Dynamic Supplier Selection and Its Optimal Strategy Considering Full

Truck Load and Fuzzy Demand using Fuzzy Expected Value-Based

Programming

PURNAWAN ADI WICAKSONO1, SUTRISNO2

1Department of Industrial Engineering,

Diponegoro University,

Jalan Prof. Soedarto, SH Tembalang, 50275 Semarang,

INDONESIA

2Department of Mathematics,

Diponegoro University,

Jalan Prof. Soedarto, SH Tembalang, 50275 Semarang,

INDONESIA

Abstract: - This article proposes a linear integer optimization model incorporating fuzzy parameters to find the

optimal solution for a dynamic supplier selection problem with uncertain demand. The uncertain demand value

is represented using a fuzzy variable. A fuzzy expected value-based linear optimization solver is used to

address the optimization problem, to minimize the total cost under fuzzy demand values. Several computational

experiments were conducted to evaluate and analyze the model. The results show that the proposed model

effectively identifies the optimal suppliers for each product. Additionally, the model determines the optimal

purchase volumes for each product type from the selected suppliers, leading to the minimal total expected cost.

Key-Words: - dynamic supplier selection problem (DSSP), full truck load, fuzzy demand, fuzzy parameters,

linear programming, order allocation, supply chain management.

Received: March 25, 2024. Revised: August 17, 2024. Accepted: September 19, 2024. Published: November 13, 2024.

1 Introduction

A manufacturing company commonly faces supplier

selection problems that involve determining the

optimal or best suppliers from several possible

alternatives. The optimal decision must satisfy the

demand constraints and provide the best quality of

purchased products or services to the manufacturer,

[1]. In large business organizations, the supplier

selection process often involves multiple products,

multiple periods, and multiple suppliers, while

maintaining quality and lead time. This is known as

a Dynamic Supplier Selection Problem (DSSP) [2],

[3] whereas a simpler problem is referred to as the

Traditional Supplier Selection Problem (TSSP). The

main difference between them is that the DSSP

approach is more realistic than the TSSP due to its

consideration of parameter dynamics over time, [4],

[5]. The impact of transportation costs on DSSP is

very significant, [6]. When a buyer splits orders

among multiple suppliers, the delivery quantities

from the suppliers to the buyer result in higher

transportation costs. However, many researchers

addressing supplier selection problems do not

consider transportation costs in their proposed

models. They usually include transportation costs

within the product price. Therefore, including

transportation costs when determining order

quantities in DSSP is important to improve

efficiency in the supply chain process, [7].

The demand value at the present time in a

supplier selection problem is commonly known with

certainty, but future demand is typically uncertain.

The optimal decision on which supplier to select and

the quantities to order from each supplier under

uncertain demand is clearly more challenging. For

certain demands, most researchers have used

mathematical models to minimize total cost, [8], [9].

There are several other approaches to solving

supplier selection problems, such as risk-optimizing

approaches, integrated supplier selection, inventory

management, and risk management, [10], [11], [12].

For supplier selection problems considering

uncertain parameters, several research papers have

been developed, most of which were solved using

stochastic programming, [13], [14], [15], [16].

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In uncertainty theory, an uncertain value can be

approached in two ways: through frequency

generated by samples (historical data or trials) and

belief degrees evaluated by the .decision-maker,

[17]. The frequency approach uses probability

theory and is applicable when samples are available,

as they can be used to determine the probability

distribution. Unfortunately, in many cases, no

samples are available to estimate the probability

distribution. In such cases, belief degree theory can

be used to estimate the uncertain value of a variable.

The simplest form of the belief degree approach is

by using a fuzzy variable. An uncertain value can be

represented by a membership function defined by

the decision maker. If an optimization problem

contains at least one fuzzy variable (or parameter),

then fuzzy programming can be used to solve it.

Fuzzy programming offers a powerful method

for handling optimization problems with fuzzy

variables. All forms of optimization, such as linear

programming, quadratic programming, and general

nonlinear programming, can be handled when they

involve fuzzy parameters, [18], [19]. A general

model of fuzzy linear optimization can be

interpreted as a linear optimization problem in

which some or all of the parameters are fuzzy

variables or fuzzy numbers. There are several

special cases: (1) the objective is crisp, (2) some or

all constraints are crisp, (3) some or all constraints

are soft constraints or a combination of these. In this

paper, type (3), where the objective function is crisp

and some constraints are soft constraints, will be

used to select the optimal supplier when the demand

value is fuzzy. To solve this, we can use the fuzzy

expected value-based approach, [17].

