EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 2, 2024, 1228-1243 ISSN 1307-5543 – ejpam.com Published by New York Business Global The Stochastic Transportation Problem with Imprecise Data Using Lomax Distribution Pullooru Bhavana1, D Kalpana Priya1,∗ 1 Department of Mathematics, SAS School, VIT University, Vellore, TN, India Abstract. The stochastic transportation problem with imprecise data is a probabilistic chance- constrained programming (CCP) problem in which the objective function is fuzzy and the supply and demand are random. Three models for the STP with ID with mixed-type restrictions that follow the Lomax distribution (LD) are created in this research. Optimising the transportation cost in FTP under probabilistic mixed constraints is the goal of the research project. To do this, the probabilistic mixed constraints are transformed into deterministic form using the LD, and the cost coefficient of the fuzzy objective function is changed with alpha cut representation. Numerical examples are presented to demonstrate the suggested models. 2020 Mathematics Subject Classifications: 90B36, 90C70 Key Words and Phrases: Stochastic Transportation Problem, Imprecise, Lomax, Mixed con- straints 1. Introduction Making decisions is essential in many different fields. The main goal of the trans- portation problem (TP) is to reduce the cost of transferring goods and materials between producers and consumers, enabling the manufacturer to better satisfy consumer demands. An uneven transportation issue with mixed constraints (TPMC) arises when there is a no- ticeable expansion or reduction in both the capacity of a supply and the need of a demand in a typical transportation system. Many researchers TP with mixed constraints intro- duced first by Brigden [7], had proposed different models by V. Adlakha [2], A. Das, [11], S. Agarwal [3],S. Gupta, [19], V. Vidhya, [32], Rashid, [31], and extended multiobjectived in fuzzy by Gupta [20]. The terms for TP that are modelled under such circumstances are fuzzy transportation problems (FTP) and stochastic transportation problems (STP). Comparision of the TP with different authors defined on Gessesse [16], Jerbi [21], Al Qah- tani et al. [30], Nasseri and Bavandi [28], Dutta et al. [13], Mahapatra et al. [26], Agrawal and Ganesh [5], Das and Lee ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5162 Email addresses: pullooru.bhavana2020@vitstudent.ac.in (P. Bhavana), dkalpanapriya@vit.ac.in (D K Priya) https://www.ejpam.com 1228 © 2024 EJPAM All rights reserved. P. Bhavana, D. Kalpana / Eur. J. Pure Appl. Math, 17 (2) (2024), 1228-1243 1229 The Lomax distribution is utilised extensively these days in numerous domains, in- cluding decision-making. This paper’s goal is to use the SFTPMC model to optimise the overall cost of transportation under ambiguous situations. Generally speaking, fuzzy or probability theory is used to characterise uncertainty. However, because it requires suffi- cient knowledge, using fuzzy theory or probability theory with stochastic to show every indeterminacy is not always feasible. In actuality, the uncertainty theory is applied be- cause low-frequency occurrences happen in our day-to-day lives. It becomes difficult to schedule an appropriate transport plan under these conditions in order to reduce the over- all expenditures. This study offers a useful paradigm that the decision making may use to handle some unpredictable variables without compromising customer reliability. Agrawal [4], Aruna Chalam [9], Giri et al. [18][19], Acharya et al. [1], Gessesse [17], Maity et al. [27] were solved the STP articles that involve both fuzziness and randomness. This study might be expanded to include additional areas where plans or decisions must be made under unknown circumstances. There are four sections to this paper: section 2 provides an overview of the funda- mental terms and concepts, while section 3 presents the suggested lomax distribution, the mathematical formulation of the SFTPMC, the steps involved in solving it, and a descrip- tion of the suggested strategy supported by numerical examples. Moreover, the conclusion in section 4 signifies the end of the paper. 