A practical weight sensitivity algorithm for goal and multiple objective programming

doi:10.1016/j.ejor.2011.03.012

European Journal of Operational Research

Volume 213, Issue 1, 16 August 2011, Pages 238-245

https://doi.org/10.1016/j.ejor.2011.03.012 Get rights and content

Abstract

This paper presents a weight sensitivity algorithm that can be used to investigate a portion of weight space of interest to the decision maker in a goal or multiple objective programme. The preferential information required from the decision maker is an initial estimate of their starting solution, with an equal weights solution being used as a default if this is not available, and preference information that will define the portion of weight space on which the sensitivity analysis is to be conducted. The different types of preferential information and how they are incorporated by the algorithm are discussed. The output of the algorithm is a set of distinct solutions that characterise the portion of weight space searched. The possible different output requirements of decision makers are detailed in the context of the algorithm.

The methodology is demonstrated on two examples, one hypothetical and the other relating to predicting cinema-going behaviour. Conclusions and avenues for future research are given.

Highlights

► A weight sensitivity algorithm for goal and multi-objective programming is presented. ► Solutions from portion of weight space of interest to the decision maker are generated. ► The algorithm is demonstrated on two examples from the literature.

Introduction

A multiple objective programme involves the simultaneous optimisation of a number of normally conflicting objective functions. A subset of the field of MOP is that of distance metric based techniques, where the distance between an ideal or desired solution and solutions that are achievable in practice is minimised. Goal programming (Jones and Tamiz, 2010), compromise programming (Yu, 1973), and the reference point method (Wierzbicki, 1982) are examples of distance metric based techniques.

One issue arising in distance metric based techniques is that of choice of model parameters and consequent sensitivity analysis to determine the stability of the solutions produced. A key example of parameters to be set is the preferential weights associated with the objectives or the deviational variables in the achievement function. Kettani et al. (2004) note that the weighting factor can be divided into two components, a preferential component and a normalising component. In this paper the term ‘weight’ shall henceforth be used to refer to the preferential component of the weighting term. The most basic method of weight assignment is to assume equal decision maker preference for all objectives and hence set all weights to an equal value. This may be a valid starting point for an exploration of possibilities in the absence of any firm decision maker preference(s) or a genuine belief that all goals are indeed equally weighted. However, as a single iteration producing a final solution it is a somewhat crude way of modelling decision maker preference(s). Another possibility is to assign weights based on a priori decision maker information. This is a valid approach if such information exists and is quantifiable. Pairwise comparison methods have also been used to determine a set of weights for the goal programming model (Gass, 1986, Kahraman and Buyukozkan, 2008, Li et al., 2009, Wey and Wu, 2007). Penalty structures (Jones and Tamiz, 1995; Chang and Lin, 2009) and methodology from the field of multi criteria decision analysis such as the Promethee method (Martel and Aouni, 1990) can be used to refine the solutions produced.

All of the above methods should be seen as a starting point for a formal or informal process of weight determination. In many cases the means of interaction with the decision maker in order to produce and refine a weighting scheme is not reported in the literature so it is difficult to make assertions as to its nature. It is therefore hypothesised that the methodology of weight elicitation and exploration and reporting of the weight space is in general currently somewhat random and ad hoc in nature. Hence, Jones and Tamiz (2010, pp. 70–72) develop a systematic methodology for weight space analysis. This algorithm takes an initial solution (equal weight or decision maker specified) and searches the entire weighting space to produce a set of solutions for consideration by the decision maker. However, it many cases it is not appropriate to explore the entire weight space as it may contain areas that lead to solutions that are not in accord with the decision maker’s preferences. Generating such solutions involves an unnecessary computational effort and generates a larger than needed number of solutions for consideration by the decision maker. This paper presents a revised weight space exploration algorithm that allows the decision maker to give additional preference information in order to more effectively guide the bounds of the search and produce a set of more solutions that are in accord with their preferences.

This paper investigates the issues involved in designing a methodology for the revised weight space exploration algorithm. The remainder of the paper is divided into five sections. Section 2 defines the three spaces of the decision problem as well as discussing some theory related to the nature of weight space. Section 3 discusses the requirements of the decision maker in order to make their decision. Section 4 develops the revised weight space exploration algorithm. Section 5 illustrates the developed theory and methodology by means of two examples. Finally Section 6 draws conclusions.

