Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 4s (2024) 

 

540 
https://internationalpubls.com 

Analyze the Stochastic Solid Fuzzy Transportation Problem with 

Mixed Constraints through Weibull Distribution 

 

Rashi Arya1, Dr. Vipin Kumar2, Dr. Abhinav Saxena3  
1,2,3Department of Mathematics, Faculty of Engineering, Teerthanker Mahaveer University, Moradabad, India 

 

Article History: 

Received: 17-04-2024 

Revised: 07-06-2024 

Accepted: 17-06-2024 

Abstract:  

This paper proposed a general formulation of the stochastic solid transportation 

problem (SSTP) with mixed constraints such as supply, demand and conveyance 

capacity taken as uncertain under stochastic environment, following the Weibull 

distribution (WD). The aim of this study is to minimize the transportation cost 

includes probabilistic constraints have inequalities of stochastic solid transportation 

problem (SSTP). SSTP with probabilistic constraints is represented as a chance 

constrained programming problem. Obtain alpha cut representation from cost 

coefficient of the fuzzy objective function. We have developed four models for 

stochastic solid transportation problem. The suggested models are demonstrated by 

taken as numerical example. A sensitivity analysis is performed to understand 

parameter’s sensitivity in the proposed model. 

Introduction:  In system of transportation, goods are moved from various sources 

to destinations using different vehicles and organizational systems, involving both 

technology and human efforts. Efficient resource allocation in transportation system 

is crucial for industries and imprecision from factors like fluctuating demand, 

unreliable supply chains and unpredictable traffic. To address these complexities, 

advanced mathematical models are needed to manage stochasticity, fuzziness and 

mixed constraints. The study explores the stochastic solid fuzzy transportation 

problem with mixed constraint by utilizing the Weibull distribution to model 

uncertainties inherent in transportation systems. This research addresses the 

complexity introduced by stochastic variables and fuzzy parameters, particularly in 

situations where demand, supply and cost of transportation are not deterministic.     

Objectives: The aim of this study is to minimize the cost of transportation includes 

probabilistic constraints have inequalities of stochastic solid transportation problem 

(SSTP).  

Methods: Obtain alpha cut representation form the cost coefficient of the fuzzy 

objective function and four models are developed for stochastic solid transportation 

problem. These models are demonstrating by using a numerical example and a 

sensitivity analysis is conducted to understand the sensitivity of the parameters in 

the propose model. 

Results: Obtained optimal solutions for developed four models of SFSTPMC and 

sensitivity analysis shows that cost of transportation and flow of unit are sensitive to 

change in probabilities of demand. Improve transportation system by understanding 

sensitivity patterns that help decision maker choose appropriate supply availability 

probabilities.  

Conclusions: This study presented an approach for solving the SFSTPMC using the 

Weibull distribution for probabilistic constraints and fuzzy objective functions for 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 4s (2024) 

 

541 
https://internationalpubls.com 

transportation cost. Developed and optimized four models, focusing on stochastic 

parameters. Sensitivity analysis demonstrated the impact of these parameters on 

transportation cost and unit flow. The results validate the model’s effectiveness in 

practical resource allocation and decision making under uncertainty.  

Keywords: SFSTPMC (Stochastic Fuzzy Solid Transportation Problem with Mixed 

Constraints), Weibull distribution, mixed constraints.  

1. Introduction 

In transportation, goods are carrying from various sources to various destinations by using types of 

vehicles within various organizational systems. It encompasses not just various technologies such as 

vehicles, energy, and infrastructure but also involves the time and efforts of individuals. In essence, 

the transportation problem is a specific category within linear programming, linked to everyday 

activities in our lives, primarily focusing on logistic. Issues arise during distribution or transportation 

of goods between different locations, this help in optimizing these issues. To fulfil specific 

requirements, items are transported from various sources to various destinations. The objective is to 

fulfil destination’s demand by minimizing cost of transportation while adhering to the limitations of 

the exits supply.  

[1] introduced the transportation problem. System of transportation includes mixed transportation 

modes for delivery of items, it might be possible to minimize cost of transportation through various 

transportation ways. In such type of cases solid transportation is appropriate which includes supply, 

demand and conveyance capacity. Develop a solution methodology to solve solid transportation 

problem and comparison between solid transportation problem and classical transportation problem 

done by [2], [3]. H. Isermann proposed an approach to solve mixed constraint solid transportation 

problem. [4] firstly stated fuzzy set theory. [5] suggested a solution methodology to solve solid 

transportation problem including mixed constrain in different environment. [6] formulated several 

models and solution methodologies of fixed charge fuzzy stochastic solid transportation problem. [7] 

proposed solution procedure of multi-objective solid transportation problem with multiple items. [8] 

developed a new heuristic approach to obtain general initial results to optimize transportation problem 

and to demonstrate their method they use 35 examples. [9] they showed that expected cost 

minimization and chance constraint uncertain model can be converted into their deterministic form 

through standard optimization solver Gurobi. [10] suggested new approach to optimize solid 

transportation problem included multiple choices cost, supply as stochastic and demand. [11] 

formulated solid transportation problem with fixed charge under budget constraint by using genetic 

algorithm and swarm optimization. [12] hybrid intelligent algorithm to solve the constructed three 

models for solid transportation problem with fixed charge under uncertainty. [13] proposed different 

process to obtain optimal solution of transportation problem with mixed constraint. [14] proposed 

solution methodology to obtain minimum cost of transportation problem including special cases of 

fixed transportation problem along truck load constraint. [15] proposed constraint programming model 

and four mixed integer linear programming models to optimize shop scheduling problem with 

distributed flexible job. [16] suggested a better approach to optimize fuzzy transportation problem with 

cost, demand and supply as fuzzy parameters. [17] developed a new algorithm of fuzzy transportation 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 4s (2024) 

 

542 
https://internationalpubls.com 

problem to obtain optimal solution. [18] suggested solution methodology to solve multi-objective 

stochastic solid transportation and obtain minimum cost and time. [19] their study aims to introduce a 

robust mechanism for addressing transportation challenges of solid transportation problem. [20] 

proposed a methodology to optimize intuitionistic transportation problem through specially multiply 

two fuzzy intuitionistic transportation issue.   

The efficient allocation of resources in transportation systems is paramount for smooth functioning of 

various industries and economies. However, real world transportation scenarios often involve 

uncertainty and imprecision due to factors such as fluctuating demand, unreliable supply chains, and 

unpredictable traffic conditions. Addressing these complexities requires sophisticated mathematical 

models capable of handling stochasticity, fuzziness, and mixed constraints.  

On the basis of above information, the main contribution of this paper is described in section wise. 

Section-2 presents assumptions and notations used in this research. Section-3 represents preliminaries 

which are relevant to research. Section-4 formulate the problem of SFSTPMC and transform the 

uncertain mathematical models into equivalent deterministic models. Section-5 described by the 

SFSTPMC Models. Section-6 develop an algorithm for optimization of generated models and also 

illustrate a numerical example to demonstrate the effectiveness of develop algorithm. Section-7 

evaluates the computational results of sensitivity analysis. Section-8 summarize the research findings 

and in last section-9 shows the limitations and future scope of the research. 

2. Assumptions and notations 

SFSTPMC (Stochastic Fuzzy Solid Transportation Problem with Mixed Constraints) is associated with 

cost, supply, demand and conveyance capacity. Here source and destination are considered as mixed 

constraints. To establish mathematical models for SFSTPMC, subsequent symbols are presented as 

below: 

m: number of sources for supply.  

n: number of destinations for demands. 

l: number of various different conveyance. 

𝑎𝑖: quantity of homogeneous items available at source i. 

𝑏𝑗: quantity of homogeneous items demands at destination j.  

𝑒𝑘: quantity of conveyance capacity for the kth modes of transport. 

�̃�𝑖𝑗𝑘: per unit cost of transportation of items from source i to destination j through kth modes of 

transportation.  

𝑥𝑖𝑗𝑘: quantity of items shipping from source i to destination j through kth modes of transportation.  

𝑃𝑎𝑖 : possibilities associated with 𝑎𝑖. 

𝑃𝑏𝑖: possibilities associated with 𝑏𝑗. 

𝑃𝑒𝑘: possibilities associated with 𝑒𝑘. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 4s (2024) 

 

543 
https://internationalpubls.com 

𝛾𝑎𝑖: For 𝑎𝑖 parameter of shape which following WD. 

𝛾𝑏𝑗: For 𝑏𝑗 parameter of shape which following WD. 

𝛾𝑒𝑘: For 𝑒𝑘 parameter of shape which following WD. 

𝛼𝑎𝑖: For 𝑎𝑖  parameter of scale which following WD. 

𝛼𝑏𝑗: For 𝑏𝑗  parameter of scale which following WD. 

𝛼𝑒𝑘: For 𝑒𝑘  parameter of scale which following WD. 

𝛽𝑎𝑖: For 𝑎𝑖  parameters of location which following WD. 

𝛽𝑏𝑗: For 𝑏𝑗  parameters of location which following WD. 

𝛽𝑒𝑘: For 𝑒𝑘  parameters of location which following WD. 