1.1. Research Gap and Motivation SFTPMC was used to discover and resolve uncertainty in linear programming prob- lems (LPPs). Fuzzy transposition problems (FTP) are unique circumstances in which many academics have devised various algorithms and ranking functions to turn fuzzy data into crisp data in order to handle the FTP. Neutrosophic Transportation Problems (NTP) have recently been proposed as a way to use optimization techniques with unknown and indeterminate variables. Furthermore, many researchers have proposed Fuzzy Opti- mization Techniques (FOT), the Single-valued Trapezoidal Neutrosophic Transportation Problem (SVTNTP) [22], the Commercial Traveler Problem (CTP) [29], and a two-stage conventional transportation model to distribute relief aid to victims in uncertain scenarios [15] and multi-objective stochastic solid transportation problem (MOSSTP) uncertainity with weibull distribution [12]. In this paper proposed the SFTPMC, the stochastic trans- portation problem with imprecise data using the Lomax distribution provides a valuable tool for addressing the challenges of transportation planning and logistics in an uncertain environment, and it helps decision makers make more accurate and robust decisions by considering the stochastic nature of various factors and incorporating the Lomax distri- bution for modeling imprecise data. 2. Preliminaries Let we recall first of all the fuzzy set theory and triangular fuzzy number and alpha-cut concepts used in this paper. Now a days the imprecise information modelling are done by P. Bhavana, D. Kalpana / Eur. J. Pure Appl. Math, 17 (2) (2024), 1228-1243 1230 using fuzzy concepts [33]. Definition 1. (Fuzzy set). Let X be a crisp set and let à ⊆ F be a fuzzy set where F is a fuzzy space. The fuzzy set à is defined on crisp set X with a membership function µÃ , can be expressed as follows à = {(x, µÃ(x)), x ∈ X} where µÃ :X → [0, 1]. Definition 2. (Triangular Fuzzy Number). A fuzzy number ã is a triangular fuzzy number denoted by (a1, a2, a3) and its membership function µã is given below µÃ =  x−a1 a2−a1 , if a1 ≤ x ≤ a2 a3−x a3−a2 , if a2 ≤ x ≤ a3 0, otherwise Definition 3. (α - cut). The cut or level of a fuzzy set A is a crisp set defined by Aα = {x/µÃ(x) ≥ α}, 0 < α < 1. A triangular fuzzy number (a,b,c) can be represented as an interval number form as follows. (a, b, c) = [a+ (b− a)α, c− (c− b)α] Figure 1: α-cut Definition 4. (Linear Membership Function). [6] A linear membership function can be defined as µR(X) =  0 if xij < x ¯ ij x̄ij−xij x̄ij−x ¯ ij if x ¯ ij < xij < x̄ij 1 if xij > x̄ij In order to transform the fuzzy system to a deterministic set, the alpha cut representation using linear membership function is x̄ij − xij x̄ij − x ¯ ij = α such that xij = (1− α)x̄ij + αx ¯ ij , ∀α ∈ [0, 1] P. Bhavana, D. Kalpana / Eur. J. Pure Appl. Math, 17 (2) (2024), 1228-1243 1231 Definition 5. (Feasible solution). Any set of {xij ≥ 0, i = 1, 2, ...,m; j = 1, 2, ...n} that satisfies all the constraints is called a feasible solution to the problem. Definition 6. (Optimal solution).A feasible solution to the problem which minimizes the total shipping cost is called an optimal solution to the problem. 