In this article, we propose a mathematical model

for dynamic supplier selection with a full truckload

transport scheme under uncertain demand values in

a linear fuzzy optimization framework. The fuzzy

expected value-based approach is used to solve the

minimization problem. Numerical experiments will

be presented to demonstrate how the problem is

solved using the proposed approach.

2 Materials and Methods

The developed model considers the multi-product,

multi-supplier, and multi-period cases. We

introduce the notations used in the model below:

indices:

P

:

Set of product sets

S

:

Set of suppliers

T

:

Set of time periods

Decision variables:

tsp

X

:

Amount (unit) of product p purchased

from supplier s in time period t

ts

Z

:

The binary number that represents

whether the supplier s is charged for

order cost at period t, it will be 1 if yes

or 0 if not

s

W

:

Binary number representing whether

supplier s is chosen as a new supplier

(1) or not (0)

ts

S

:

The number of truck that delivers

product from supplier s in period t

tp

i

:

Inventory level of product p in time

period t

tp

i

:

Shortage level of product p in time

period t

Parameters:

sp

UP

:

Unit price of product p at supplier s in

each time period

s

TC

:

FTL cost from supplier s to the buyer

in each time period

s

NC

:

Cost occurs when a new supplier is

selected to be contracted.

tp

SOC

:

Cost of shortage unit product p in time

period t

C

:

Full truck load maximum capacity

tp

D

:

Demand value (unit) of product p in

period t

tsp

SC

:

Maximum capacity of supplier s to

supply product p in period t

sp

l

:

Late on delivery rate (in percentage) of

ordered product from suppliers of

product p

sp

de

:

Percentage of rejected product p from

supplier s

l

p

P

:

Penalty cost for late delivery of

product p in each time period

d

p

P

:

Penalty cost for defective product p

s

O

:

Cost that occurred while ordering

products from supplier s

p

h

:

Cost for storing a unit product p per

one time period

tp

MS

:

Maximum warehouse capacity to store

product p in time period t

tp



:

Service level value of supplier in time

period t for product p. The value (1-



)

means the proportion of unsatisfied

demand.

Figure 1 shows the solution procedure

implemented in this study. In the first 3 steps, the

DM has a significant role especially since he has to

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define the membership function value for each

fuzzy parameter. This defining process is using

intuition from the DM based on his experience.

Let

tp

D

be the fuzzy variable of the demand

value of product p in review period t. The solution

must satisfy the demand value, meaning the total

purchased product must be greater than or equal to

the demand. However, if the demand is uncertain

and represented by a fuzzy variable, the purchase

quantity must be greater than the fuzzy variable of

the demand. This condition can be referred to as a

not-well-defined problem since the feasible set does

not produce a crisp value. As a result, the optimal

strategy cannot be determined as a crisp value. To

address this, the authors approach the crisp value of

the fuzzy demand by using the fuzzy expected

value. In fact, there are several formulas to calculate

the expected value of a fuzzy number. Our proposed

approach used the expectation value defined in [20]

where in any time period t and for each product p,

the expectation of

tp

D

is given by:

   

0

tp tp tp

E D Cr D r dr Cr D r dr







   





(1)

provided that at least one result of these two integral

terms is finite where

 

Cr 

is denoting the

credibility value.

Fig. 1: The solution procedure

The formula can be used to calculate the

expectation of any fuzzy number according to its

membership function. For a special case where the

discrete fuzzy number/variable



having the

membership function given by:

11

22

, if

()

, if

mm

xx

x

xx





















(2)

with x1, x2, …, xm distinct and

1 2 1mm

x x x x



   

, the expectation of



will be

 

1

mii

i

E w x







(3)

where

1

21 1 1

max max max max

i j j j j

j i j i i j m j m

w

   

       



   





(4)

for

1,2, , .im

For a trapezoidal fuzzy number

 

, , ,T a b c d

, the expectation value is

.