3. Preliminaries 

In 1965 Zadeh gives improved fuzzy theory. Mathematical techniques were proposed from fuzzy 

theory with fuzzy concepts and problems containing many possible optimal solutions. 

• Let us consider S be a collection of sets, µ𝑆(𝑥) is a function from R to [1,0]. A fuzzy set �̃� along 

with membership function  µ𝑆(𝑥) is defined by  

�̃� = {(𝑥, µ𝑆(𝑥); 𝑥 ∈ 𝐴 𝑎𝑛𝑑 µ𝑆(𝑥) ∈ [0,1]}. 

• Triangular Fuzzy number (TFN) is the type of fuzzy number, a continuous mapping: µ�̃�(𝑥): 𝑅 →

[0,1] . Its membership function µ�̃�(𝑥) in Figure 1 as 

µ�̃�(𝑥) =

{
 
 

 
 
0,        𝑓𝑜𝑟 −∞ < 𝑥 < 𝑞1,
𝑥−𝑞1

𝑞2−𝑞1
,    𝑓𝑜𝑟 𝑞1 ≤ 𝑥 ≤ 𝑞2,

𝑞3−𝑥

𝑞3−𝑞2
,    𝑓𝑜𝑟 𝑞2 ≤ 𝑥 ≤ 𝑞3,

0,            𝑓𝑜𝑟 𝑞3 ≤ 𝑥 ≤ ∞.

                                                                                  (1) 

 

µ𝑠(𝑥) 

                1 

 

 

 

0 

                  𝑞1                       𝑞2                     𝑞3          x 

Figure 1. Triangular Fuzzy number (TFN) 

• Here �̃� is a fuzzy number which is subset of the R real number with membership function µ�̃� fulfil 

the following conditions: 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 4s (2024) 

 

544 
https://internationalpubls.com 

1. µ�̃� is a continuous function from 𝑅 → [0,1]. 

2. µ�̃� is equal to zero ∀ (−∞, 𝑞1]. 

3. µ�̃� is equal to zero ∀ [𝑞3, ∞). 

4. µ�̃� is strictly increasing on [𝑞1, 𝑞2]. 

5. µ�̃� is strictly decreasing on [𝑞2, 𝑞3]. 

6. µ�̃� is equal to one ∀ 𝑞 ∈ 𝑞2 where 𝑞1 ≤ 𝑞2 ≤ 𝑞3. 

• A fuzzy set S is defined on X and 𝛼 ∈ [0,1], then the alpha cut is defined as  

𝑆(𝛼) = {𝑥/µ(x) ≥ 𝛼}. 

• A linear membership function can be defined as follows 

    µ𝑆(𝑋) =

{
 

 
0,        𝑖𝑓 𝑥𝑖𝑗𝑘 ≤ 𝑥𝑖𝑗𝑘,

𝑥𝑖𝑗𝑘−𝑥𝑖𝑗𝑘

𝑥𝑖𝑗𝑘−𝑥𝑖𝑗𝑘
,    𝑖𝑓 𝑥𝑖𝑗𝑘 < 𝑥𝑖𝑗𝑘 < 𝑥𝑖𝑗𝑘 ,

1,            𝑖𝑓 𝑥𝑖𝑗𝑘 > 𝑥𝑖𝑗𝑘.

                                                                         (2) 

To obtain crisp set from the fuzzy system t, 𝛼-cut for linear membership function can be written as ∀ 

𝛼 ∈ [0,1], (
𝑥𝑖𝑗𝑘−𝑥𝑖𝑗𝑘

𝑥𝑖𝑗𝑘−𝑥𝑖𝑗𝑘
) = 𝛼 

Such that 𝑥𝑖𝑗𝑘 = (1 − 𝛼)𝑥𝑖𝑗𝑘 + 𝑥𝑖𝑗𝑘                                                                                          (3) 

 

4. Problem’s Formulation and Optimal Solution 

Our main objective is minimization of cost of transportation for SFSTPMC problem. In practical 

scenarios, uncertainties arise in parameters like cost, cost, supply, demand, and conveyance capacity 

which creates challenges for decision makers in achieving optimal solutions. To address this, fuzzy 

and random variables are employed. Cost is treated as triangular fuzzy variables and constraints are 

consider as random variables in this model. On the basis of conditions of decision makers, occurrence 

of uncertainty in demand, supply and conveyance capacity may be occur. On the basis constraint’s 

uncertainty formulate four SFSTPMC models.  

4.1. Weibull Distribution  

Weibull distribution of three parameter is followed by probability density function and cumulative 

density function of random variable t given as 

𝑓(𝑡) =
𝛾

𝛿
(
𝑡−𝛽

𝛿
)𝛾−1𝑒−(

𝑡−𝛽

𝛿
)𝛾 ,                                                                                                      (4)                                          

and 𝐹(𝑡) = 1 − 𝑒−(
𝑡−𝛽

𝛿
)
𝛾

,                                                                                                                      (5) 

Here ( ) 0, 0f t t  or  . Note that shape parameter 𝛾 > 0, scale parameter 𝛿 > 0 and location 

parameter −∞ < 𝛽 < ∞. 

4.2.  Chance Constraint Programming  

The closed form of quantiles for a probability distribution function can be achieved by applying the 

constraint from the proposed SP model to the deterministic constraints. The utilization of the Weibull 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 4s (2024) 

 

545 
https://internationalpubls.com 

distribution is also motivated by its closed form for quartiles. This paper outlines the CCP model for 

SFSTPMC as: 

(R) Minimize �̃� = ∑ ∑ ∑ �̃�𝑖𝑗𝑘𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑛
𝑗=1

𝑚
𝑖=1  

Subject to 

1 1

( ) ,
i

n l

ijk i a

j k

P x a P
= =

                𝑖 ∈ 𝑟1,                                                                                       (6)

1 1

( ) ,
i

n l

ijk i a

j k

P x a P
= =

=                𝑖 ∈ 𝑟2,                                                                                      (7)

1 1

( ) ,
i

n l

ijk i a

j k

P x a P
= =

                𝑖 ∈ 𝑟3,                                                                                        (8) 

1 1

( ) ,
j

m l

ijk j b

i k

P x b P
= =

                𝑗 ∈ 𝑞1,                                                                                       (9)

1 1

( ) ,
j

m l

ijk j b

i k

P x b P
= =

=                𝑗 ∈ 𝑞2,                                                                                   (10)

1 1

( ) ,
j

m l

ijk j b

i k

P x b P
= =

                𝑗 ∈ 𝑞3,                                                                                                 (11) 

1 1

( ) ,
k

m n

ijk k e

i j

P x e P
= =

                𝑘 ∈ 𝑢1,                                                                                                (12) 

1 1

( ) ,
k

m n

ijk k e

i j

P x e P
= =

=                𝑘 ∈ 𝑢2,                                                                                                (13) 

1 1

( ) ,
k

m n

ijk k e

i j

P x e P
= =

                𝑘 ∈ 𝑢3,                                                                                                 (14)                                                                 

𝑥𝑖𝑗𝑘 ≥ 0,     ∀ 𝑖 = 1, 2, . . . , 𝑚, 𝑗 = 1, 2, . . . , 𝑛 and 𝑘 = 1, 2, . . . , 𝑙                                                           (15)       

Where 𝑃𝑎𝑖 , 𝑃𝑏𝑖and 𝑃𝑒𝑘 are the given probabilities. Assumed that these random variables (𝑎𝑖, 𝑏𝑗  and 𝑒𝑘) 

follows the Weibull distribution. The Weibull distribution for 𝑎𝑖 characterized by three parameters: 

ia as shape parameter, 
ia as scale parameter, and 

ia as location parameters. Similarly, for 𝑏𝑗 and 𝑒𝑘 

Weibull distribution also have their respective specified parameters. Constraints (6) to (8) represents 

the probabilistic constraints for the quantity of supply at source i. These constraints establish, with a 

specified probability 𝑃𝑎𝑖 , the total quantities of shipping of products from supply origin i must be 

distribute either be exactly ia units, at least ia units, or at most ia units. Here 𝑟1, 𝑟2, and  𝑟3 are partitions 

of i, where i takes values from 1 to m. Similarly, constraints (9) to (11) represents the demand at 

destination j must be receive either be exactly jb units, at least jb units, or at most jb units. Here 𝑞1, 𝑞2, 

and  𝑞3are partitions of j, where j takes values from 1 to n. Similarly, constraint (12) to (14) represents 

required conveyance capacity for k modes of transportation either be exactly 𝑒𝑘 units, at least 𝑒𝑘 units, 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 4s (2024) 

 

546 
https://internationalpubls.com 

or at most 𝑒𝑘 units. Here are partitions of k, where k takes values from 1 to l. Fuzzy transportation cost 

for transportation of goods from i origins to j destinations, represented as �̃�𝑖𝑗𝑘. 

 In order to satisfy mixed type of constraints such as supply and demand, the goal is to characterise the 

quantity 𝑥𝑖𝑗𝑘 delivered from i origin to destination j while reducing the total transportation cost. In 

three cases we consider only one random variable as uncertain among 𝑎𝑖, 𝑏𝑗 and 𝑒𝑘. In fourth case we 

consider all these random variables (𝑎𝑖, 𝑏𝑗 and 𝑒𝑘) are uncertain. 

