A Goal Programming Approach to Multichoice Multiobjective Stochastic Transportation Problems with Extreme Value Distribution

Abstract

This paper presents the study of a multichoice multiobjective transportation problem (MCMOTP) when at least one of the objectives has multiple aspiration levels to achieve, and the parameters of supply and demand are random variables which are not predetermined. The random variables shall be assumed to follow extreme value distribution, and the demand and supply constraints will be converted from a probabilistic case to a deterministic one using a stochastic approach. A transformation method using binary variables reduces the MCMOTP into a multiobjective transportation problem (MOTP), selecting one aspiration level for each objective from multiple levels. The reduced problem can then be solved with goal programming. The novel adapted approach is significant because it enables the decision maker to handle the many objectives and complexities of real-world transportation problem in one model and find an optimal solution. Ultimately, a mixed-integer mathematical model has been formulated by utilizing GAMS software, and the optimal solution of the proposed model is obtained. A numerical example is presented to demonstrate the solution in detail.

1。Introduction

运输问题是线性编程的众所周知的特定应用，其中可以从$运送物品m$ sources to $n$ destinations [1]. The availability of the product at the $i^th$ source is denoted by $a_i$, where 那and the demand required at the $j^th$ destination is $b_j$, where 。The penalty $c_ij$是客观函数的成本系数，可以代表从来源到目的地的商品的费用，这是希望最小化的[1].

问题可能存在多于一个目的，并且它们可能是冲突的，例如，最小化运输成本以及最小化运输时间。这里，两个目标具有相同的方向，即最小化，但有一个权衡。例如，使用汽车作为运输方式可能比空气运输成本低，但需要更长时间。因此，引入了目标编程，使得决策者（DM）可以在运输问题中至少一个目标中为抽吸水平设定多个选择，定义多漏水级别编程运输问题。此外，供应和需求参数可以是随机变量，因此它成为随机多尺度水平目标编程运输问题。

mahapatra [2] considers a model of a multichoice stochastic transportation problem (MCSTP), where the supply and demand parameters of the constraints follow extreme value distribution. Some of the cost coefficients of the objective function are a multichoice type. In an optimal solution, the number of units to be transported should be determined while satisfying source and destination demands to ensure minimum transportation costs.

在本文中,我们将看的问题other angle, by including the concept of goal programming to allow the model to deal with more than one conflicting objective and set multiaspiration levels to certain goals. The new model becomes a stochastic multiaspiration level goal programming transportation problem with an extreme value distribution.

Often, one cannot determine an exact value of any parameters in the problem because of uncertainty in supply or demand parameters for a number of reasons. For example, fluctuating markets or service output levels from suppliers, raw material defects, machine performances, delivery delays, and transportation issues are among the factors which cause uncertainty in supply assumptions. Similarly, unknown customer demand for products or services offered by the buyer, customer preferences, competition, and an unpredictable economy are among the factors that contribute to demand uncertainty. A stochastic problem can be formulated to overcome these uncertainties by considering that random variables follow a specific distribution instead of assuming fixed values. Here, an extreme value distribution will be assumed to convert the constraints from probabilistic to deterministic with the disjoint chance-constrained method. Extreme value distribution is used when there is a requirement for a limiting distribution to the maximum or minimum of a sample of independent and identically distributed random variables. The probability density function of extreme value distribution type I [3.] is as follows:

Goal programming is an extension of linear programming which handles multiobjective optimization where the individual objectives are often conflicting. Every one of these measures is assigned a goal or target value to be accomplished. Undesirable deviations from this arrangement of target values are then minimized through an achievement function. This can be a vector or a weighted sum depending on the goal programming variant adopted or the DM’s requirements.

The type of goal programming model employed is determined by the nature of the DM’s goals. The initial goal programming formulations order the undesirable deviations into a hierarchy by criticality, which enables more priority to be given to minimizing deviation of the more important factors. This is known as lexicographic (preemptive) or non-Archimedean goal programming.

Lexicographic goal programming can be used when prioritization is relevant to the goals. In preemptive goal programming, the objectives can be separated into various priority classes. Here, it is assumed that no two goals have equal priorities. Each will then be satisfied sequentially from most important to least important. The DMs can set multichoice aspiration levels (MCALs) for each goal to avoid underestimating, accounting for the “more/higher is better” and “less/lower is better” aspirations. To handle these multiple aspiration levels, multiplicative terms of binary variables are utilized, where all binary variables constitute mutually exclusive choices and only one variable is selected. The number of binary variables required for a constraint is equivalent to the total number of options of that constraint.