4

a b c d

ET   







(5)

In the supplier selection problem with fuzzy

demand that we are discussing, the objective is to

minimize the total procurement cost, with

constraints to satisfy the fuzzy demand and other

related conditions. The mathematical model is as

follows:

1 1 1 1 1

1 1 1 1

1 1 1

min T S P T S

tsp sp s ts

t s p t s

T S T P

s s p tp

t s t p

T P T S

p tp s ts

t p t s

T S P d

p sp tsp

t s p

T S P l

p sp tsp

t s p

Z X UP O Z

NC W h i

SOC i TC S

P de X

P l X

    



   



   

  

   

   

  

 

 



(6)

Subject to:

( 1) ( 1)

11

( 1)

11

, , ;

SS

t p tp tsp ps t sp

ss

SS

ps tsp ps tsp t p

ss

tp tp

i i X l X

l X d X i

i E D t T p P













  

  



     





(7)

, , ;

(1 )

tp tp

tp

iE D p P t T







    





(8)

The Decision Maker (DM) identifies the fuzzy

parameters appeared in the problem

Calculating the expected value of each fuzzy

parameter

DM selects the membership function form

(discrete or trapezoidal) for each fuzzy

parameter

Substituting all parameters to the mathematical

model

Solving the optimization and determining the

optimal strategy

DM defines the membership function value for

each fuzzy parameter

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1, , ;

P

s

ptp

ts

X

s S t TS

C





   









(9)

, , , ;

tsp tsp

X SC p P s S t T      

(10)

1

, , ;

P

tsp ts

p

X M Z s S t T



     



(11)

, , ;

tp tp

i MS p P t T

    

(12)

1

,;

T

ts s

t

Z M W s S



   



(13)

, , integer;

tsp tp tp

X i i



(14)

 

, 0,1 .

ts s

ZW

(15)

The objective function Z represents the total

operational cost whereas constraints (7)-(15) can be

explained respectively as follows. The first

constraint is used to manage the inventory and for

demand satisfaction. The second one is the service

level requirement whereas the third one is the full

truck load condition. The fourth to the ninth

constraints respectively represent the supplier

capacity, ordering cost, storage capacity, new

supplier indicator, integer constraint, and binary

constraint for the decision variables.

3 Results and Discussion

3.1 Results

In this section, we evaluate the optimization model

(6) using data given in Table 1, Table 2, Table 3,

Table 4, Table 5 and Table 6. We consider three

products, four suppliers over a planning horizon of

ten periods. Table 1 provides the unit price (𝑈𝑃𝑡𝑠𝑝)

that is offered by each supplier. Table 2 shows the

supplier capacity of each supplier. Table 3 presents

parameter values related to products i.e. the values

of storage capacities, defect penalties, late delivery

penalties, holding costs, and shortage costs. Table 4

provides parameter values related to suppliers that

consist of ordering costs, contract costs and

transportation costs. Defect rates at all supplier are

shown in Table 5. Table 6 presents late rates at all

suppliers.

Table 1. Unit price for all time periods

Supplier

Products

P1

P2

P3

S1

40

82

61

S2

42

83

62

S3

41

82

62

S4

41

81

61

Table 2. Supplier capacity for all time periods

Supplier

Products

P1

P2

P3

S1

1200

400

750

S2

1000

350

650

S3

950

300

800

S4

900

450

850

Table 3. Product’s parameter value for all periods

Parameter

P1

P2

P3

Storage capacity (unit)

1200

1000

Defect penalty ($)

1

2

1

Late delivery penalty ($)

0.5

0.01

0.02

Holding cost ($)

0.2

0.8

0.4

Shortage cost ($)

1

2

Table 4. Suppliers’ parameters in all periods

Supplier

Ordering cost

Contract

cost

Transportation

cost

S1

12

45

120

S2

10

50

120

S3

14

45

120

S4

12

40

120

Table 5. Defect rates in all time periods

Supplier

P1

P2

P3

S1

0.04

0.02

0.03

S2

0.04

0.00

S3

0.04

0.00

0.05

S4

0.03

0.02

0.05

Table 6. Late rates in all periods

Supplier

P1

P2

P3

S1

0.02

0.01

0.03

S2

0.00

0.04

0.05

S3

0.02

0.00

S4

0.04

0.03

0.02

Example 1 (Discrete membership function).