2.1. Lomax Distribution The Lomax distribution, also known as the Pareto Type II distribution [25], is a probability distribution commonly used to model heavy-tailed and skewed data. Definition 7. (Lomax Distribution). [23] [24] The Lomax distribution expressed as the random variable X has parameters scale parameter β and shape parameter α as X ∼ Lomax(β, α), A Lomax random variable X with scale parameter β and shape parameter α has probability density function f(x) = α β (1 + x β )−(α+1), x > 0 for β > 0, α > 0. The cumulative distribution function is as follows by F (x) = 1− (1 + x β ) −α where α and β are the shape and scale parameters, respectively, and x > 0, β > 0, α > 0. One specific aspect that has gained attention in stochastic transportation modeling is the involvement of the Lomax distribution. The real motivation behind incorporating the Lomax distribution into stochastic transportation modeling lies in its ability to capture the variability and uncertainty in transportation parameters such as supply and demand. By using the Lomax distribution, researchers aim to better represent the range of possible values for these parameters and incorporate their probabilistic nature into the modeling process. By doing so, they can obtain more realistic and robust transportation models that account for the inherent uncertainty in the system. Additionally, the Lomax distribu- tion provides flexibility in handling mixed-type restrictions and fuzzy objective functions, allowing for a more comprehensive analysis of the transportation problem under uncertain conditions. Probabilistic programming is a mathematical programming approach that is utilized when some or all of the model parameters are random and follow a probability distribution. Charnes and Cooper [10] established the chance-constrained programming technique for individual probabilistic constraints. Rasha [14] revealed how to turn chance constraints into similar deterministic linear constraints in the lomax distribution, as presented here, offers equal deterministic constraints for individual and joint constraints. P. Bhavana, D. Kalpana / Eur. J. Pure Appl. Math, 17 (2) (2024), 1228-1243 1232 The goal of the paper is to optimize the transportation cost in the Stochastic Trans- portation Problem under probabilistic mixed constraints. To achieve this, transforms the probabilistic mixed constraints into a deterministic form using the Lomax distribu- tion. Overall, the real motivation of Lomax distribution involvement in stochastic trans- portation modeling is to enhance the accuracy and reliability of transportation models by incorporating the probabilistic nature of transportation parameters and capturing the variability and uncertainty present in real-world transportation systems. 3. Main results 3.1. Formulation of the Stochastic Transportation Problem by Using Lomax Distribution The following is our definition of the transportation issue with mixed constraints. Let us assume that there are m origins, Oi, {i = 1,2,,,,,,,m} divided into sets I1, I2, I3, such that the origin Oi(i ∈ I1) must distribute at least at ai supply units, Oi(i ∈ I2) must distribute precisely at ai supply units, and Oi(i ∈ I3) may distribute at most at ai supply units. Assume also that there are n destinations, Dj {j=1,2,...,n}, divided into sets J1, J2, J3, where Dj(j ∈ J1) is required to receive a minimum of bj units of demand, Dj(j ∈ J2) an exact amount of bj units of demand, and Dj(j ∈ J3) a maximum of bj units of demand. The main goal is to reduce the overall cost of shipping, where cij represents the cost of shipping from OitoDj and xij represents the amount of shipping from OitoDj . All requests and