4.2.1. Case I: - Only 𝒂𝒊 (i is equal to 1 to m) is uncertain 

(i) Here proof for 𝑃(∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑛
𝑗=1 ≥ 𝑎𝑖) ≥ 𝑃𝑎𝑖 , 𝑖 ∈ 𝑟1 shown as follows. 

It is considered that independent random variable 𝑎𝑖 follows the WD along with three parameters 
ia , 

ia , and 
ia .  Equation (6) can be reconstructed as  

1 1

( )
i

n l

i ijk a

j k

P a x P
= =

  ,   𝑖 = 𝑟1.                                                                                                            (16) 

Assuming that 
1 1

,
i

n l

ijk a

j k

x 
= =

= equation (16) can be represented as ( ) ,
i ii a aP a P  1i ,r  for 

probability function we use Weibull distribution which can be written as  

∫
𝛿𝑎𝑖

𝛾𝑎𝑖

𝜉𝑎𝑖
−∞

(
𝑎𝑖−𝛽𝑎𝑖

𝛾𝑎𝑖
)
𝛿𝑎𝑖−1

𝑒
{−(

𝑎𝑖−𝛽𝑎𝑖
𝛾𝑎𝑖

)

𝛿𝑎𝑖
}

𝑑𝑎𝑖 ≥ 𝑃𝑎𝑖 ,                   𝑖 ∈ 𝑟1                                                        (17)  

Simultaneously, 𝛽𝑎𝑖 ≤ 𝑎𝑖, 𝑖 ∈ 𝑟1, equation (17)’s integration yields the following form: 

 ∫
𝛿𝑎𝑖

𝛾𝑎𝑖

𝜉𝑎𝑖
𝛽𝑎𝑖

(
𝑎𝑖−𝛽𝑎𝑖

𝛾𝑎𝑖
)
𝛿𝑎𝑖−1

𝑒
{−(

𝑎𝑖−𝛽𝑎𝑖
𝛾𝑎𝑖

)

𝛿𝑎𝑖
}

𝑑𝑎𝑖 ≥ 𝑃𝑎𝑖 ,                   𝑖 ∈ 𝑟1                                                       (18)  

After solving equation (18), we get 

1 − 𝑒
{−(

𝜉𝑎𝑖
−𝛽𝑎𝑖
𝛾𝑎𝑖

)

𝛿𝑎𝑖
}

≥ 𝑃𝑎𝑖 .                                                                                                                 (19) 

Logarithm applied on equation (19) then we get, 

𝜉𝑎𝑖 ≥ 𝛽𝑎𝑖 + 𝛾𝑎𝑖{− ln(1 − 𝑃𝑎𝑖)}
1

𝛿𝑎𝑖 ,       𝑖 ∈ 𝑟1. Then, this can be expressed as deterministic constraint:  

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑛
𝑗=1 ≥ 𝛽𝑎𝑖 + 𝛾𝑎𝑖{− ln(1 − 𝑃𝑎𝑖)}

1

𝛿𝑎𝑖 , 𝑖 ∈ 𝑟1.         

Here proof of 𝑃(∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑛
𝑗=1 ≤ 𝑎𝑖) ≥ 𝑃𝑎𝑖 , 𝑖 = 𝑟3, 1 2( 1, 2,..., )i m m m= + + for obtaining the 

deterministic values from probabilistic values through utilizing Weibull distribution was obtained in 

[18].  

 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 4s (2024) 

 

547 
https://internationalpubls.com 

4.2.2. Case II: - Only 𝒃𝒋 (i is equal to 1 to n) is uncertain 

(i) Proof of 𝑃(∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑚
𝑖=1 ≥ 𝑏𝑗) ≥ 𝑃𝑏𝑗 , 𝑗 ∈ 𝑞1, (𝑗 = 1, 2, 3, … , 𝑛) for obtaining the deterministic 

values from probabilistic values through utilizing Weibull distribution was obtained in [18]. 

(ii) Proof of 𝑃(∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑚
𝑖=1 ≤ 𝑏𝑗) ≥ 𝑃𝑏𝑗 , 𝑗 ∈ 𝑞3, shows as below 

It is considered that independent random variable 𝑏𝑗 follows the WD along with three parameters 
jb , 

jb , and 
jb .  Equation (11) can be reconstructed as  

1 1

(b )
j

m l

j ijk b

i k

P x P
= =

  ,        𝑗 = 𝑞3.                                                                                                     (20) 

Assuming that 
1 1

,
j

m l

ijk b

i k

x 
= =

= equation (20) can be represented as (b ) ,
j jj b bP P  3 ,j q  for 

probability function we use Weibull distribution which can be written as  

∫
𝛿𝑏𝑗

𝛾𝑏𝑗

∞

𝜉𝑏𝑗
(
𝑏𝑗−𝛽𝑏𝑗

𝛾𝑏𝑗
)

𝛿𝑏𝑗−1

𝑒
{−(

𝑏𝑗−𝛽𝑏𝑗

𝛾𝑏𝑗

)

𝛿𝑏𝑗
}

𝑑𝑏𝑗 ≥ 𝑃𝑏𝑗 ,                   𝑗 ∈ 𝑞3                                                      (21)  

After solving equation (21), we get 

𝑒
{−(

𝜉𝑏𝑗
−𝛽𝑎𝑗

𝛾𝑏𝑗

)

𝛿𝑏𝑗
}

≥ 𝑃𝑏𝑗 .                                                                                                                        (22) 

Logarithm applied on equation (22) then we get, 

𝜉𝑏𝑗 ≤ 𝛽𝑏𝑗 + 𝛾𝑏𝑗 {− ln (𝑃𝑏𝑗)}

1

𝛿𝑏𝑗 ,       𝑗 ∈ 𝑞3. Then, this can be expressed as deterministic constraint:  

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑚
𝑖=1 ≤ 𝛽𝑏𝑗 + 𝛾𝑏𝑗 {− ln (𝑃𝑏𝑗)}

1

𝛿𝑏𝑗 , 𝑗 ∈ 𝑞3.      

4.2.3. Case III: - Only 𝒆𝒌 (k is equal to 1 to l) is uncertain 

(i) Proof of 𝑃(∑ ∑ 𝑥𝑖𝑗𝑘
𝑛
𝑗=1

𝑚
𝑖=1 ≥ 𝑒𝑘) ≥ 𝑃𝑒𝑘 , 𝑘 ∈ 𝑢1 shows as below 

It is considered that independent random variable 𝑏𝑗 follows the WD along with three parameters 
ke , 

ke , and 
ke .  Equation (12) can be reconstructed as  

1 1

(e )
k

m n

k ijk e

i j

P x P
= =

  ,                      (𝑘 ∈ 𝑢1)                                                                                       (23) 

Assuming that 
1 1

,
k

m n

ijk e

i j

x 
= =

= equation (23) can be represented as (e ) ,
k kk e eP P  𝑘 ∈ 𝑢1 for 

probability function we use Weibull distribution which can be written as  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 4s (2024) 

 

548 
https://internationalpubls.com 

∫
𝛿𝑒𝑘

𝛾𝑒𝑘

𝜉𝑒𝑘
−∞

(
𝑒𝑘−𝛽𝑒𝑘

𝛾𝑒𝑘
)
𝛿𝑏𝑘−1

𝑒
{−(

𝑒𝑘−𝛽𝑒𝑘
𝛾𝑒𝑘

)

𝛿𝑒𝑘
}

𝑑𝑒𝑘 ≥ 𝑃𝑒𝑘 ,                    𝑘 ∈ 𝑢1                                                   (24)  

Simultaneously, 𝛽𝑒𝑘 ≤ 𝑒𝑘, 𝑘 ∈ 𝑢1, equation (24)’s integration yields the following form: 

 ∫
𝛿𝑒𝑘

𝛾𝑒𝑘

𝜉𝑒𝑘
𝛽𝑒𝑘

(
𝑒𝑘−𝛽𝑒𝑘

𝛾𝑒𝑘
)
𝛿𝑒𝑘−1

𝑒
{−(

𝑒𝑘−𝛽𝑒𝑘
𝛾𝑒𝑘

)

𝛿𝑒𝑘
}

𝑑𝑒𝑘 ≥ 𝑃𝑒𝑘 ,                   𝑘 ∈ 𝑢1                                                   (25)  

After solving equation (25), we get 

1 − 𝑒
{−(

𝜉𝑒𝑘
−𝛽𝑒𝑘
𝛾𝑒𝑘

)

𝛿𝑒𝑘
}

≥ 𝑃𝑒𝑘 .                                                                                                                    (26)                                                                 

Logarithm applied on equation (26) then we get, 

𝜉𝑒𝑘 ≥ 𝛽𝑒𝑘 + 𝛾𝑒𝑘{− ln(1 − 𝑃𝑒𝑘)}
1

𝛿𝑒𝑘 , 𝑘 ∈ 𝑢1. Then, this can be expressed as deterministic constraint:  

∑ ∑ 𝑥𝑖𝑗𝑘
𝑛
𝑗=1

𝑚
𝑖=1 ≤ 𝛽𝑒𝑘 + 𝛾𝑒𝑘{− ln(1 − 𝑃𝑒𝑘)}

1

𝛿𝑒𝑘 , 𝑘 ∈ 𝑢1.                                                                       (27)                               