随后本文组织如下。在部分2那a problem overview will be considered; the mathematical model will be presented in Section3.那and Section4.will discuss the transformation of the goal constraint involving multiple aspiration levels into an equivalent form. Finally, a case study to demonstrate the model will be presented in Section5.。

2。Problem Overview

Contini [4.] considered the first formulation of the stochastic goal programming model. He set the goals as uncertain normally distributed variables. The technique for solving the probabilistic programming model was to convert it into an equivalent deterministic model. Many approaches have been proposed to solve the probabilistic programming model, of which the most common approach is chance-constrained programming (CCP), developed by Charnes and Cooper [5.-7.].

Chang [8.] proposed a new idea for modelling the multichoice goal programming problem using multiplicative terms of binary variables to handle multiple aspiration levels. Biswal and Acharya [9.建议的转换技术将多体线性编程问题转换为等同数学模型，其中约束与多相参数相关联。

Many researchers have extensively studied the MCSTP. Barik et al. [10.] presented a stochastic transportation model involving Pareto distribution. Roy et al. [11.] presented an equivalent deterministic model of MCSTP by assuming that both availabilities $a_i$ and demands $b_j$ are random variables following an exponential distribution. Biswal and Samal [12.] obtained an equivalent deterministic model of MCSTP in which they considered that both $a_i$ and $b_j$ follow Cauchy distribution. Mahapatra [2]考虑了具有极端值分布的MCSTP，作为这项研究的灵感的基础。本文的新颖贡献是在模型中包括多目标问题，并且在抽吸水平而不是成本系数参数方面表示多体问题。mahapatra [2] also studied an MCSTP model involving Weibull distribution, whereas Quddoos et al. [13.] presented an MCSTP involving a general form of distribution. Roy [14.]引入拉格朗日的插值多项​​式，以处理多体运输问题。他随后发表了一份运输问题的论文，具有多种成本和需求参数和随机供应[15.),他利用拉格朗日插值polynomial to select an appropriate value for the cost coefficients of the objective function and the demand of the constraints in the transportation problem. By adopting stochastic programming, the stochastic supply constraints were transformed into deterministic constraints. One of the key publications of Maity and Roy [16.]提出了经修订的多相目标编程（RMCGP）和实用功能作为MOTP的方法的技术。在另一篇论文中，他们介绍了将多种介意间隔问题（MCITP）转换为确定性运输问题的过程，以解决[17.]. In an additional publication, the same authors demonstrated solving a fuzzy transportation problem (FTP) using a multichoice goal programming approach [18.]. Roy et al. [19.还提出了与RMCGP组合将锥形标定功能引入MOTP的技术。

在这项研究中，我们将提出一种新的运输问题方法，在极值分布后供应和需求参数是随机变量。无论是最小化运输问题的成本系数，而不是最小化运输问题的成本系数，而不是最小化运输时间，最大限度地减少运输物品的风险等。作为附加功能，每个目标可以具有多个抽吸级别而不是一个。现在这个问题成为多体多目标随机运输问题。为了克服这种困难，首先我们将使用随机方法将概率约束转变为确定性的方法。其次，应用了由二进制变量组成的一般转换，以选择来自多个级别的每个目标的一个抽吸级别。减少的问题然后成为MOTP，它将通过目标编程来解决。

3.。Mathematical Model

Initially, the classical transportation problem is considered. Ifx_ijrepresents the amount transported from the source to the destination, then the transportation model can be defined as follows.

Model 1 哪里is the transportation cost per unit,is the amount shipped,is the amount of supply at sourcei那andis the amount of demand at destinationj[20.].

Now, we consider a mathematical model for a stochastic multiaspiration level goal programming transportation problem with an extreme value distribution as follows.

Model 2 哪里is the linear function of thegoal,is the aspiration levels of thegoal,is the amount shipped,is the amount of supply at sourcei那is the amount of demand at destinationj那is the negative deviational variable, andis the positive deviational variable.