Section snippets

Three spaces of the multi-objective problem

The generic multiple objective programming model (Steuer, 1986) is defined as: $\begin{matrix} Min z = (f_{1} (\underset{̲}{x}), f_{2} (\underset{̲}{x}), \dots . ., f_{k} (\underset{̲}{x})) \\ Subject to : \underset{̲}{x} \in F, \end{matrix}$ where x is a set of m decision variables over which the decision maker(s) has control. These can be continuous, discrete, integer, or binary in nature. They define the m-dimensional decision space X. The set F ⊆ X defines the subset of the decision space with feasible or implementable solutions. F is defined by a number of constraints given as functions of x. These may be

Aiding the decision maker(s)

Attention is now turned on how analysis of the parameter space can be used to aid the decision maker(s) in arriving at a solution in line with their preferences. As noted Section 1, sensitivity analysis for goal programming and or justification of weighting schemes is an area for potential improvement in the goal programming modelling process. Another factor to consider is the means of engagement and level of familiarity with decision making techniques of the decision maker. As is the

A revised weight sensitivity algorithm

This section presents a revised version of the weight space exploration algorithm given by Jones and Tamiz (2010, pp. 70–72). The Jones and Tamiz framework allows an exploration of the entire weight space to take place. However, many decisions already have some preference information a priori in addition to their initial weighting estimate. This preference information can restrict the area in weight space that needs to be explored by the algorithm and hence makes exploration of the entire

Illustrative examples

The revised weight sensitivity analysis algorithm is applied to two goal programmes for the purposes of illustration.

Example 1

The first goal programme is taken from Jones and Tamiz, 2010, Chapter 3 and relates to a hypothetical manufacturing situation. The goal programme has four goals and the generic form of the weighted version using percentage normalisation has the following algebraic structure. $Min a = \frac{w_{1} p_{1}}{120} + \frac{w_{2} n_{2}}{7000} + \frac{w_{3} n_{3}}{40} + \frac{w_{4} n_{4}}{40}$ $Subject to 4 x_{1} + 3 x_{2} + n_{1} - \underset{̲}{p_{1}} = 120$ $100 x_{1} + 150 x_{2} + \underset{̲}{n_{2}} - p_{2} = 7000,$ $x_{1} + \underset{̲}{n_{3}} - p_{3} = 40,$ $x_{2} + \underset{̲}{n_{4}} - p$

Conclusions

A revised weight sensitivity algorithm has been developed and illustrated on two examples in this paper. The algorithm has been illustrated on two goal programming models but the methodology also can be applied to any convex distance-metric based multiple objective programme. Although for some types of model this may require adaptation to include a number of other key parameters, such as the maximum permissible deviations in fuzzy goal programming (Yaghoobi et al., 2008); the parameter

References (19)

C.T. Chang et al.
Interval goal programming for s-shaped penalty function
European Journal of Operational Research
(2009)
R.B. Flavell
A new goal programming formulation
Omega
(1976)
D.F. Jones et al.
Improving the flexibility of goal programming via preference modelling techniques
Omega
(1995)
D.F. Jones et al.
A classification model based on goal programming with non-standard preference functions with application to prediction of cinema-going behaviour
European Journal of Operational Research
(2007)
O. Kettani et al.
The double role of the weight factor in the goal programming model
Computers and Operations Research
(2004)
J.P. Oddoye et al.
Combining simulation and goal programming for healthcare planning in a medical assessment unit
European Journal of Operational Research
(2009)
C. Romero
Extended lexicographic goal programming: a unifying approach
Omega
(2001)
W.M. Wey et al.
Using ANP priorities with goal programming in resource allocation in transportation
Mathematical And Computer Modelling
(2007)
A.P. Wierzbicki
A mathematical basis for satisficing decision making
Mathematical Modelling
(1982)

There are more references available in the full text version of this article.

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Decision SupportA practical weight sensitivity algorithm for goal and multiple objective programming

Abstract

Highlights

Introduction

Section snippets

Three spaces of the multi-objective problem

Aiding the decision maker(s)

A revised weight sensitivity algorithm

Illustrative examples

Conclusions

European Journal of Operational Research

Omega

Omega

European Journal of Operational Research

Computers and Operations Research

European Journal of Operational Research

Omega

Mathematical And Computer Modelling

Mathematical Modelling

Decision Support
A practical weight sensitivity algorithm for goal and multiple objective programming