(ii) Proof of 𝑃(∑ ∑ 𝑥𝑖𝑗𝑘
𝑛
𝑗=1

𝑚
𝑖=1 ≤ 𝑒𝑘) ≥ 𝑃𝑒𝑘 , 𝑘 ∈ 𝑢3 shows as below 

It is considered that independent random variable 𝑏𝑗 follows the WD along with three parameters 
ke , 

ke , and 
ke . Equation (14) can be reconstructed as  

1 1

(e )
k

m n

k ijk e

i j

P x P
= =

  ,                      , 𝑘 ∈ 𝑢3                                                                                            (28) 

Assuming that 
1 1

,
k

m n

ijk e

i j

x 
= =

= equation (28) can be represented as (e ) ,
k kk e eP P    𝑘 ∈ 𝑢3 for 

probability function we use Weibull distribution which can be written as  

∫
𝛿𝑒𝑘

𝛾𝑒𝑘

∞

𝜉𝑒𝑘
(
𝑒𝑘−𝛽𝑒𝑘

𝛾𝑒𝑘
)
𝛿𝑒𝑘−1

𝑒
{−(

𝑒𝑘−𝛽𝑒𝑘
𝛾𝑒𝑘

)

𝛿𝑒𝑘
}

𝑑𝑒𝑘 ≥ 𝑃𝑒𝑘 ,    𝑘 ∈ 𝑢3                                                                   (29)  

After solving equation (29), we get 

𝑒
{−(

𝜉𝑒−𝛽𝑒𝑘
𝛾𝑒𝑘

)

𝛿𝑒𝑘
}

≥ 𝑃𝑒𝑘 .                                                                                                                            (30)                                                                                                  

Logarithm applied on equation (30) then we get, 

𝜉𝑒𝑘 ≤ 𝛽𝑒𝑘 + 𝛾𝑒𝑘{− ln(𝑃𝑒𝑘)}
1

𝛿𝑒𝑘 . Then, this can be expressed as deterministic constraint:  

∑ ∑ 𝑥𝑖𝑗𝑘
𝑛
𝑗=1

𝑚
𝑖=1 ≤ 𝛽𝑒𝑘 + 𝛾𝑒𝑘{− ln(𝑃𝑒𝑘)}

1

𝛿𝑒𝑘  , 𝑘 ∈ 𝑢3.                                                                           (31) 

 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 4s (2024) 

 

549 
https://internationalpubls.com 

4.2.4. Case IV: Equality constraint   

Assuming these types of equality supply, demand and conveyance capacity constraints

1 1

( ) ,
i

n l

ijk i a

j k

P x a P
= =

=   2 1 1 21, 2,..., ,i r m m m = + +  
1 1

( ) ,
j

m l

ijk j b

i k

P x b P
= =

=  2 1 1 21,n 2,..., nj q n = + +  

and 
1 1

( ) ,
k

m n

ijk k e

i j

P x e P
= =

=   𝑘 ∈ 𝑢2 = 𝑙1 + 1, 𝑙1 + 2,… , 𝑙2. Change these equality type constraints to 

inequality type of constraints, for supply constraints can be written as 𝑃(∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑛
𝑗=1 ≥ 𝑎𝑖) ≥ 𝑃𝑎𝑖 

and 𝑃(∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑛
𝑗=1 ≤ 𝑎𝑖) ≥ 𝑃𝑎𝑖, for the demand constraints can be written  as  𝑃(∑ ∑ 𝑥𝑖𝑗𝑘

𝑙
𝑘=1

𝑚
𝑖=1 ≥

𝑏𝑗) ≥ 𝑃𝑏𝑗 and 𝑃(∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑚
𝑖=1 ≤ 𝑏𝑗) ≥ 𝑃𝑏𝑗, for the conveyance capacity constraints can be written 

as 𝑃(∑ ∑ 𝑥𝑖𝑗𝑘
𝑛
𝑗=1

𝑚
𝑖=1 ≥ 𝑒𝑘) ≥ 𝑃𝑒𝑘 and 𝑃(∑ ∑ 𝑥𝑖𝑗𝑘

𝑛
𝑗=1

𝑚
𝑖=1 ≤ 𝑒𝑘) ≥ 𝑃𝑒𝑘 . Selecting the probability value 

at the 50-percentage level within the supply and demand inequality constraint by utilizing Weibull 

distribution results in obtaining an identical deterministic value for both types of inequalities in supply 

and demand constraints. Then the equality constraints for supply can be written in deterministic form 

as ∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑛
𝑗=1 ≥ 𝛽𝑎𝑖 + 𝛾𝑎𝑖{− ln(1 − 𝑃𝑎𝑖)}

1

𝛿𝑎𝑖 , 𝑖 ∈ 𝑟2 or ∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑛
𝑗=1 ≥ 𝛽𝑎𝑖 +

𝛾𝑎𝑖{− ln(𝑃𝑎𝑖)}
1

𝛿𝑎𝑖 , 𝑖 ∈ 𝑟2, for demand constraints can be written as ∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑚
𝑖=1 ≤ 𝛽𝑏𝑗 +

𝛾𝑏𝑗 {− ln (𝑃𝑏𝑗)}

1

𝛿𝑏𝑗 , 𝑗 ∈ 𝑞2 or ∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑚
𝑖=1 ≤ 𝛽𝑏𝑗 + 𝛾𝑏𝑗 {− ln (1 − 𝑃𝑏𝑗)}

1

𝛿𝑏𝑗 , 𝑗 ∈ 𝑞2 and for the 

conveyance capacity constraints can be written as 𝜉𝑒𝑘 ≥ 𝛽𝑒𝑘 + 𝛾𝑒𝑘{− ln(1 − 𝑃𝑒𝑘)}
1

𝛿𝑒𝑘 , 𝑘 ∈ 𝑢2 or 

𝜉𝑒𝑘 ≤ 𝛽𝑒𝑘 + 𝛾𝑒𝑘{− ln(𝑃𝑒𝑘)}
1

𝛿𝑒𝑘 , 𝑘 ∈ 𝑢2. 

5. MODELLING 

In this section, deterministic mathematical programming models for SFSTPMC are introduced. These 

models incorporate the quantiles of the Weibull distribution as constraints and utilize a fuzzy linear 

membership function for the cost function. In practical situations, only specific elements of supply and 

demand might be subject to uncertainty and other parameters as certain. Therefore, deterministic form 

of SFSTPMC can be adjusted on the basis of circumstances. There are three models in this section. In 

model 1, 2 and 3 includes any one of the parameters such as 𝑎𝑖, 𝑏𝑗 , 𝑜𝑟 𝑒𝑘 as uncertain respectively. In 

model 3 introduces a general model in where all random variables are subject to uncertainty. 

Model 1: Modelling for deterministic SFSTPMC can be converted to stochastic one when supply is 

uncertain, demand and conveyance capacity constraints as certain. 

(R1) Minz = ∑ ∑ ∑ 𝑐𝑖𝑗𝑘 ((1 − 𝛼)𝑥𝑖𝑗𝑘 + 𝛼𝑥𝑖𝑗𝑘)
𝑙
𝑘=1

𝑛
𝑗=1

𝑚
𝑖=1 .                                                                   (32) 

Subject to 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑛
𝑗=1 ≥ 𝛽𝑎𝑖 + 𝛾𝑎𝑖{− ln(1 − 𝑃𝑎𝑖)}

1

𝛿𝑎𝑖 ,        𝑖 ∈ 𝑟1,                                                                   (33) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 4s (2024) 

 

550 
https://internationalpubls.com 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑛
𝑗=1 = 𝛽𝑎𝑖 + 𝛾𝑎𝑖{− ln(1 − 𝑃𝑎𝑖)}

1

𝛿𝑎𝑖 ,       𝑖 ∈ 𝑟2,                                                                    (34) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑛
𝑗=1 ≤ 𝛽𝑎𝑖 + 𝛾𝑎𝑖{− ln(𝑃𝑎𝑖)}

1

𝛿𝑎𝑖 ,        𝑖 ∈ 𝑟3,                                                                           (35) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑚
𝑖=1 ≥ 𝑏𝑗 ,           𝑗 ∈ 𝑞1,                                                                                                         (36) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑚
𝑖=1 = 𝑏𝑗 ,           𝑗 ∈ 𝑞2,                                                                                                          (37) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑚
𝑖=1 ≤ 𝑏𝑗 ,           𝑗 ∈ 𝑞3,                                                                                                        (38) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑛
𝑗=1

𝑚
𝑖=1 ≥ 𝑒𝑘,           𝑘 ∈ 𝑢1,                                                                                                        (39) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑛
𝑗=1

𝑚
𝑖=1 = 𝑒𝑘,           𝑘 ∈ 𝑢2,                                                                                                         (40) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑛
𝑗=1

𝑚
𝑖=1 ≤ 𝑒𝑘,           𝑘 ∈ 𝑢3,                                                                                                        (41)   

𝑥𝑖𝑗𝑘 ≥ 0, 𝑖, 𝑗 𝑎𝑛𝑑 𝑘,                                                                                                                               (42) 

Model 2: Modelling for deterministic SFSTPMC can be converted to stochastic one when demand is 

uncertain, supply and conveyance capacity constraints as certain. 