supply are considered to be non-negative. Applying the constraints in the proposed TP model to the deterministic constraints is required to obtain the quantiles of a probability distribution function in a closed form. The fact that the Lomax distribution has the quantiles in their closed form is also another incentive to use it. Mathematically, the transportation issue with mixed constraints may be expressed as follows: Minimizez = m∑ i=1 n∑ j=1 cijxij , subject to constraints, P ( ∑ j=J xij ≥ ai) ≥ P (ai), i ∈ I1 = 1, 2, ....m1 P ( ∑ j=J xij = ai) ≥ P (ai), i ∈ I2 = m1 + 1,m1 + 2, ....m2 P ( ∑ j=J xij ≤ ai) ≥ P (ai), i ∈ I3 = m2 + 1,m2 + 2, ....m P ( ∑ i=I xij ≥ bj) ≥ P (bj), j ∈ J1 = 1, 2, ....n1 P. Bhavana, D. Kalpana / Eur. J. Pure Appl. Math, 17 (2) (2024), 1228-1243 1233 P ( ∑ i=I xij = bj) ≥ P (bj), j ∈ J2 = n1 + 1, n1 + 2, ....n2 P ( ∑ i=I xij ≤ bj) ≥ P (bj), j ∈ J3 = n2 + 1, n2 + 2, ....n and xij ≥ 0, i ∈ I, j ∈ J where P (ai) and P (bj) are the probabilities of the random variables of supply and demand respecively and also which follows the lomax distribution. The parameters of the lomax distribution for supply ai has shape parameter αai and scale parameter βai . Like wise the parameters of the lomax distribution for demand bj has shape parameter αbj and scale parameter βbj . The following cases are to be considered: (i) Only ai, i = 1, 2, . . . , m follows LD. (ii) Only bj , j = 1, 2, . . . , n follows LD. (iii) Both ai and bj , i=1,2,.....m and j=1,2,....n follows LD. Case 1. Only ai follows LD For P ( n∑ j=1 xij ≥ ai) ≥ P (ai), i ∈ I1, P (ai ≤ n∑ j=1 xij) ≥ P (ai), i ∈ I1 Let us consider ∑n j=1 xij = δai and ai ≥ ξaithen P (ai ≤ δai) ≥ P (ai), i ∈ I1 Now by using Lomax distribution PDF, integrating from∫ δai ξai αai βai (1 + ai − ξai βai )−(αai+1)dai ≥ P (ai) [−(1 + ai − ξai βai )−αai ]δaiξai ≥ P (ai) 1 + δai − ξai βai ≤ −(P (ai)− 1) − 1 αai δai ≤ ξai − βai[1 + (P (ai)− 1) − 1 αai ] n∑ j=1 xij ≤ ξai − βai[1 + (P (ai)− 1) − 1 αai ] Remarks. P. Bhavana, D. Kalpana / Eur. J. Pure Appl. Math, 17 (2) (2024), 1228-1243 1234 (i) For P ( n∑ j=1 xij ≤ ai) ≥ P (ai), i ∈ I3, then n∑ j=1 xij ≥ ξbj + βbj [(P (bj) + 1) − 1 αbj − 1] (ii) For P ( n∑ j=1 xij = ai) ≥ P (ai), i ∈ I2, then n∑ j=1 xij = ξai − βai[1 + (P (ai)− 1) − 1 αai ] We get the same deterministic result for both inequality types for supply and demand constraints by using lomax distribution to select the probability value at the 50% level in the supply and demand inequality constraint. Case 2. Only bj follows LD For P ( m∑ i=1 xij ≤ bj) ≥ P (bj), j ∈ J3, P (bj ≥ m∑ i=1 xij) ≥ P (bj), j ∈ J3 Let us consider ∑m i=1 xij = δbj and bj ≥ ξbjthen P (bj ≤ δbj) ≥ P (bj), j ∈ J3 Now by using Lomax distribution PDF, integrating from∫ ξbj δbj αbj βbj (1 + bj − ξbj βbj )−(αbj+1)dbj ≥ P (bj) [−(1 + bj − ξbj βbj )−αbj ] ξbj δbj ≥ P (bj) [1 + δbj − ξbj βbj ]−αbj ≥ P (bj) + 1 δbj ≥ ξbj + βbj [(P (bj) + 1) − 1 αbj − 1] m∑ i=1 xij ≥ ξbj + βbj [(P (bj) + 1) − 1 αbj − 1] P. Bhavana, D. Kalpana / Eur. J. Pure Appl. Math, 17 (2) (2024), 1228-1243 1235 Remarks. (i) For P ( m∑ i=1 xij ≤ bj) ≥ P (bj), j ∈ J1, then m∑ i=1 xij ≥ ξai − βai[1 + (P (ai)− 1) − 1 αai ] (ii) For P ( m∑ i=1 xij = bj) ≥ P (bj), j ∈ J2, then m∑ i=1 xij ≥ ξbj + βbj [(P (bj) + 1) − 1 αbj − 1] Case 3. Both ai and bj follows LD By using the above cases (i) and (ii) we follows. 