4617 4213 9134 3803 5924

9 5723 3686 1773 6784 4036 4382

10 5214 3498 2189 6048 3006 3861

Table 3. The center of symmetric fuzzy warehouse capacities and operating costs

(adapted

from Mete and Zabinsky, 2010)

Warehouse Capacity (103 units) Cost ($106) Cost/capacity ($103/ unit)

1 20 25 1.25

2 25 20 0.80

3 30 12 0.40 4 10 6 0.60

5 5 12 2.40

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Table 4. The center of symmetric fuzzy transportation times for scenarios (adapted

from Mete and Zabinsky, 2010)

Warehouse Hospital Seattle fault

Cascadia fault

W R N W R N

1

1 77 210 44 44 90 11

2 105 210 60 60 90 15

3 27 27 18 18 18 9

4 15 15 10 10 10 5

5 105 210 60 60 90 15

6 112 210 64 64 90 16

7 147 245 84 84 105 21

8 18 18 12 12 12 6

9 24 24 16 16 16 8

10 18 18 12 12 12 6

2

1 20 40 10 30 60 20

2 14 14 7 21 21 14

3 133 133 76 76 57 19

4 126 245 72 72 105 18

5 26 26 13 39 39 26

6 32 50 16 48 75 32

7 42 60 21 63 90 42

8 133 245 76 76 105 19

9 140 245 80 80 105 20

10 119 245 68 68 105 17

3

1 98 245 56 56 105 14

2 112 175 64 64 75 16

3 112 245 64 64 105 16

4 98 245 56 56 105 14

5 14 14 7 21 21 14

6 8 8 4 12 12 8

7 24 24 12 36 36 24

8 45 105 30 30 70 15

9 51 105 34 34 70 17

10 15 15 10 10 10 5

4

1 24 24 12 36 36 24

2 34 50 17 51 75 34

3 119 119 68 68 51 17

4 119 119 68 68 51 17

5 34 34 17 51 51 34

6 30 50 15 45 75 30

7 40 70 20 60 105 40

8 54 90 36 36 60 18

9 57 105 38 38 70 19

10 51 51 34 34 34 17

5

1 147 210 84 84 90 21

2 56 56 28 84 84 56

3 154 154 88 88 66 22

4 66 66 44 44 44 22

5 108 81 27 189 189 108

6 96 72 24 168 168 96

7 48 36 12 84 84 48

8 69 69 46 46 46 23

9 75 75 50 50 50 25

10 63 90 42 42 60 21

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It should be noted that for a fair comparison, we extended the original SP model of Mete

and Zabinsky (2010) to an extended weighted model (ESP) by considering different weights

for total transportation time and the penalty cost of unfulfilled demands similar to the

equation (3.5). Both models (i.e., the proposed FSBSP and extended SP model) have been

coded in GAMS and solved by using the OSL solver using a PC with Intel Dual Core CPU,

2.53 GHz using 4 GB of RAM. The basic results are shown in Table 5 in which the weight

vector and feasibility degree level (α) are set to (0.5, 0.5) and 0.8, respectively.

Table 5. Comparison of solutions in the two approaches

Model Selected

warehouse Inventory level

(103 units) Objective value

(106 units) unfulfilled

demand

FSBSP

1 19.4

55.770 0 2 24.25

3 29.1

ESP

1 20

57.511 0 2 25

3 30

According to Table 5; the warehouses 1, 2 and 3 are selected in both models. Although the

warehouse 1 has the second highest cost/capacity ratio, it is selected because it is very close

to the down part hospitals. The lowest cost/capacity ratio and the location of warehouse 3 (in

the northern part) are the reasons of its selection. The other main variable which plays a vital

role in HRC models is unfulfilled demand which is zero in all scenarios and for all hospitals.

In addition, the objective function value in the FSBSP model is less than that of ESP

model which can be related to the flexibility of the proposed FSBSP model. Flexibility is

resulted from assigning distributed supplies and satisfying demand smoothly. Furthermore,

fuzzy parameters and soft constraints which are presented by considering feasibility level (α)

are the other aspects of flexibility. Moreover, highly utilization of warehouses' capacity and

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fewer inventory levels are the other reasons regarding better performance of the FSBSP

model from the cost reduction viewpoint.

5. 2. Evaluation method

To compare the FSBSP and ESP models, the following indicators are adapted from Torabi

et al. (2010) and Mula et al. (2008):

1. Objective function value

2. Maximum inventory level

3. Unfulfilled demand

4. Computational efficiency involving the following sub-criteria:

– number of iterations to find the optimal solution;

– number of continuous variables;

– number of integer variables;

– number of constraints;

– CPU run time (in seconds); and

– work space allocated.