(R2) Minz = ∑ ∑ ∑ 𝑐𝑖𝑗𝑘 ((1 − 𝛼)𝑥𝑖𝑗𝑘 + 𝛼𝑥𝑖𝑗𝑘)
𝑙
𝑘=1

𝑛
𝑗=1

𝑚
𝑖=1 .                                                                       (43) 

Subject to 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑛
𝑗=1 ≥ 𝑎𝑖,        𝑖 ∈ 𝑟1,                                                                                                           (44) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑛
𝑗=1 = 𝑎𝑖,       𝑖 ∈ 𝑟2,                                                                                                             (45) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑛
𝑗=1 ≤ 𝑎𝑖,        𝑖 ∈ 𝑟3,                                                                                                              (46) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑚
𝑖=1 ≥ 𝛽𝑏𝑗 + 𝛾𝑏𝑗 {− ln (1 − 𝑃𝑏𝑗)}

1

𝛿𝑏𝑗 ,           𝑗 ∈ 𝑞1,                                                          (47) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑚
𝑖=1 = 𝛽𝑏𝑗 + 𝛾𝑏𝑗 {− ln (𝑃𝑏𝑗)}

1

𝛿𝑗 ,           𝑗 ∈ 𝑞2,                                                                      (48) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑚
𝑖=1 ≤ 𝛽𝑏𝑗 + 𝛾𝑏𝑗 {− ln (𝑃𝑏𝑗)}

1

𝛿𝑗 ,           𝑗 ∈ 𝑞3,                                                                       (49) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑛
𝑗=1

𝑚
𝑖=1 ≥ 𝑒𝑘,           𝑘 ∈ 𝑢1                                                                                                        (50) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑛
𝑗=1

𝑚
𝑖=1 = 𝑒𝑘,           𝑘 ∈ 𝑢2                                                                                                        (51) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑛
𝑗=1

𝑚
𝑖=1 ≤ 𝑒𝑘,           𝑘 ∈ 𝑢3                                                                                                        (52)   

𝑥𝑖𝑗𝑘 ≥ 0, 𝑖, 𝑗 𝑎𝑛𝑑 𝑘,                                                                                                                              (53) 

Model 3: Modelling for deterministic SFSTPMC can be converted to stochastic one when conveyance 

capacity is uncertain, supply and demand constraints as certain. 

(R3) Minz = ∑ ∑ ∑ 𝑐𝑖𝑗𝑘 ((1 − 𝛼)𝑥𝑖𝑗𝑘 + 𝛼𝑥𝑖𝑗𝑘)
𝑙
𝑘=1

𝑛
𝑗=1

𝑚
𝑖=1 .                                                                    (54) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 4s (2024) 

 

551 
https://internationalpubls.com 

Subject to 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑛
𝑗=1 ≥ 𝑎𝑖,         𝑖 ∈ 𝑟1,                                                                                                            (55) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑛
𝑗=1 = 𝑎𝑖,         𝑖 ∈ 𝑟2,                                                                                                          (56) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑛
𝑗=1 ≤ 𝑎𝑖,         𝑖 ∈ 𝑟3,                                                                                                         (57) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑚
𝑖=1 ≥ 𝑏𝑗 ,          𝑗 ∈ 𝑞1,                                                                                                           (58) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑚
𝑖=1 = 𝑏𝑗 ,          𝑗 ∈ 𝑞2,                                                                                                        (59) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑚
𝑖=1 ≤ 𝑏𝑗 ,          𝑗 ∈ 𝑞3,                                                                                                            (60) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑛
𝑗=1

𝑚
𝑖=1 ≥ 𝛽𝑒𝑘 + 𝛾𝑒𝑘{− ln(1 − 𝑃𝑒𝑘)}

1

𝛿𝑒𝑘 , 𝑘 ∈ 𝑢1,                                                                             (61) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑛
𝑗=1

𝑚
𝑖=1 = 𝛽𝑒𝑘 + 𝛾𝑒𝑘{− ln(1 − 𝑃𝑒𝑘)}

1

𝛿𝑒𝑘 , 𝑘 ∈ 𝑢2,                                                                           (62) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑛
𝑗=1

𝑚
𝑖=1 ≤ 𝛽𝑒𝑘 + 𝛾𝑒𝑘{− ln(𝑃𝑒𝑘)}

1

𝛿𝑒𝑘 ,            𝑘 ∈ 𝑢3,                                                                             (63) 

𝑥𝑖𝑗𝑘 ≥ 0, 𝑖, 𝑗 𝑎𝑛𝑑 𝑘,                                                                                                                                  (64) 

Model 4: Modelling for deterministic SFSTPMC can be converted to stochastic one when supply, 

demand and conveyance capacity constraints as uncertain. 

(R4) Minz = ∑ ∑ ∑ 𝑐𝑖𝑗𝑘 ((1 − 𝛼)𝑥𝑖𝑗𝑘 + 𝛼𝑥𝑖𝑗𝑘)
𝑙
𝑘=1

𝑛
𝑗=1

𝑚
𝑖=1 .                                                               (65) 

Subject to 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑛
𝑗=1 ≥ 𝛽𝑎𝑖 + 𝛾𝑎𝑖{− ln(1 − 𝑃𝑎𝑖)}

1

𝛿𝑎𝑖 ,         𝑖 ∈ 𝑟1,                                                                 (66) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑛
𝑗=1 = 𝛽𝑎𝑖 + 𝛾𝑎𝑖{− ln(1 − 𝑃𝑎𝑖)}

1

𝛿𝑎𝑖 ,         𝑖 ∈ 𝑟2,                                                                   (67) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑛
𝑗=1 ≤ 𝛽𝑎𝑖 + 𝛾𝑎𝑖{− ln(𝑃𝑎𝑖)}

1

𝛿𝑎𝑖 ,         𝑖 ∈ 𝑟3,                                                                             (68) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑚
𝑖=1 ≥ 𝛽𝑏𝑗 + 𝛾𝑏𝑗 {− ln (1 − 𝑃𝑏𝑗)}

1

𝛿𝑏𝑗 ,          𝑗 ∈ 𝑞1,                                                            (69) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑚
𝑖=1 = 𝛽𝑏𝑗 + 𝛾𝑏𝑗 {− ln (𝑃𝑏𝑗)}

1

𝛿𝑏𝑗 ,          𝑗 ∈ 𝑞2,                                                                     (70) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑙
𝑘=1

𝑚
𝑖=1 ≤ 𝛽𝑏𝑗 + 𝛾𝑏𝑗 {− ln (𝑃𝑏𝑗)}

1

𝛿𝑏𝑗 ,          𝑗 ∈ 𝑞3,                                                                        (71) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑛
𝑗=1

𝑚
𝑖=1 ≥ 𝛽𝑒𝑘 + 𝛾𝑒𝑘{− ln(1 − 𝑃𝑒𝑘)}

1

𝛿𝑒𝑘 , 𝑘 ∈ 𝑢1,                                                                     (72) 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑛
𝑗=1

𝑚
𝑖=1 = 𝛽𝑒𝑘 + 𝛾𝑒𝑘{− ln(1 − 𝑃𝑒𝑘)}

1

𝛿𝑒𝑘 , 𝑘 ∈ 𝑢2,                                                                          (73) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 4s (2024) 

 

552 
https://internationalpubls.com 

∑ ∑ 𝑥𝑖𝑗𝑘
𝑛
𝑗=1

𝑚
𝑖=1 ≤ 𝛽𝑒𝑘 + 𝛾𝑒𝑘{− ln(𝑃𝑒𝑘)}

1

𝛿𝑒𝑘 ,            𝑘 ∈ 𝑢3,                                                                        (74) 

𝑥𝑖𝑗𝑘 ≥ 0, 𝑖, 𝑗 𝑎𝑛𝑑 𝑘,                                                                                                                                  (75) 

6. Algorithm for Optimization   

Method for optimize the SFSTPMC model is shows as below: 

Step 1: - By utilizing alpha cut method, we convert cost which is given as triangular fuzzy problem 

into deterministic equivalent form.  

Step 2: - Then, get 4 models by transformed problem and its constraints. 

Step 3: - In model 1 we take supply constraint 𝑎𝑖 as uncertain which is probabilistic, 𝑒𝑘 & 𝑏𝑗 as certain 

constrains. To obtain deterministic form of 𝑎𝑖 supply constraints (33) to (35), we use Weibull 

distribution. With the minimum deterministic cost, deterministic supply constraints (33) to (35), certain 

demand constraint (36) to (38) and certain conveyance capacity constraint (39) to (41), formulate the 

problem (R1). To get the optimal cost of transportation and flow of unit we optimize the problem (R1) 

at zero level of alpha, and then use Lingo software. 

Step 4: - In model 2 we take only demand constraint 𝑏𝑗 as uncertain which is probabilistic and 

remaining constraints as certain. To obtain deterministic form of 𝑏𝑗 (47) to (49), we use Weibull 

distribution. With the minimum deterministic cost, deterministic demand constraint (47) to (49), 

certain supply constraint (44) to (46) and certain conveyance capacity constraint (50) to (52), formulate 

the problem (R2). To get the optimal cost of transportation and flow of unit we optimize the problem 

(R2) at zero level of alpha, and then use Lingo software. 