3.2. Formulation of the Modulations of the Stochastic Transportation Problem with the Imprecise Data by Using Lomax Distribution The nature of the parameters (cost, supply, and demand) changes in many real-life scenarios, making it difficult for decision to make the best choice. It is possible to manage this scenario with fuzzy and random variables. We treat restrictions as random variables and cost as triangular fuzzy variables in our model. Depending on the circumstances around decision, there may or may not be uncertainty regarding supply or demand limits. As a result, we develop three STP models depending on the degree of uncertainty in the requirements. Here the objective function can be used the alpha cut to transform the provided tri- angular fuzzy cost of problem into an equal deterministic cost. Minimizez = m∑ i=1 n∑ j=1 cij((1− α̂)x̄ij + α̂xij) (1) The following models are to be considered: (i) Only ai, i = 1, 2, . . . , m follows uncertainty. (ii) Only bj , j = 1, 2, . . . , n follows uncertainty. (iii) Both ai, i = 1, 2, . . . , m and bj , j = 1, 2, . . . , n follow uncertainty. Model 1. Only ai follows uncertainty For modulation of the STP with the imprecise data by using LD is used for probabilistic only for supply constraints and demand constraints are certain. P. Bhavana, D. Kalpana / Eur. J. Pure Appl. Math, 17 (2) (2024), 1228-1243 1236 The mathematics formulation can be represented as follows as Minimizez = m∑ i=1 n∑ j=1 cij((1− α̂)x̄ij + α̂xij) subject to constriants, n∑ j=1 xij ≤ ξai − βai[1 + (P (ai)− 1) − 1 αai ], i ∈ I1 (2) n∑ j=1 xij = ξai − βai[1 + (P (ai)− 1) − 1 αai ], i ∈ I2 (3) n∑ j=1 xij ≥ ξbj + βbj [(P (bj) + 1) − 1 αbj − 1], i ∈ I3 (4) m∑ i=1 xij ≥ bj , j ∈ J1 (5) m∑ i=1 xij = bj , j ∈ J2 (6) m∑ i=1 xij ≤ bj , j ∈ J3 (7) and xij ≥ 0 Model 2. Only bj follows uncertainty For modulation of the STP with the imprecise data by using LD is used for probabilistic only for demand constraints and supply constraints are certain. The mathematics formulation can be represented as follows as Minimizez = m∑ i=1 n∑ j=1 cij((1− α̂)x̄ij + α̂xij) subject to constriants, n∑ j=1 xij ≥ ai, i ∈ I1 (8) n∑ j=1 xij = ai, i ∈ I2 (9) P. Bhavana, D. Kalpana / Eur. J. Pure Appl. Math, 17 (2) (2024), 1228-1243 1237 n∑ j=1 xij ≤ ai, i ∈ I3 (10) m∑ i=1 xij ≤ ξai − βai[1 + (P (ai)− 1) − 1 αai ], j ∈ J1 (11) m∑ i=1 xij = ξbj + βbj [(P (bj) + 1) − 1 αbj − 1], j ∈ J2 (12) m∑ i=1 xij ≥ ξbj + βbj [(P (bj) + 1) − 1 αbj − 1], j ∈ J3 (13) and xij ≥ 0 Model 3. Both ai and bj follows uncertainty For modulation of the STP with the imprecise data by using LD is used for probabilistic both supply and demand constraints. The mathematics formulation can be represented as follows as Minimizez = m∑ i=1 n∑ j=1 cij((1− α̂)x̄ij + α̂xij) subject to constriants, n∑ j=1 xij ≤ ξai − βai[1 + (P (ai)− 1) − 1 αai ], i ∈ I1 (14) n∑ j=1 xij = ξai − βai[1 + (P (ai)− 1) − 1 αai ], i ∈ I2 (15) n∑ j=1 xij ≥ ξbj + βbj [(P (bj) + 1) − 1 αbj − 1], i ∈ I3 (16) m∑ i=1 xij ≤ ξai − βai[1 + (P (ai)− 1) − 1 αai ], j ∈ J1 (17) m∑ i=1 xij = ξbj + βbj [(P (bj) + 1) − 1 αbj − 1], j ∈ J2 (18) m∑ i=1 xij ≥ ξbj + βbj [(P (bj) + 1) − 1 αbj − 1], j ∈ J3 (19) and xij ≥ 0. P. Bhavana, D. Kalpana / Eur. J. Pure Appl. Math, 17 (2) (2024), 1228-1243 1238 Example 1. This section offers an example to show the effectiveness and applicability. By taking from the weibull distribution [8]. There are three factories and four repositories, and the coal plant produces a homogeneous output. The manufacturing capacity of coal plant A is precisely a1 units, the manufacturing capacity of coal plant B is at least a2 units, and the manufacturing capacity of coal plant C is the greatest amount of a3 units. Similarly, demand capacity for repository 1 is at least b1 units; demand capacity for repository 2 is at most b2 units; and demand capacity for repository 3 is at least b3 units. Repository 4 can hold precisely b4 units in demand. If the cost of transportation for each unit from every coal plant to every deposit is cij, it is in the form of imprecise data 1 2 