Solving multi-choice multi-objective transportation problem: a utility function approach

Transportation problem is an integral part of this competitive world. Minimizing transportation cost of a product from sources to destinations is the main objective of traditional transportation problems. In the present context single objective transportation problem fails to meet the managerial decision making requirements which demands the multi objective transportation problems. Many real life problems can be formulated as multi objective transportation problem, and their solution requires consideration of conflicting objectives. Naturally, these problems have a number of potentially Pareto-optimal solutions. Multi-objective transportation problem refers to a special class of vector minimum linear programming problem, which usually involves multiple, conflicting, and incommensurable objective functions. Solving these complexities, the decision maker has to develop a utility function. In this paper, we develop a utility function to attain the targeted goal for a multi objective transportation problem. This utility function is developed using fuzzy membership functions considering deviations of objective function from the goal value where deviation variable has chosen as fuzzy type. Moreover, fuzzy max-min operator has been included for taking limiting value of the objective functions. Performance of the utility function to achieve proper goal is evaluated and justified by solving numerical examples using fuzzy goal programming.

The real world conditions of transportation pose new challenges for development of intelligent assistive systems for effective decision making. The ignorance of relationships among various attributes and parameters is one of the major causes of uncertainty in existing models of transportation problems. Fuzzy sets have been explored by various investigators to address the uncertainty issues due to relationships among various attributes. However fuzzy sets are not fully capable of dealing all kinds of uncertainty to provide intelligent solutions of transportation problems. A soft set is explored in this paper to model the uncertainty arising from the relationships of attributes with the parameters in a multimodal and multi objective transportation problem. The three modes of transportation incorporated in this model are road, rail and air. The real data set is prepared using the tariff, distance and time duration of transport available on websites of transport service agencies. The objectives are to minimize the cost and time duration of transportation. The proposed model evaluates multiple criteria using various combinations of different modes of transport to optimize the objectives of the problem. The model has been illustrated using the real data set using existing methods of solution. The soft set approach is found to be quite effective in dealing with the relationships of attributes with the parameters arising due to multi criteria decision making and leads to intelligent optimal solutions. The proposed model can be used as intelligent assistive system for decision making of multi objective and multimodal transportation problems.

This chapter presents solution procedures for solving unbalanced multi-objective multi-choice stochastic transportation problems in a hybrid fuzzy uncertain environment. In this chapter, various types of unbalanced multi-objective fuzzy stochastic transportation models are considered with the assumption that the parameters representing supplies of the products at the origins and demands of the products at the destinations, capacity of the conveyances, associated with the system constraints are either fuzzy numbers (FNs) or fuzzy random variables (FRVs) with some known continuous fuzzy probability distributions. The multi-choice cost parameters are considered as FNs. In this chapter, two objectives are considered: total transportation cost and total transportation time. As the transportation cost mainly depends on fuel prices and since fuel prices are highly fluctuating, the cost parameters are taken as multi-choice cost parameters with possibilistic uncertain nature. The time of transportation mainly depends on vehicle conditions, quality of roads, and road congestion. Due to these uncertain natures, the parameters representing time of transportation are also taken as fuzzy uncertain multi-choice parameters. In this transportation model, these objectives are minimized satisfying the constraints: product availability constraints, requirement of the product constraints, and capacity of the conveyance constraints. Numerical examples are provided for the sake of illustration of the methodology presented in this chapter, and also achieved solutions are compared with the solutions obtained by some existing methodologies to establish its effectiveness.

Goal programming approach to solving network design problem with multiple objectives and demand uncertainty

Disasters are the unfortunate suddenly occurring events which collapse a living society within a moment. Facebook, the most trending social media have initiated an active role to response the crisis developing a safety check feature to know about the demand of victims in an area. Through this feature the Facebook authority can confirm the requirement of the affected people in a specific area. This research have introduced a mathematical model for multi-objective solid transportation problem to quick delivery of the relief products to the devastating society according to the most prior demand of an area acquiring the best information through safety check feature launched by Facebook with limited resources. The two main objectives of the model are to minimize the total cost and minimize the total time so that relief can be executed under a limited budget responding quickly to the affected area. In this paper, fuzzy goal programming is adopted to solve the multi-objective solid transportation problem. To contemplate the performance of the model, a numerical example is presented in this paper. The numerical example is solved with LINGO optimization solver.

In this problems is shown that the problem of allocating units of blood from the central depots of a regional blood transfusion centre to the hospital blood banks of its area, can be formulated as a multiobjective transportation problem. Various objectives can be taken into consideration such as the minimisation of the expected expiries, the minimisation of the variance of the expiries, transportation costs or distances etc.. An application of the model is examined.

In classical fractional transportation programming problem, we want to optimize the objective function in the form of one or several ratios subject to some linear constraints. If in multi-objective transportation problem, objective function is in ration of two linear function under some linear restrictions, then the problem is called multi-objective linear fractional transportation problem (MOLFTP). In this paper we propose a new method to solve multi-objective linear fractional transportation problem which is extension of Nomani et al. (Int J Manag Sci Eng Manag (2016) [9]). Two numerical problems are presented to validate the proposed algorithm.