Step 5: - In model 3 we take conveyance capacity constraint 𝑒𝑘 as uncertain which is probabilistic and 

remaining constraint as certain. To obtain deterministic form of 𝑒𝑘 (61) to (63), we use Weibull 

distribution. With the minimum deterministic cost, deterministic conveyance capacity (61) to (63), 

certain supply constraint (55) to (57) and certain demand constraint (58) to (60), formulate the problem 

(R3). To get the optimal cost of transportation and flow of unit we optimize the problem (R3) at zero 

level of alpha, and then use Lingo software. 

Step 6: - In model 4 we take 𝑎𝑖, 𝑏𝑗 and 𝑒𝑘 as probabilistic uncertain variables. To obtain deterministic 

form of supply, demand and conveyance capacity constraint (66) to (74), we use Weibull distribution. 

With the minimum deterministic cost, deterministic constraints (66) to (74), formulate the problem 

(R4). To get the optimal cost of transportation and flow of unit we optimize the problem (R4) at zero 

level of alpha, and then use Lingo software. 

Step 7: - For varying alpha values, repeat above steps from 3 to 6, to determine the optimal cost of 

transportation and flow of unit. 

6.1. Computational analysis 

To show the applicability and effectiveness of SFSTPMC and its different versions, offers a numerical 

example in this section. The food factory supplies homogeneous items, and there are three food 

factories and four warehouses. Food factory A has capacity of production at least 𝑎1 units, food factory 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 4s (2024) 

 

553 
https://internationalpubls.com 

B has capacity of production exactly 𝑎2units, and Food factory C has capacity of production at most 

𝑎3 units. Similarly, warehouse 1 has demand’s capacity at most 𝑏1 units, warehouse 2 has demand’s 

capacity at least 𝑏2 units, warehouse 3 has demand’s capacity at exactly 𝑏3 units, and warehouse 4 has 

demand’s capacity at least 𝑏4 units. Appropriate conveyance is accommodated by food factory 

supplier. Also, address two conveyances in this problem along with their different loading capacities 

which are denoted as for first conveyance has loading capacity at least 𝑒1 units and for second 

conveyance has loading capacity at most 𝑒2 units. The cost of transportation per unit from every food 

factory to each warehouse represented as triangular fuzzy numbers which is denoted as �̃�𝑖𝑗𝑘, and cost 

for two conveyance is represented as 𝑐𝑖𝑗1and 𝑐𝑖𝑗2 shown in table 1 and 2 respectively. 

The supply of the fresh food is uncertain because of unpredictable circumstances at food factory such 

as availability of labour and severity of weather. It is easy to think of scenario in which the supply is 

uncertain, implementing probability on such availability of production. Therefore, 𝑃𝑎1 is the 

probability for food factory A that the needed quantity of goods is available. Similarly, 𝑃𝑎2 and 𝑃𝑎3 are 

the probabilities for food factories B and C respectively.  

The demand for fresh food is also naturally unpredictable due to inaccurate demand forecast, erratic 

delivery schedules. Thus, 𝑃𝑏1 is the probability for warehouse 1 that the needed expected demand. 

Similarly, 𝑃𝑏2 and 𝑃𝑏3 are the probabilities for warehouses 2 and 3 respectively. Similarly, blockages 

of roads and traffic jams make the capacity of conveyance is unpredictable or uncertain. 𝑃𝑒1 and 𝑃𝑒2 

are the probabilities that the availability of capacities of two conveyances.  

In the numerical example, the nominal values of uncertain constants are taken as 𝑎1 = 19, 𝑎2 = 17, 

𝑎3 = 23, 𝑏1 = 12, 𝑏2 = 13, 𝑏3 = 18, 𝑏4 = 15, 𝑒1 = 27 and 𝑒2 = 28.  Arbitrary probabilities are 

𝑃𝑎1 = 0.90, 𝑃𝑎2 = 0.50, 𝑃𝑎3 = 0.85, 𝑃𝑏1 = 0.25, 𝑃𝑏2 = 0.30, 𝑃𝑏3 = 0.28, 𝑃𝑏4 = 0.35, 𝑃𝑒1 = 0.07 and 

𝑃𝑒2 = 0.06. Here 𝑎𝑖, & 𝑏𝑗 are uncertain and follows the Weibull distribution. So, consider different 

values for parameters of the Weibull distribution as 𝛽𝑎1 = 25, 𝛽𝑎2 = 18, 𝛽𝑎3 = 13, 𝛽𝑏1 = 11, 𝛽𝑏2 =

10, 𝛽𝑏3 = 16, 𝛽𝑏4 = 14, 𝛽𝑒1 = 2, 𝛽𝑒2 = 3, 𝛿𝑎𝑖 = 𝛿𝑏𝑗 = 𝛿𝑒𝑘 = 2, 𝛾𝑎𝑖 = 𝛾𝑏𝑗 = 𝛾𝑒𝑘 = 2. Deterministic 

form of constraints which are probabilistic can be obtain by using equations (66) to (74). 

Warehouse 

Food factory  

A B C D 𝑎𝑖  

1 (2.5,3,3.5) (4,4.5,5) (2.5,3,3.5) (4.5,5,5.5) ≥ 𝑎1  

2 (0,0.5,1) (1.5,2,3) (2,2.5,3) (2.5,3,3.5) = 𝑎2  

3 (4.5,5,5.5) (7,7.5,8) (6.5,7,7.5) (5.5,6,6.5) ≤ 𝑎3  

𝑏𝑗  ≤ 𝑏1  ≥ 𝑏2  = 𝑏3   ≥ 𝑏4   

Table 1: Cost of transportation for 𝑒1 conveyance. 

Warehouse 

Food factory  

A B C D 𝑎𝑖  

1 (4.5,5,5.5) (8.5,9,9.5) (6.5,7,7.5) (8,8.5,9) ≥ 𝑎1  

2 (2.5,3,3.5) (4.5,5,5.5) (6.5,7,7.5,) (5,5.5,6) = 𝑎2  

3 (3.5,4,4.5) (6.5,7,7.5) (4.5,5,5.5) (5,5.5,6) ≤ 𝑎3  

𝑏𝑗  ≤ 𝑏1  ≥ 𝑏2  = 𝑏3   ≥ 𝑏4   

Table 2: Cost of transportation for 𝑒2 conveyance. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 4s (2024) 

 

554 
https://internationalpubls.com 

Through step 1, deterministic form of triangular fuzzy cost can be obtained by utilizing linear 

membership function. Cost of the problem shows as alpha cut representation for conveyance capacity 

𝑒1 & 𝑒2 in table 3 & 4 respectively.  

Warehouse 

Food factory  

A B C D 𝑎𝑖  

1 (1 − 𝛼)3.5�̅�111
+ 2.5𝛼𝑥111 

(1 − 𝛼)5�̅�121
+ 4𝛼𝑥121 

(1 − 𝛼)3.5�̅�131
+ 2.5𝛼𝑥131 

(1 − 𝛼)5.5�̅�141
+ 4.5𝛼𝑥141 

≥ 𝑎1  

2 (1 − 𝛼)1�̅�211
+ 0𝛼𝑥211 

(1 − 𝛼)3�̅�221
+ 1.5𝛼𝑥221 

(1 − 𝛼)3�̅�231
+ 2𝛼𝑥231 

(1 − 𝛼)3.5�̅�241
+ 2.5𝛼𝑥241 

= 𝑎2  

3 (1 − 𝛼)5.5�̅�311
+ 4.5𝛼𝑥311 

(1 − 𝛼)8�̅�321
+ 7𝛼𝑥321 

(1 − 𝛼)7.5�̅�331
+ 6.5𝛼𝑥331 

(1 − 𝛼)6.5�̅�341
+ 5.5𝛼𝑥341 

≤ 𝑎3  

𝑏𝑗  ≤ 𝑏1  ≥ 𝑏2  = 𝑏3   ≥ 𝑏4   

Table 3: Transportation cost 𝑐𝑖𝑗1 alpha cut representation for conveyance capacity 𝑒1. 

Warehouse 

Food factory  

A B C D 𝑎𝑖  

1 (1 − 𝛼)5.5�̅�111
+ 4.5𝛼𝑥111 

(1 − 𝛼)9.5�̅�121
+ 8.5𝛼𝑥121 

(1 − 𝛼)7.5�̅�131
+ 6.5𝛼𝑥131 

(1 − 𝛼)9�̅�141
+ 8𝛼𝑥141 

≥ 𝑎1  

2 (1 − 𝛼)3.5�̅�211
+ 2.5𝛼𝑥211 

(1 − 𝛼)5.5�̅�221
+ 4.5𝛼𝑥221 

(1 − 𝛼)7.5�̅�231
+ 6.5𝛼𝑥231 

(1 − 𝛼)6�̅�241
+ 5𝛼𝑥241 

= 𝑎2  

3 (1 − 𝛼)4.5�̅�311
+ 3.5𝛼𝑥311 

(1 − 𝛼)7.5�̅�321
+ 6.5𝛼𝑥321 

(1 − 𝛼)5.5�̅�331
+ 4.5𝛼𝑥331 

(1 − 𝛼)6�̅�341
+ 5𝛼𝑥341 

≤ 𝑎3  

𝑏𝑗  ≤ 𝑏1  ≥ 𝑏2  = 𝑏3   ≥ 𝑏4   

Table 4: Transportation cost 𝑐𝑖𝑗2 as alpha cut representation for conveyance capacity 𝑒2. 