3 4 ai A (0,0.5,1) (2,4,6) (1.5,2,3) (2,4,5) =a1 B (3,5,7) (1.5,2,3) (0,0.5,1) (4,5.8,6) ≥ a2 C (7,8.5,9) (2.5,3,4) (3,4,5) (2,3,4) ≤ a3 bj ≥ b1 ≤ b2 ≥ b3 = b4 Table 1: Fuzzy Data i.e., TFN by using the linear membership function of alpha cut representation it will be convert into crisp data. By using these formula with alpha value as ”0”, it represent in given table into crisp data, xij = ((1− α̂)x̄ij + α̂xij)∀[0, 1] 1 2 3 4 ai A 1 6 3 5 =a1 B 7 3 1 6 ≥ a2 C 9 4 5 4 ≤ a3 bj ≥ b1 ≤ b2 ≥ b3 = b4 Table 2: Crisp data Here the following arbitrary nominal values of certain constants are supplied as a1 = 20, a2 = 16 a3 = 25, and demand as b1 = 11, b2 = 13, b3= 17, b4= 14 in the following sections. Furthermore, probabilities are given as Pa1 = 0.50, Pa2= 0.96, Pa3= 0.95, Pb1= 0.26, Pb2= 0.29, Pb3= 0.25, Pb4= 0.28. Since ai and bj are presumed to follow Lomax distribution, the distinct values for the parameters are ξa1 = 19, ξa2 =13, ξa3= 24, ξb1= 10, ξb2 = 11, ξ − b3= 16, ξb4 = 13 and also the scale parameter βai and βbj both are as 2 and the shape parameter αai and αbj are 3 as taken. Now the modulations of the stochastic transportation problem with the imprecise data by using lomax distribution as Model-1. For modulation of the STP with the imprecise data by using LD is used for probabilistic only for supply constraints and demand constraints are certain. Then By using the Lingo software, we obtain the optimal transportation cost as 92.52 and x11= 11, x14= 8.52, x23= 17, x34=5.48 and unit flow as 42. Model-2. For modulation of the STP with the imprecise data by using LD is used for probabilistic only for demand constraints and supply constraints are certain. P. Bhavana, D. Kalpana / Eur. J. Pure Appl. Math, 17 (2) (2024), 1228-1243 1239 1 2 3 4 ai A 1 6 3 5 =19.52 B 7 3 1 6 ≥ 16.85 C 9 4 5 4 ≤ 23.6 bj ≥ 11 ≤ 13 ≥ 17 = 14 Table 3: Only ai are uncertain 1 2 3 4 ai A 1 6 3 5 =20 B 7 3 1 6 ≥ 16 C 9 4 5 4 ≤ 25 bj ≥ 10.15 ≤ 10.83 ≥ 16.2 = 12.84 Table 4: Only bj are uncertain Then By using the Lingo software, we obtain the optimal transportation cost as 87.56 and x11= 10.15, x14= 9.85, x23= 16.2, x34=2.99 and unit flow as 39.19. Model-3. For modulation of the STP with the imprecise data by using LD is used for probabilistic both supply and demand constraints. 1 2 3 4 ai A 1 6 3 5 =19.52 B 7 3 1 6 ≥ 16.85 C 9 4 5 4 ≤ 23.6 bj ≥ 10.15 ≤ 10.83 ≥ 16.2 = 12.84 Table 5: Both ai and bj are uncertain Then By using the Lingo software, we obtain the optimal transportation cost as 87.08 and x11= 10.15, x14= 9.37, x23= 16.2, x34=3.47 and unit flow as 39.19. 3.3. Result and Discussion with comparison The ideal results are compared to different distributions for the SFTPMC problem minimizes transportation costs relative to prior techniques. The table below illustrates the comparison of findings. The table 6 shows the results obtained from different distribution functions. It justified the lomax distribution of the approach which attains very minimum when compare with other different distributions. REFERENCES 1240 Distribution Optimization Method Model 1 Model 2 Model 3 Weibull Optimal Transportation cost 93.67 95.75 96.42 Normal 102.68 105.8 107.4 Poisson 98.45 100.2 99.89 Lomax 92.52 87.56 87.08 Weibull Units flow 42 42.84 42.84 Normal 49 49.85 49.85 Poisson 45 45.6 45.6 Lomax 42 39.19 39.19 Table 6: Comparing results with different distributions 4. Conclusion This article presents a methodology for solving an SFTPMC that has probabilistic constraints together with the lomax distribution and a fuzzy integer for the objective function’s cost coefficient. Alpha cut representation is used to transform the fuzzy objec- tive value into a corresponding consistent