**Physical Programming: A Review of the State of the Art**

**Mehmet Ali ILGIN ^{1}, Surendra M. GUPTA^{2}**

^{ 1}Celal Bayar University, Muradiye Campus,

Manisa, 45140, Turkey

mehmetali.ilgin@cbu.edu.tr

^{2}Northeastern University, MIE Dept.,

360 Huntington Avenue, Boston, MA, 02115, USA

gupta@neu.edu

**Abstract**: Most traditional multi-criteria optimization techniques require that the decision maker construct an aggregate objective function using the weights determined as a result of a trial and error process. Physical programming (PP) eliminates this tedious weight assignment process by providing decision makers with a flexible and more natural problem formulation. In PP, the decision maker specifies ranges of different degrees of desirability instead of defining weights. In this paper, we present a comprehensive review of PP studies by classifying them into four major categories (viz., methodological papers, industrial engineering applications, mechanical engineering applications and other applications). Insights from the review and future research directions conclude the paper.

**Keywords**: Physical programming, multi-criteria optimization, review.

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**CITE THIS PAPER AS**:

Mehmet Ali ILGIN, Surendra M. GUPTA, **Physical Programming: A Review of the State of the Art**, *Studies in Informatics and Control*, ISSN 1220-1766, vol. 21 (4), pp. 349-366, 2012. https://doi.org/10.24846/v21i4y201201

**1. Introduction**

Most real world decision making and design problems are inherently multi-objective. Yet, many mathematical programming methods (e.g., goal programming, analytical hierarchy process) require the decision maker (DM) to assign physically meaningless weights to express his (her) preferences. Physical programming (PP) avoids the weight assignment by providing preference functions. In PP, DM determines a suitable preference function and specifies ranges of different degrees of desirability (desirable, tolerable, undesirable, etc.) for each criterion. There are eight preference functions classified into 8 classes, 4 soft and 4 hard.

*Soft Classes:*

Class 1S (smaller-is-better, i.e., minimization)

Class 2S (larger-is-better, i.e., maximization)

Class 3S (value-is-better)

Class 4S (range-is-better)

*Hard Classes:*

Class 1H (must be smaller)

Class 2H (must be larger)

Class 3H (must be equal)

Class 4H (must be in range)

It must be noted that selection of hard or soft classes depends on the sharpness of the preference defined by the DM. The properties of class functions are listed below Kongar and Gupta [14]:

- A lower value of a class function is preferred over a higher value thereof.
- A class function is strictly positive.
- Class function is continuous, piecewise linear and convex.
- The value of a class function, zu, at a given ranges-intersection (say, desirable-tolerable) is the same for any class-type.

After defining class functions for each objective, the following minimization is performed for soft classes:

subject to

*fi (x) ≤ fi5 * (for Class 1S objectives)

*fi (x) ≥ fi5* (for Class 2S objectives)

*fi5L ≤ fi (x) ≤ fi5R* (for Class 3S objectives)

*fi5L ≤ fi (x) ≤ fi5R* (for Class 4S objectives)

and for *hard *classes, invoke constraint

*f _{i} (x)* ≤

*f*(for Class 1H objectives)

_{iM}*f*≥

_{i}(x)*f*(for Class 2H objectives)

_{im}*f*=

_{i}(x)*f*(for Class 3H objectives)

_{iv}*f*≤

_{im}*f*≤

_{i}(x)*f*(for Class 4H objectives)

_{iM}*x*≤

_{jm}*x*≤

_{j}*x*(for design var. constraints)

_{jM}where

*f*,

_{im}*f*,

_{iM}*x*and

_{jm}*x*represent minimum and maximum values,

_{jM}*f*helps define the equality constraints; the range limits are provided by the designer (see Figure 1), and

_{iv}*n*is the number of soft objectives that the problem comprises. The above problem model conforms to the framework of most nonlinear programming codes, with possible minor rearrangements.

_{sc}The purpose of this paper is to provide an overview of the PP literature. The literature is organized into four main areas: methodological papers, industrial engineering applications, mechanical engineering applications and other applications. Papers are classified into subcategories in each main area. Section 2 presents the papers which make methodological contributions by modifying the original physical programming methodology. The papers investigating the application of PP to Industrial Engineering and Mechanical Engineering related problems are discussed in Sections 3 and 4, respectively. Section 5 reviews the other application papers. Finally, some concluding remarks and future research directions are presented in Section 6.

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