Through step 2, obtain 4 models by transformed problem (R) and its constraints. Through step 3, in 

model 1 we take supply constraint 𝑎𝑖 as uncertain which is probabilistic, 𝑒𝑘 & 𝑏𝑗 as certain constrains. 

Function of cost at zero level of alpha, utilizing Weibull distribution on supply constraint which is 

probabilistic due to uncertainty in supply.  Now, supplies are 𝑎1 = 28.035, 𝑎2 = 19.665, 𝑎3 =

13.8063, demands are 𝑏1 = 12, 𝑏2 = 13, 𝑏3 = 18, 𝑏4 = 15, and conveyance capacities are 𝑒1 = 27, 

𝑒2 = 28. Through lingo software, obtain optimal cost of transportation as 176.27 and 𝑥131 =

18.00, 𝑥141 = 10.035, 𝑥211 = 1.70, 𝑥221 = 13.00, 𝑥241 = 4.965 & unit flow as 47.7.  

Through step 4, in model 2 we take only demand constraint 𝑏𝑗 as uncertain which is probabilistic and 

remaining constraints as certain. Function of cost at zero level of alpha, utilizing Weibull distribution 

on demand constraint which is probabilistic due to uncertainty in demand. Now, supplies are 𝑎1 = 19, 

𝑎2 = 17, 𝑎3 = 23, demands are 𝑏1 = 13.36, 𝑏2 = 11.195, 𝑏3 = 17.665, 𝑏4 = 15.313, and 

conveyance capacities are 𝑒1 = 27, 𝑒2 = 28. Through lingo software, obtain optimal cost of 

transportation as 168.024 and 𝑥131 = 17.665, 𝑥141 = 9.508, 𝑥221 = 11.195, 𝑥241 = 5.805 & unit 

flow as 44.173. 

Through step 5, in model 3 we take conveyance capacity constraint 𝑒𝑘 as uncertain which is 

probabilistic and remaining constraint as certain. Function of cost at zero level of alpha, utilizing 

Weibull distribution on conveyance capacity constraint which is probabilistic due to uncertainty in 

conveyance capacity. Now, supplies are 𝑎1 = 19, 𝑎2 = 17, 𝑎3 = 23, demands are 𝑏1 = 12, 𝑏2 = 13, 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 4s (2024) 

 

555 
https://internationalpubls.com 

𝑏3 = 18, 𝑏4 = 15, and conveyance capacities are 𝑒1 = 2.539, 𝑒2 = 6.3547. Through lingo software, 

obtain optimal cost of transportation as 176.50 and 𝑥131 = 18.00, 𝑥141 = 11.00, 𝑥221 = 13.00, 𝑥241 =

4.00 & unit flow as 46. 

Through step 6, in model 4 we take 𝑎𝑖, 𝑏𝑗 and 𝑒𝑘 as probabilistic uncertain variables. Now, supplies 

are 𝑎1 = 28.035, 𝑎2 = 19.665, 𝑎3 = 13.8063, , demands are 𝑏1 = 13.36, 𝑏2 = 11.195, 𝑏3 =

17.665, 𝑏4 = 15.313, and conveyance capacities are 𝑒1 = 2.539, 𝑒2 = 6.3547. Through lingo 

software, obtain optimal cost of transportation as 173.275 and 𝑥131 = 17.665, 𝑥141 = 10.370, 𝑥211 =

3.527, 𝑥221 = 11.195, 𝑥241 = 4.943 & unit flow as 47.7. Similarly, solve four cases of constraint for 

varying alpha values. Table 5 shows the computational results for 4 models.  

Constructed four models for SFSTPMC. Model 1 depicts supply constraint as probabilistic values, 

demand and conveyance capacity as certain values, model 2 depicts demand as probabilistic values 

and other constraints as certain values, model 3 depicts conveyance capacity as probabilistic value and 

other constraints as certain values, model 4 depicts supply, demand and conveyance capacity as 

probabilistic values. The objective function’s cost coefficient converts to alpha cut representation and 

mixed constraints which are stochastic are converts into crisp form by utilize Weibull distribution to 

solve the models. It’s notable that these models are constructed from different perspectives. As the 

choice of model depends on the decision-maker’s preference, determining the superior model for 

decision-making is inconclusive.   

 Optimal solution Flow of units  Shipment 𝑥𝑖𝑗𝑘 

Model 1 176.27 47.7 𝑥131 = 18, 𝑥141 = 10.035, 𝑥211 = 1.70, 𝑥221 =

13.00, 𝑥241 = 4.965. 

Model 2 168.024 44.173 𝑥131 = 17.665, 𝑥141 = 9.508, 𝑥221 = 11.195, 𝑥241 =

5.805.  

Model 3 176.50 46 𝑥131 = 18, 𝑥141 = 11, 𝑥221 = 13.00, 𝑥241 = 4.00.  

Model 4 173.275 47.7 𝑥131 = 17.665, 𝑥141 = 10.35, 𝑥211 = 11.195, 𝑥221 =

13.00, 𝑥241 = 4.943.  

Table 5: Computational results of 4 models. 

7. Sensitivity analysis and Discussion  

 In this section, SFSTPMC conducted a sensitivity analysis to assess optimality concerning variation 

in probabilities related to uncertain parameters like source, demand and conveyance capacity. Model 

4th has chosen for this analysis, with probabilities ranging from 0 to 1 for parameters 𝑎𝑖, 𝑏𝑗 & 𝑒𝑘. The 

analysis involved, holding one probability parameter (𝑃𝑎𝑖or 𝑃𝑏𝑗 or 𝑃𝑒𝑘) as constant at 0.5 while varying 

the probabilities of other parameters (𝑃𝑎𝑖or 𝑃𝑏𝑗 or 𝑃𝑒𝑘). For equality constraints, the probabilities of 

that parameter remain 0.5 by using case IV. Each parameter which is stochastic and cost which is fuzzy 

in model 4 we obtained and listed optimal results as units of entire shipping (flow) and cost of 

transportation. The lingo software aided in solving the optimization problem. Results of sensitivity 

analysis for probabilistic demand 𝑏𝑗 presented in Table 6. Graphical representation of cost of 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 4s (2024) 

 

556 
https://internationalpubls.com 

transportation & flow of units for probabilistic demand in model 4th shown in figures 1 and 2. It’s 

important that change in the probability of demand requirements significantly affect cost of 

transportation. Figure 1 shows that the cost of transportation increase gradually for probabilistic 

demand parameter and obtain least cost of transportation when 𝑃𝑏𝑗 = 0.01. It’s important that change 

in the probability of demand requirements significantly affect cost of transportation and flow of unit. 

Figure 2 shows that the flow of unit largest when 𝑃𝑏𝑗 = 0.99. When 0 ≤ 𝑃𝑏𝑗 ≥ 0.70, the flow of unit 

unchanged. When 𝑃𝑏𝑗 > 0.70, the flow of unit increases.   

The sensitivity analysis reveals that the cost of transportation cost and flow of unit in this problem are 

sensitive through changes in the probabilities of demand parameter 𝑏𝑗. This insight aids decision 

makers in selecting suitable probabilities for the availability of supply. Understanding these sensitivity 

patterns regarding uncertain parameters empowers decision maker with valuable insights and the 

ability to enhance the system of transportation.  

Sr. No.  Probability for 

𝑎𝑖(𝑃𝑎𝑖) 

Probability for 

𝑏𝑗(𝑃𝑏𝑗) 

Probability for 

𝑒𝑘(𝑃𝑒𝑘) 

Cost of       

transportation 

Flow of unit 

1   

 

 

 

 

 

 

 

 

   0.50 

 

0.99  

 

 

 

 

 

 

 

 

0.50 

194.562 50.2488 

2 0.95 185.845 48.5883 

3 0.90 181.363 47.7347 

4 0.85 178.422 47.1744 

5 0.80 176.139 46.7396 

6 0.75 174.223 46.3746 

7 0.70 173.368 46.33 

8 0.65 172.714 46.33 

9 0.60 172.108 46.33 

10 0.55 171.535 46.33 

11 0.50 170.985 46.33 

12 0.45 170.415 46.33 

13 0.40 169.925 46.33 

14 0.35 169.400 46.33 

15 0.30 168.868 46.33 

16 0.25 168.32 46.33 

17 0.20 167.744 46.33 

18 0.15 167.121 46.33 

19 0.10 166.141 46.33 

20 0.05 165.531 46.33 

21 0.01 164.3948 46.33 

Table 6: Sensitivity analysis of SFSTPMC 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 4s (2024) 

 

557 
https://internationalpubls.com 

 

Figure1: Sensitivity analysis of transportation cost.  