objective function, and LD is used to transform each stochastic constraint into an analogous deterministic constraint. The Lomax distribu- tion is a crucial aspect of stochastic transportation modeling, as it captures variability and uncertainty in transportation parameters like supply and demand. This paper aim to rep- resent these values realistically and incorporate their probabilistic nature, resulting in more robust models and more comprehensive analysis under uncertain conditions.Additionally, three SFTPMC models—models 1, 2, and 3—are constructed, and Lingo software has been used to determine each model’s ideal answer.A numerical example illustrating the models’ performance is provided. Due to this issue, SFTPMC is essential in many circumstances involving managerial decision-making, such as the planning of several intricate resource allocation issues in the context of industrial production, where supply and demand are essentially random variables. This model will function as an effective tool for the best planning in such circumstances. When compared with different distributions, the LD optimum solution is superior in this instance, due to minimum. References [1] S. Acharya, N. Ranarahu, J. K. Dash, and M. M. Acharya. Computation of a mul- tiobjective fuzzy stochastic trans- portation problem. International Journal of Fuzzy Computation and Modelling, 1(2):212, 2014. [2] V. Adlakha, K. Kowalski, and B. Lev. Solving transportation problems with mixed con- straints. International Journal of Management Science and Engineering Man- agement, 1(1):47–52, 2006. [3] S. Agarwal and S. Sharma. A shootout method for time minimizing transportation REFERENCES 1241 problem with mixed constraints. merican Journal of Mathematical and Management Sciences, 39(4):299–314, 2020. [4] P. Agrawal, K. Alnowibet, T. Ganesh, A. F Alrasheedi, H. Ahmad, and A. Wagdy Mohamed. An artificial intelligence approach for solving stochastic transportation problems. Computers, Materials Continua, 70(1):817–829, 2022. [5] P. Agrawal and T. Ganesh. Solving transportation problem with stochastic demand and non-linear multi-choice cost. Research Gate, 2019. [6] S. K. Behera and J. R. Nayak. Solution of multi-objective mathematical program- ming problems in fuzzy approach. International Journal of Computer Science and Engineering, 3(12), 2011. [7] M. E. B. Brigden. A variant of the transportation problem in which the constraints are of mixed type. Operational Research Society, 25(3):437–445, 1977. [8] T.K. Buvaneshwari and D. Anuradha. Solving stochastic fuzzy transportation prob- lem with mixed constraints using the weibull distribution. ournal of Mathematics, 2022. [9] G. Aruna Chalam. Fuzzy goal programming (fgp) approach to a stochastic trans- portation problem under budgetary constraint. Fuzzy Sets and Systems, 66(3):293– 299, 1994. [10] A. Charnes and W. W. Cooper. Chance-constrained programming. Management Science, 6(1):73–79, 1959. [11] A. Das, U. K. Bera, and B. Das. A solid transportation problem with mixed constraint in different environment. Journal of Applied Analysis Computation, 6(1):179–195, 2016. [12] 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, 9:1757, 2021. [13] S. Dutta, S. Acharya, and R. Mishra. Genetic algorithm based fuzzy stochastic trans- portation programming problem with continuous random variables. Operational Re- search, 53(4):835–872, 2016. [14] Rasha Ebaid and Afaf El-Dash. Probabilistic programming technique for bivariate lomax random parameters. Far East Journal of Mathematical Sciences, 130(2):151– 164, 2021. [15] J. L. Edward and K. Palanivel. Two-stage transportation model for distributing relief aids to the affected regions