Suppose a manufacturer faces a supplier selection

problem involving three products: P1, P2, and P3,

and four suppliers: S1, S2, S3, and S4, where the

demand value for all products is uncertain. Assume

that the decision-maker deals with uncertainty in

demand values, which can be represented by fuzzy

variables, with the membership functions being

discrete and defined by:

1,4, 1,2,3

0.25 if 480;0.40 if 490;

0.70 if 510;1.00 if 530;

0.90 if 550;0.88 if 570;

0.75 if 590;0.60 if 610;

0.50 if 630;0.45 if 650;

tp

tp tp

D

tp tp

DD











  













,

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2,5, 1,2,3

0.25 if 130;0.40 if 140;

0.70 if 150;1.00 if 160;

0.90 if 170;0.88 if 180;

0.75 if 190;0.60 if 200;

0.50 if 210;0.45 if 220;

tp

tp tp

D

tp tp

DD











  













,

3, 1,2,3

0.25 if 370;0.40 if 395;

0.70 if 410;1.00 if 430;

0.90 if 450;0.88 if 460;

0.75 if 465;0.60 if 475;

0.50 if 480;0.45 if 490;

tp

tp tp

D

tp tp

DD











  













,

6,9, 1,2,3

0.22 if 430;0.58 if 440;

0.65 if 460;0.75 if 480;

1.00 if 500;0.99 if 520;

0.62 if 540;0.50 if 560;

0.35 if 580;0.25 if 600;

tp

tp tp

D

tp tp

DD











  













7,10, 1,2,3

0.25 if 80;0.54 if 90;

0.60 if 100;0.72 if 110;

0.85 if 120;0.92 if 130;

1.00 if 140;0.85 if 150;

0.75 if 160;0.40 if 170;

tp

tp tp

D

tp tp

DD











  













,

8, 1,23

0.10 if 320;0.54 if 345;

0.72 if 370;0.75 if 380;

1.00 if 400;0.90 if 410;

0.80 if 415;0.65 if 425;

0.52 if 430;0.45 if 440.

tp

tp tp

D

tp tp

DD











  













.

We solved (6) for 10 time periods in LINGO®

17.0 with a daily used personal computer with the

Operating System Windows 8, RAM 4 GB, and

Processor AMD A6 2.7 GHz. The solution or the

optimal decision obtained by the calculation is

shown in Figure 2 (Appendix). It describes the

optimal solution, which specifies the amount of each

product that the manufacturer should purchase from

each corresponding supplier to achieve the

minimum expected total cost. For example, in

period 1, P1 is supplied with 423 units by Supplier

1, 70 units by Supplier 3, and 71 units by Supplier 4.

For P2, the orders in period 1 are split as 51 units to

Supplier 1, 4 units to Supplier 2, and 121 units to

Supplier 3. Orders for P3 are fulfilled with 126 units

from Supplier 1, 326 units from Supplier 2, and 8

units from Supplier 3. The total cost for this solution

is $ 613,468.

Example 2 (Trapezoidal membership function).

For this example, let

tp

D

be the fuzzy demand value

with a trapezoidal membership function illustrated

by Figure 3 (Appendix) where the values of

,,,

tp tp tp tp

a b c d

are given in Table 7. By evaluating

the optimization problem (6) over 10 time periods,

where the demand is represented by a trapezoidal

membership function shown in Figure 3

(Appendix), we derive the optimal strategy

illustrated in Figure 3 (Appendix). The optimal

strategy consists of the unit volumes of all products

that the manufacturer should order from each

supplier in each time period (1, 2, …, 10) to achieve

the minimum expected total cost. From Figure 4

(Appendix), we can see that in time period 1, 273

units of product P1 and 1 unit of product P2 should

be purchased from Supplier 1, 180 units of P3 must

be purchased from Supplier 2, and 1 unit of P1 and

76 units of P2 must be purchased from Supplier 4.