 

Figure2: Sensitivity analysis of flow of unit. 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 4s (2024) 

 

558 
https://internationalpubls.com 

8. Conclusion  

The purpose of this study was to introduced an approach for solving the SFSTPMC under probabilistic 

constraints such as supply, demand & conveyance capacity described as Weibull distribution and also 

includes fuzzy objective function such as cost of transportation. Obtain alpha cut representation from 

cost coefficient of the fuzzy objective function and through Weibull distribution, obtain deterministic 

form of stochastic constraints. Developed four models of SFSTPMC and optimize these models 

through Lingo software. In this study we address solid transportation problem which includes three 

mixed constraint stochastic parameters (supply, demand and conveyance). Its’s noted that several 

factors significantly influence the overall cost of transportation and unit of shipping, when distributing 

harmful goods such as vaccines and fragile items. Consequently, its’s crucial to incorporate these 

factors as parameters for STP’s and implement the relevant constraints accordingly.  An illustrative 

numerical example is provided to demonstrate the efficacy of these models. Results of sensitivity 

analysis for model 4 Supply, demand and conveyance capacity, presented. Figures 1 & 2 visually 

represents the impact of change in demand parameters on entire cost of transportation and flow of unit 

in model 4 as part of sensitivity analysis. Similarly, sensitivity analysis is conducted for the supply and 

conveyance capacity parameters. Moreover, this analysis underscores the importance of understanding 

the sensitivity of mixed constraint in the condition of increase uncertainty, aiding decision makers in 

determining the appropriate level of uncertainty for uncertain parameters. The computed results clearly 

demonstrate the validity of the designed model across various parameters. SFSTPMC problem shows 

important role in various decision makers managerial scenarios, particularly in problem including 

planning of resource allocation in industries production where demand, conveyance capacity and 

supply are inherently random variables. For optimization, this tool is effective on these types of 

problems.  

9. Limitations and Future Scope 

The current study’s limitations are outlined, a SFTP with mixed constraints is examined within 

stochastic environment, focusing only on cost of transportation as a single objective. However, in real-

world decision-making processes, dealing with complex organizational contexts cannot apply only 

single criterion. Therefore, it’s necessary to recognize the presence of multiple criteria that can enhance 

multi criteria decision making. Consequently, our future research aims to explore SFSTPMC scenarios 

by incorporating multiple objectives including multiple item parameters. Employing the methodology 

introduces in the study, variations in supply, demand and conveyance can be factored into the 

economic order quantity model. Additionally, in forthcoming research, we intent to collect real world 

data from authoritative sources and utilize statistical regularity criteria to establish its probability 

distribution, thereby expanding the applicability of the Weibull distribution.  

References 

[1] F. L. Hitchcock, “The Distribution of a Product from Several Sources to Numerous Localities,” Journal of 

Mathematics and Physics, vol. 20, no. 1–4, pp. 224–230, Apr. 1941, doi: 10.1002/SAPM1941201224. 

[2] K. B. Haley, “New Methods in Mathematical Programming—The Solid Transportation Problem,” 

https://doi.org/10.1287/opre.10.4.448, vol. 10, no. 4, pp. 448–463, Aug. 1962, doi: 10.1287/OPRE.10.4.448. 

[3] K. B. Haley, “The Multi-Index Problem,” Oper Res, vol. 11, no. 3, pp. 368–379, Jun. 1963, doi: 

10.1287/OPRE.11.3.368. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 4s (2024) 

 

559 
https://internationalpubls.com 

[4] L. A. Zadeh, “Fuzzy sets as a basis for a theory of possibility,” Fuzzy Sets Syst, vol. 1, no. 1, pp. 3–28, 1978, doi: 

10.1016/0165-0114(78)90029-5. 

[5] A. Das, U. Kumar Bera, and B. Das, “A SOLID TRANSPORTATION PROBLEM WITH MIXED CONSTRAINT 

IN DIFFERENT ENVIRONMENT,” Journal of Applied Analysis and Computation, vol. 6, no. 1, pp. 179–195, 

2016, doi: 10.11948/2016015. 

[6] P. K. Giri, M. K. Maiti, and M. Maiti, “Fuzzy stochastic solid transportation problem using fuzzy goal 

programming approach,” Comput Ind Eng, vol. 72, pp. 160–168, Jun. 2014, doi: 10.1016/J.CIE.2014.03.001. 

[7] N. Güzel, S. Alp, and E. Geçici, “SOLVING SOLID TRANSPORTATION PROBLEMS UNDER UNCERTAIN 

ENVIRONMENT USING GOAL PROGRAMMING,” Journal of Industrial Engineering, vol. 33, no. 1, pp. 130–

144, Apr. 2022, Accessed: Apr. 15, 2024. [Online]. Available: 

https://dergipark.org.tr/en/pub/endustrimuhendisligi/issue/69524/928699 

[8] B. Amaliah, C. Fatichah, and E. Suryani, “A new heuristic method of finding the initial basic feasible solution to 

solve the transportation problem,” Journal of King Saud University - Computer and Information Sciences, vol. 34, 

no. 5, pp. 2298–2307, May 2022, doi: 10.1016/J.JKSUCI.2020.07.007. 

[9] Y. Gao and S. Kar, “Uncertain Solid Transportation Problem with Product Blending,” International Journal of 

Fuzzy Systems, vol. 19, no. 6, pp. 1916–1926, Dec. 2017, doi: 10.1007/S40815-016-0282-X/METRICS. 

[10] S. K. Roy and D. R. Mahapatra, “Solving Solid Transportation Problems with Multi-Choice Cost and Stochastic 

Supply and Demand,” https://services.igi-global.com/resolvedoi/resolve.aspx?doi=10.4018/978-1-4666-7272-

7.ch023, pp. 397–428, Jan. 1AD, doi: 10.4018/978-1-4666-7272-7.CH023. 

[11] P. Kumar Giri, M. Kumar Maiti, and M. Maiti, “Profit maximization solid transportation problem under budget 

constraint using fuzzy measures,” Iranian Journal of Fuzzy Systems, vol. 13, no. 5, pp. 35–63, Oct. 2016, doi: 

10.22111/IJFS.2016.2732. 

[12] B. Zhang, J. Peng, S. Li, and L. Chen, “Fixed charge solid transportation problem in uncertain environment and its 

algorithm,” Comput Ind Eng, vol. 102, pp. 186–197, Dec. 2016, doi: 10.1016/J.CIE.2016.10.030. 

[13] V. Vidhya, P. Uma Maheswari, and K. Ganesan, “An alternate method for finding more for less solution to fuzzy 

transportation problem with mixed constraints,” Soft comput, vol. 25, no. 18, pp. 11989–11996, Sep. 2021, doi: 

10.1007/S00500-021-05664-X/METRICS. 

[14] A. N. Balaji, J. Mukund Nilakantan, I. Nielsen, N. Jawahar, and S. G. Ponnambalam, “Solving fixed charge 

transportation problem with truck load constraint using metaheuristics,” Ann Oper Res, vol. 273, no. 1–2, pp. 207–

236, Feb. 2019, doi: 10.1007/S10479-017-2692-Z/TABLES/26. 

[15] L. Meng, C. Zhang, Y. Ren, B. Zhang, and C. Lv, “Mixed-integer linear programming and constraint programming 

formulations for solving distributed flexible job shop scheduling problem,” Comput Ind Eng, vol. 142, p. 106347, 

Apr. 2020, doi: 10.1016/J.CIE.2020.106347. 

[16] S. Muthuperumal, P. Titus, and M. Venkatachalapathy, “An algorithmic approach to solve unbalanced triangular 

fuzzy transportation problems,” Soft comput, vol. 24, no. 24, pp. 18689–18698, Dec. 2020, doi: 10.1007/S00500-

020-05103-3/METRICS. 

[17] M. Sam’an and Farikhin, “A new fuzzy transportation algorithm for finding fuzzy optimal solution,” International 

Journal of Mathematical Modelling and Numerical Optimisation, vol. 11, no. 1, pp. 20–36, 2021, doi: 

10.1504/IJMMNO.2021.111715. 

[18] A. Das and G. M. Lee, “A Multi-Objective Stochastic Solid Transportation Problem with the Supply, Demand, and 

Conveyance Capacity Following the Weibull Distribution,” Mathematics 2021, Vol. 9, Page 1757, vol. 9, no. 15, 

p. 1757, Jul. 2021, doi: 10.3390/MATH9151757. 

[19] N. Qiuping, T. Yuanxiang, S. Broumi, and V. Uluçay, “A parametric neutrosophic model for the solid 

transportation problem,” Management Decision, vol. 61, no. 2, pp. 421–442, Mar. 2023, doi: 10.1108/MD-05-

2022-0660. 

[20] G. Gupta, D. Rani, and M. K. Kakkar, “Optimal Solution of Fully Intuitionistic Fuzzy Transportation Problems A 

Modified Approach,” Manufacturing Technologies and Production Systems: Principles and Practices, pp. 302–

316, Jan. 2023, doi: 10.1201/9781003367161-28/OPTIMAL-SOLUTION-FULLY-INTUITIONISTIC-FUZZY-

TRANSPORTATION-PROBLEMS-GOURAV-GUPTA-DEEPIKA-RANI-MOHIT-KUMAR-KAKKAR.