in an emergency response under uncertainty. Intelligent and Fuzzy Systems, 758, 2023. REFERENCES 1242 [16] A. Gessesse, R. Mishra, and M. M. Acharya. Solving multi-objective linear fractional stochastic transportation problems involving normal distribution using simulation- based genetic algorithm. nternational Journal of Engineering and Advanced Technol- ogy, 9(2):9–17, 2019. [17] A. Abebaw Gessesse, R. Mishra, M. M. Acharya, and K. N. Das. Genetic algorithm based fuzzy programming approach for multi- objective linear fractional stochas- tic transportation problem involving four-parameter burr distribution. International Journal of System Assurance Engineering and Management, 11(1):93–109, 2020. [18] P. K. Giri, M. K. Maiti, and M. Maiti. Fuzzy stochastic solid transportation problem using fuzzy goal programming approach. Computers Industrial Engineering, 72:160– 168, 2014. [19] S. Gupta, I. Ali, and A. Ahmed. Multi-objective capacitated transportation problem with mixed constraint: a case study of certain and uncertain environment. Operation Research, 55(2):447–477, 2018. [20] S. Gupta, I. Ali, and A. Ahmed. An extended multi-objective capacitated transporta- tion problem with mixed constraints in fuzzy environment. International Journal of Operational Research, 37(3):345–376, 2020. [21] B. Jerbi. A fuzzy multi-objective polynomial time algorithm to solve the stochas- tic transportation formulation of a hospitalbed rearrangement problem. Journal of MultiCriteria Decision Analysis, 28:34–44, 2021. [22] K. Kalaivani and Palanivel Kaliyaperumal. A neutrosophic approach to the trans- portation problem using single-valued trapezoidal neutrosophic numbers. Proyec- ciones Journal of Mathematics, 42(2):533–547, 2023. [23] Parmil Kumar, Kirandeep Kour, and Jaspreet Kour. Estimation of the probability density function of lomax distribution. nternational Journal of Statistics and Eco- nomics, 19(2), 2018. [24] Jubran Abdulameer Labban. On 2-parameter estimation of lomax distribution. 2nd International Science Conference, Journal of Physics: Conf. Series, page 1294, 2019. [25] K. S. Lomax. Business failures: Another example of the analysis of failure data. Journal of the American Statistical Association, 49(268):847–852, 1954. [26] D. R. Mahapatra, S. Panda, and S. S. Sana. Multi-choice and stochastic programming for transportation problem involved in supply of foods and medicines to hospitals with consideration of logistic distribution. BAIRO Operational Research, 54(4), 2020. [27] G. Maity, V. F. Yu, and S. K. Roy. Optimum intervention in transportation networks using multimodal system under fuzzy stochastic environment. ounal of Advanced Transportation, 2022. REFERENCES 1243 [28] S. H. Nasseri and S. Bavandi. Solving multi-objective multi-choice stochastic trans- portation problem with fuzzy programming approach. 8th Iranian Joint Congress on Fuzzy and intelligent Systems (CFIS), 2020. [29] K. Palanivel. Fuzzy commercial traveler problem of trapezoidal membership functions within the sort of optimum solution using ranking technique. Afrika Matematika, 27(1-2):263, 2016. [30] H. Al Qahtani, A. El-Hefnawy, M. M. El-Ashram, and A. Fayomi. A goal program- ming approach to multichoice multiobjective stochastic transportation problems with extreme value distribution. Advances in Operations Research, 2019. [31] F. Rashid, A. R. Khan, and S. Uddin. Mixed constraints cost minimization trans- portation problem an effective algorithmic approach. American Journal of Operations Research, 11(1), 2021. [32] 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 Computing, 25(18), 2021. [33] L. A. Zadeh. Fuzzy sets. Information and Control, 8(3):338–353, 1965.