3.。1。Converting the Probabilistic Constraint into a Deterministic Constraint Using the Disjoint Chance-Constrained Method

From Mahapatra [2), 3例随机性在右手side of the supply and demand constraints were considered:（1）Onlyfollows extreme value distribution(2)Onlyfollows extreme value distribution(3)Bothandfollow extreme value distribution

这导致了三种不同的模型(德泰ls, see Mahapatra [2]. The final transformed constraint is then considered here as the probabilistic constraint (4.) transformed into a deterministic linear constraint:

The probabilistic constraint (5.) transformed into a deterministic linear constraint:

现在，将获得与极值分布模型的确定性多泊素级别目标编程运输问题如下。

Model 3 哪里is the feasibility condition.

4.将涉及多尺度水平的目标约束转变为等同的形式

Considering the goal constraint with multiple aspiration levels,

A binary variable will be utilized to select a single aspiration level, and we can utilize the relationto determinate the number of binary variables needed withaspiration levels under the given linearized constraint [21.]. Let 那哪里satisfies the following inequalities:

不平等是确定的：(i)If然后(from (12.))(ii)If然后(from (13.） - （15.))

And so, the new goal constraint will be 哪里represents the function of the binary serial number.

A stochastic multiaspiration level goal programming transportation problem with extreme value distribution model will be as follows.

Model 4 哪里is the feasibility condition.

5.。Case Study

In this section, a case study from Mahapatra [2] is considered with modifications and the assumption of extreme value distribution instead of Weibull distribution. In this case study, a cold drink supply company transports cold drinks from three product centres at Dankuni, Howrah, and Asansol to four destination centres at Jhargram, Kharagpur, Tarkeshwar, and Contai. In the summer season, the cold drinks are in high demand at each of the four destination centres. The transportation time cost is an essential factor in a transportation planning programme as well as the transportation cost. The manufacturing time at production centres depends on the availability of current supply, machine condition, skilled labour, etc. Delivery time is related to the transporting means and seamless distribution of a product in due time to destination centres. The transportation time costand cost coefficientfrom each source to each destination are considered in Table1。

The cold drinks supply company is seeking to reach the following goals: goal 1 is to minimize the transportation time cost and goal 2 is to minimize the cost of transportation. The target values are 112,000 or 113,000 hours and $150,000 or $160,000, respectively.

Due to the fluctuation of the above factor, a stochastic multiaspiration level goal programming transportation problem approach has been considered, in which the supply and demand parameters follow extreme value distribution. The specified probability levels with shape and scale parameters for supply are listed in Table2那and the specified probability levels with shape and scale parameters of demand parameters are provided in Table3.。

Utilizing the data in Tables1-3.那the deterministic multiaspiration level goal programming transportation problem is formulated as follows:

Checking that the feasibility condition is satisfied:

The deterministic linear mixed-integer problem is then solved using GAMS (software), where the optimal solution is obtained:

The remaining decision variables are zero. The results show that goal 1 has an aspiration level of 113,000 hours and zero positive deviation, which means that the transportation time cost achieved the aspiration level exactly, and goal 2 has an aspiration level of $160,000 and zero positive deviation, which means that transportation cost also reached the desired aspiration level exactly.

6.。Conclusion

In this paper, we have explored a study of problem when the supply and demand parameters are the stochastic type and follow extreme value distribution. Three different approaches (the stochastic approach, binary variable approach, and goal programming approach) can be combined to reach an optimal solution to the transportation problem. This provides a new capability to handle real-life DM problems such as agricultural, managerial, economical, and industrial. One numerical example has been presented to illustrate the approach, which was solved using GAMS software. One can apply the proposed model to real-life problems in feature work or adapt other multiobjective techniques such as the -constraint method, weighting method, or fuzzy programming methods and compare their performance.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