Goal programming (GP) is a multi-objective extension of linear programming. Interval GP (IGP) is one of the earliest methods to expand the range of preferred structures in GP. The decision maker’s (DM’s) utility or preference in IGP is investigated by incorporating a widening range of underlying utility functions, commonly known as penalty functions. The basic idea of these functions is that undesirable deviations from the target levels of the goals are penalized regarding a constant or variable penalty value. The main concern with introducing the penalty functions is providing a wide range of a priori preference structures. Yet, the evaluation of how undesirable deviations are penalized based on DM’s behavioral preferences is not sufficiently addressed in the penalty function types developed in the GP literature. In real-world scenarios involving risk, the achievement levels of decision-making attributes are typically associated with the behavior of the DM. In such scenarios, the DM’s unavoidable attitude toward risk should be integrated into the decision-making process. We introduce the concept of behavioral penalty functions into the IGP approach, incorporating a risk aversion parameter tailored to the nature of each attribute to address this gap. This concept offers an innovative framework for capturing the preferences of the DMs and their various attitudes toward risk within the IGP approach. In this paper, we first introduce the concept of behavioral penalty functions. Next, we develop a behavioral utility-based IGP model. Finally, we present a portfolio selection case study to demonstrate the applicability and efficacy of the proposed procedure, followed by a post-optimality analysis and comparisons with other GP approaches.

In this study, we propose an exact method for optimizing a linear function over the efficient set of a multi-objective transportation problem (MOTP). This type of problem arises when a decision maker needs to optimize a preference function across numerous efficient solutions. We developed a cutting plane method that iteratively adds efficient cuts to the relaxation of the original problem until a feasible and optimal solution is achieved. The method is enhanced by a lower bound, enabling a more efficient search for the optimal solution.

This article addresses a special class of non-linear programming problems, viz., linear fractional programming problems having multiple objectives. In solving the real-life linear fractional optimization problems, the ambiguity and hesitation in the decision are inherent and ever-present, therefore, it is perfectly viable to formulate and solve these optimization models using the intuitionistic fuzzy environment. The purpose of this study is to propose a simple and computationally efficient approach to obtain the solution of multiple objective linear fractional programming problems having all the decision variables and parameters expressed in terms of triangular intuitionistic fuzzy numbers. The proposed solution algorithm is primarily based on the goal programming approach, fuzzy-based linearization technique, and a membership function strategy. The original linear fractional programming problem is first converted to its equivalent deterministic/crisp multi-objective linear fractional optimization problem using the weighted goal programming methodology along with the linear membership technique to resolve the intuitionistic fuzzy constraints into the crisp one. Finally, the variable transformation technique for the under- and over-deviational variables of the goal programming model is employed to linearize all the fractions involved in the problem so as to convert the original problem to an equivalent linear optimization problem. Further, this linear programming problem can be solved using any available commercial packages. Moreover, a numerical illustration is provided to demonstrate the steps of the proposed technique followed by the analysis and solution of an E-education set-up problem. The discussion and comparisons of the practical case establish the relevancy and usefulness of the proposed model.

In the present paper, a fuzzy programming model with quadratic membership functions has been developed for the solution of a Multi-Objective Transportation problem. In literature, several fuzzy programming approaches exist with various types of membership functions such as linear, exponential, hyperbolic etc. These membership functions are defined, by taking the lower and upper values of the objective functions into account. In some cases, these methods fail to obtain an integer compromise optimal solution. In the present method, two coefficients of the quadratic membership functions are determined by the lower and upper values of the objective functions. The other coefficient is taken as a variable in the fuzzy programming approach. This means that the membership curve is fixed at the two end points and set free in between. Application of the method on numerical examples proved that the approach could generate integer compromise optimal solutions.

Multi-choice goal programming with utility functions

The single-objective fuzzy or non-fuzzy transportation problem (TP) is not capable of dealing with real-life decision-making problems due to our current competitive market state. In this article, we investigate a fully integer interval multi-objective transportation problem (FIIMOTP) and a fully fuzzy integer multi-objective transportation problem (FFIMOTP). We also provide two solution approaches for solving the FIIMOTP and FFIMOTP. Numerical examples are provided to validate these two approaches. Our results show that the proposed algorithm hugely outperforms the best solution approaches.

Multi-objective optimization model for railway heavy-haul traffic: Addressing carbon emissions reduction and transport efficiency improvement

DINAS (Dynamic Interactive Network Analysis System) is an interactive system to aid in the solution of various multiobjective transportation problems with facility location. DINAS utilizes an extension of the reference point approach for interactive handling multiple objectives. In this approach the decision-maker forms his requirements in terms of aspiration and reservation levels, i.e., he specifies acceptable and required values for given objectives. A special solver was developed to provide DINAS with solutions to single-objective problems. It is based on the branch and bound scheme with a pioneering implementation of the simplex special ordered network (SON) algorithm with implicit representation of the simple and variable upper bounds (SUB & VUB). DINAS is prepared for IBM-PC XT/AT or compatibles as a menu-driven and easy in usage system equipped with a special network editor which reduces to minimum effort associated with data input for real-life problems.