In this comparison, lower levels of the indicators 1-3 are favored. In addition, the higher

levels of indicator 4 (as the minimum values of respective sub-indicators) would be preferred.

To evaluate the model, 15 test problems are generated which are close to the parameters of

the case study with the feasibility degree (α) being set to 0.8. The results reveal that the

warehouses 1, 2 and 3 are still selected in all test problems.

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Furthermore, a sensitivity analysis for the weight vector (w1, w2) has been conducted. The

mean results of all test problems for each vector are shown in Table 6. Table 7 presents the

results of other indicators.

Table 6. Sensitivity analysis regarding the weights in the extended SP and FSBSP models

Weights (w1,w2) Model Objective value (10

6 units)

Computational efficiency

Iterations CPU time (in seconds)

(0.1, 0.9) FSBSP 55.393 383 0.0872

ESP 57.106 367 0.0609

(0.2, 0.8) FSBSP 55.496 381 0.0656

ESP 57.213 366 0.0539

(0.3, 0.7) FSBSP 55.598 377 0.0462

ESP 57.319 369 0.0368

(0.4, 0.6) FSBSP 55.701 383 0.0652

ESP 57.425 366 0.0334

(0.5, 0.5) FSBSP 55.804 373 0.0394

ESP 57.531 343 0.0338

(0.6, 0.4) FSBSP 55.907 375 0.0419

ESP 57.638 351 0.0362

(0.7, 0.3) FSBSP 56.010 361 0.0516

ESP 57.744 338 0.0344

(0.8, 0.2) FSBSP 56.113 362 0.0419

ESP 57.850 343 0.0286

(0.9, 0.1) FSBSP 56.215 387 0.0546

ESP 57.956 375 0.0317

Mean results FSBSP 55.804 376 0.0549

ESP 57.531 358 0.0387

The results show that the impact of changing weights on the objective function values is

negligible, at least in our experiment. Therefore, the decision maker could consider the

preferred weights without worrying too much about the effects of weights. In this case, the

weights are set to (0.5, 0.5) arbitrarily as decision maker's preference.

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Table 7. The complementary comparative results

Model Maximum

inventory level (10

3 units)

Unfulfilled

demand

Computational efficiency

Continuous variables

Integer variables

Constraints Allocated memory

(MB)

FSBSP 29.1 0 377 5 223 0.78

ESP 30 0 377 5 163 0.68

Nonetheless, the results show that the objective function value in the FSBSP model is less

(i.e., better) than that of the extended SP one in all cases which could be interpreted by

smooth assignment of distributed supplies and soft constraints. Also, the maximum inventory

level in the FSBSP model is less than that of the ESP which could be indicated as the better

utilization of capacity. Moreover, the demands of all hospitals (destinations) are entirely

satisfied in both models.

Regarding the computational efficiency, the number of continuous and integer variables

are equal for both models. On the other hand, the number of constraints in the FSBSP model

is more than the ESP model and consequently, the number of iterations, allocated work space

and CPU time are increased. Overall, it could be concluded that the computational efficiency

indicator is slightly better in the ESP model.

It is also worth noting that the model flexibility is another important comparison factor

which is not considered here because of its qualitative nature. However, flexibility of the

FSBSP model through modeling the input data such as demand, storage capacity and

transportation time as fuzzy numbers and also considering the soft constraints instead of hard

ones provide enough flexibility which, in turn, can decrease infeasibility and inconsistency

which are the main disadvantages of crisp models in general. Based on the above

comparisons, it could be concluded that the proposed fuzzy scenario-based stochastic model

is more realistic than the ESP model at least for the concerned HRC scenario.

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6. Conclusion

While international reports state that the number and spread of natural disasters is

progressively increasing, the amount of quantitative methods in regards to logistical HRC

problems is surprisingly scarce. This paper addresses pre-positioning and distribution of

emergency supplies in a HRC. It proposes a novel two-stage fuzzy scenario-based stochastic

model which simultaneously accounts for the two major sources of uncertainty, i.e.,

randomness and fuzziness. The original FSBSP model is transformed into an equivalent crisp

model which can be solved efficiently with the current optimization packages.

The proposed model is implemented on a case study reported in the literature. The

numerical results indicate the superiority of the proposed FSBSP model when compared with

an extended version of the current approach (ESP model) in terms of different indicators.

The proposed FSBSP approach could be considered in the other phases of HRCs

especially in the response phase as a direction of further research. Moreover, since such a

problem in practice is of large-scale type which leads to a computationally intractable

problem, presenting some heuristic/metaheuristic methods seems to be necessary.

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