1. D. R. Mahapatra, S. K. Roy, and M. P. Biswal, “Multi-choice stochastic transportation problem involving extreme value distribution,”Applied Mathematical Modelling，卷。3.7.那no. 4, pp. 2230–2240, 2013.View at:Publisher Site|Google Scholar
2. D. r.Mahapatra，“涉及威布尔分布的多选择随机运输问题”An International Journal of Optimization and Control: Theories & Applications (IJOCTA)，卷。4，不。1，pp。45-55,2013。View at:Google Scholar
3. B. S.埃弗蒂特，The Cambridge Dictionary of Statistics那Cambridge University Press, Cambridge, UK, 2002.
4. B. Contini, “A stochastic approach to goal programming,”Operations Research，卷。16.那no. 3, pp. 576–586, 1968.View at:Publisher Site|Google Scholar
5. A. Charnes and W. W. Cooper, “Chance-constrained programming,”Management Science，卷。6.那no. 1, pp. 73–79, 1959.View at:Publisher Site|Google Scholar
6. A. Charnes and W. W. Cooper, “Chance constraints and normal deviates,”Journal of the American Statistical Association，卷。5.7.那no. 297, pp. 134–148, 1962.View at:Publisher Site|Google Scholar
7. A. Charnes and W. W. Cooper, “Deterministic equivalents for optimizing and satisficing under chance constraints,”Operations Research，卷。11.那no. 1, pp. 18–39, 1963.View at:Publisher Site|Google Scholar
8. C.-t.Chang, “Multi-choice goal programming,”Omega，卷。3.5.那no. 4, pp. 389–396, 2007.View at:Publisher Site|Google Scholar
9. M. P. Biswal and S. Acharya, “Transformation of a multi-choice linear programming problem,”Applied Mathematics and Computation，卷。21.0, no. 1, pp. 182–188, 2009.View at:Publisher Site|Google Scholar
10. S. K. Barik, M. P. Biswal, and D. Chakravarty, “Stochastic programming problems involving pareto distribution,”Journal of Interdisciplinary Mathematics，卷。14.那no. 1, pp. 40–56, 2011.View at:Publisher Site|Google Scholar
11. S. K. Roy, D. R. Mahapatra, and M. P. Biswal, “Multi-choice stochastic transportation problem with exponential distribution,”Journal of Uncertain Systems，卷。6.那pp. 200–213, 2012.View at:Google Scholar
12. M. P. Biswal and H. K. Samal, “Stochastic transportation problem with Cauchy random variables and multi choice parameters,”Journal of Physical Sciences，卷。17.那pp. 117–130, 2013.View at:Google Scholar
13. A. Quddoos, M. G. Ull Hasan, and M. M. Khalid, “Multi-choice stochastic transportation problem involving general form of distributions,”Springer Plus，卷。3.那no. 1, pp. 1–9, 2014.View at:Publisher Site|Google Scholar
14. S. K. Roy, “Lagrange’s interpolating polynomial approach to solve multi-choice transportation problem,”International Journal of Applied and Computational Mathematics，卷。1那no. 4, pp. 639–649, 2015.View at:Publisher Site|Google Scholar
15. S. K. Roy, “Transportation problem with multi-choice cost and demand and stochastic supply,”Journal of the Operations Research Society of China，卷。4，不。2那pp. 193–204, 2016.View at:Publisher Site|Google Scholar
16. G. Maity and S. K. Roy, “Solving multi-objective transportation problem with interval goal using utility function approach,”国际业务研究杂志，卷。27.那no. 4, pp. 513–529, 2016.View at:Publisher Site|Google Scholar
17. S. K. Roy and G. Maity, “Minimizing cost and time through single objective function in multi-choice interval valued transportation problem,”Journal of Intelligent & Fuzzy Systems，卷。3.2那no. 3, pp. 1697–1709, 2017.View at:Publisher Site|Google Scholar
18. G. Maity and S. K. Roy, “Solving fuzzy transportation problem using multi-choice goal programming,”Discrete Mathematics, Algorithms and Applications，卷。9.那no. 6, Article ID 1750076, 2017.View at:Publisher Site|Google Scholar
19. S. K. Roy，G. Maity，G.W.Weber和S.Z.A.Gök，“锥形标定方法，以间隔目标解决多项选择多目标运输问题”，“Annals of Operations Research，卷。25.3.那no. 1, pp. 599–620, 2017.View at:Publisher Site|Google Scholar
20. H. A. Taha,Operations Research: An Introduction那Pearson/Prentice Hall, Upper Saddle River, NJ, USA, 2011.
21. C.-t.Chang，“一种有效的混合整数问题的线性化方法”欧洲运营研究杂志，卷。12.3.那no. 3, pp. 652–659, 2000.View at:Publisher Site|Google Scholar

Copyright

Copyright © 2019 Hadeel Al Qahtani et al. This is an open access article distributed under the创意公共归因许可证那which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.