The optimal strategies for each subsequent time

period (2, 3, …, 10) can be obtained from Figure 4

(Appendix). The expected total cost for all time

periods is 259,288.

3.2 Discussion

From the two examples, we can draw several

interpretations. In the first example, the decision

maker needs to specify certain discrete demand

values where the membership values are positive,

while other discrete demand values have

membership values of zero. In the second example,

the decision maker must determine the membership

values of the demand, which are assumed to follow

a line segment in the corresponding piecewise linear

trapezoidal function. The first example, which uses

a discrete membership function, is easier to apply

since the decision maker only needs to determine

the demand and its membership values. In contrast,

the second example requires the decision maker to

specify the lower bound with a membership value of

0, the mid-lower bound and mid-upper bound with

membership values of 1, and the upper bound with a

membership value of 0 in the trapezoidal function.

This means that the membership values for demand

between these points are not decided by the decision

maker, as they will follow the trapezoidal function

shown in Figure 3 (Appendix). Consequently, this

approach may not fully represent the real conditions

of the problem.

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Table 7. Trapezoidal membership functions of

tp

D

Period

Product

Trapezoidal fuzzy

membership function for

tp

D

i.e.

 

, , ,

tp tp tp tp tp

Da b c d





Expectation

Value

tp

ED





tp

a

tp

b

tp

c

tp

d

1

P1

100

200

250

350

225

P2

40

60

70

120

72.5

P3

50

150

200

280

170

2

P1

100

200

250

350

225

P2

40

60

70

120

72.5

P3

50

150

200

280

170

3

P1

100

200

250

350

225

P2

40

60

70

120

72.5

P3

50

150

200

280

170

4

P1

100

200

250

350

225

P2

40

60

70

120

72.5

P3

50

150

200

280

170

5

P1

100

200

250

350

225

P2

40

60

70

120

72.5

P3

50

150

200

280

170

6

P1

100

200

250

350

225

P2

40

60

70

120

72.5

P3

50

150

200

280

170

7

P1

100

200

250

350

225

P2

40

60

70

120

72.5

P3

50

150

200

280

170

8

P1

100

200

250

350

225

P2

40

60

70

120

72.5

P3

50

150

200

280

170

9

P1

100

200

250

350

225

P2

40

60

70

120

72.5

P3

50

150

200

280

170

10

P1

100

200

250

350

225

P2

40

60

70

120

72.5

P3

50

150

200

280

170

4 Conclusion

A dynamic supplier selection problem with a full

truckload transport scheme and fuzzy demand was

considered. A fuzzy expected value-based approach

for fuzzy optimization was formulated and

successfully used to determine the optimal decisions

and calculate the optimal product volumes that the

manufacturer should purchase from the selected

suppliers for all time periods. The solution

procedure involves the following steps: first, the

decision maker (DM) identifies the fuzzy

parameters; second, the DM defines the membership

function values for each fuzzy parameter; third, the

expectation for all fuzzy parameters is calculated;

fourth, these values are substituted into the

formulated model; and finally, the corresponding

linear programming problem is solved using

LINGO. Two numerical examples were considered.

For all given problems, the proposed approach

successfully obtained the optimal decisions, i.e., the

optimal product volumes for all time periods from

each supplier were determined with minimal

expected total cost. The decision maker can then use

and apply these optimal decisions to the

manufacturing system.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed to the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

This work was supported by DRPM

KEMENRISTEK Indonesia under Penelitian

Terapan research grant contract no. 257-

99/UN7.P4.3/PP/2019.

Conflict of Interest

The authors have no conflicts of interest to declare.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

_US

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DOI: 10.37394/23202.2024.23.30

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E-ISSN: 2224-2678

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APPENDIX

Fig. 2: Optimal product volume for time periods 1 to 10

Fig. 3: Trapezoidal membership function

 

,,,

tp tp tp tp tp

Da b c d





Fig. 4: Optimal product volume for Example 2

tp

a

tp

D



 

,,,

tp tp tp tp tp

Da b c d





tp

d

tp

b

tp

D

1

tp

c

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DOI: 10.37394/23202.2024.23.30

Purnawan Adi Wicaksono, Sutrisno

E-ISSN: 2224-2678

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Volume 23, 2024