**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.

**>>Full text**

**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.

**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.

**REFERENCES**

- BARIL, C., S. YACOUT, B. CLÉMENT,
**An Interactive Multi-objective Algorithm for Decentralized Decision Making in Product Design**. Optimization and Engineering, vol. 13, no. 1, 2012, pp. 121-150. - BARIL, C., S. YACOUT, B. CLÉMENT, Design for Six Sigma through Collaborative Multi-objective Optimization. Computers & Industrial Engineering, vol. 60, no. 1, 2011, pp. 43-55.
- CHEN, W., A. SAHAI, A. MESSAC, G. SUNDARARAJ,
**Exploration of the Effectiveness of Physical Programming in Robust Design**. Journal of Mechanical Design, vol. 122, no. 2, 2000, pp. 155-163. - GULSUN, B., G. TUZKAYA, U. R. TUZKAYA, S. ONUT,
**An Aggregate Production Planning Strategy Selection Methodology based on Linear Physical Programming**. International Journal of Industrial Engineering, vol. 16, no. 2, 2009, pp. 135-146. - HERNANDEZ, G., J. K. ALLEN, F. MISTREE,
**The Compromise Decision Support Problem: Modeling the Deviation Function as in Physical Programming**. Engineering Optimization, vol. 33, no. 4, 2001, pp. 445-471. - HUANG, H.-Z., Z.-G. TIAN, M. ZUO,
**Multiobjective Optimization of Three-stage Spur Gear Reduction Units using Interactive Physical Programming**. Journal of Mechanical Science and Technology, vol. 19, no. 5, 2005a, pp. 1080-1086. - HUANG, H.-Z., X. ZHANG, Z.-G. TIAN, C.-S. LIU, Y.-K. GU,
**Optimal Design of Conic-Cylindrical Gear Reduction Unit Using Fuzzy Physical Programming**. Intelligent Information Processing II, Springer, Boston, 2005b, pp. 191-200. - HUANG, H., Z. TIAN,
**Application of Neural Network to Interactive Physical Programming**. Advances in Neural Networks – ISNN 2005, Springer Berlin / Heidelberg.2005, pp. 194-205. - HUANG, H., Z. TIAN, Y. GU,
**Reliability and Redundancy Apportionment Optimization using Interactive Physical Programming**. International Journal of Reliability, Quality and Safety Engineering, vol. 11, no. 3, 2004, pp. 213-222. - ILGIN, M. A., S. M. GUPTA,
**Remanufacturing Modeling and Analysis**. Boca Raton, Florida, CRC Press, 2012. - IMTANAVANICH, P., S. M. GUPTA,
**Evolutionary Computation with Linear Physical Programming for Solving a Disassembly-to-Order System**. Proceedings of the SPIE International Conference on Environmentally Conscious Manufacturing VI, Boston, Massachusetts, 2006a, pp. 30-41. - IMTANAVANICH, P., S. M. GUPTA,
**Linear Physical Programming Approach for a Disassembly-to-Order System under Stochastic Yields and Product’s Deterioration**. Proceedings of the 2006 POMS Meeting, Boston, MA 2006b, Paper no. 004-0213. - KONGAR, E., S. M. GUPTA,
**Disassembly-to-Order System using Linear Physical Programming**. Proceedings of 2002 IEEE International Symposium on Electronics and the Environment, San Francisco, California, 2002, pp. 312-317. - KONGAR, E., S. M. GUPTA,
**Solving the Disassembly-to-Order Problem using Linear Physical Programming**. International Journal of Mathematics in Operational Research, vol. 1, no. 4, 2009, pp. 504-531. - KONGSUWAN, P., S. SANGMUN,
**Integrating Physical Programming to Information Security System Management**. Proceedings of the 11th International Conference on Advanced Communication Technology, Phoenix Park, South Korea, 2009, pp. 143-148. - KOVACH, J., B. CHO, J. ANTONY,
**Development of an Experiment-based Robust Design Paradigm for Multiple Quality Characteristics using Physical Programming**. The International Journal of Advanced Manufacturing Technology, vol. 35, no. 11, 2008, pp. 1100-1112. - KUMAR, V., M. TRIPATHI, M. PANDEY, M. TIWARI,
**Physical Programming and Conjoint Analysis-based Redundancy Allocation in Multistate Systems: a Taguchi Embedded Algorithm Selection and Control (TAS&C) Approach**. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, vol. 223, no. 3, 2009, pp. 215-232. - LAI, X., M. XIE, K.-C. TAN,
**QFD Optimization using Linear Physical Programming**. Engineering Optimization, vol. 38, no. 5, 2006, pp. 593-607. - LAMBERT, A. J. D., S. M. GUPTA,
**Disassembly Modeling for Assembly, Maintenance, Reuse, and Recycling**. Boca Raton, Florida, CRC Press, 2005. - LI, H., M. MA, Y. JING,
**A New Method based on LPP and NSGA-II for Multiobjective Robust Collaborative Optimization**. Journal of Mechanical Science and Technology, vol. 25, no. 5, 2011a, pp. 1071-1079. - LI, L., X. CHU, L. GAO, Q. BO, X. SHAO,
**Analytical Target Cascading based on Physical Programming**. Proceedings of IEEE International Conference on Systems Man and Cybernetics, Istanbul, Turkey 2010, pp. 3060-3067. - LI, L., Y.-H. ZHANG,
**Study on Multi-objective Optimization Based on the Integration of Linear Physical Programming within Collaborative Optimization**. Proceedings of the International Conference on Fuzzy Systems and Knowledge Discovery, Sichuan, China, 2012, pp. 2711-2714. - LI, W., M. J. ZUO, R. MOGHADDASS,
**Optimal Design of a Multi-State Weighted Series-Parallel System Using Physical Programming and Genetic Algorithms**. Asia-Pacific Journal of Operational Research, vol. 28, no. 4, 2011b, pp. 543-562.

- LI, X., M. XIE, N. SZU HUI,
**Multi-Objective Optimization Approaches to Software Release Time Determination**. Asia-Pacific Journal of Operational Research, vol. 29, no. 3, 2012, pp.1-19. - LIN, J., Z. LUO, L. TONG,
**A New Multi-objective Programming Scheme for Topology Optimization of Compliant Mechanisms**. Structural and Multidisciplinary Optimization, vol. 40, no. 1, 2010, pp. 241-255. - LUO, Y., G. TANG, G. PARKS,
**Multi-objective Optimization of Perturbed Impulsive Rendezvous Trajectories using Physical Programming**. Journal of Guidance, Control, and Dynamics, vol. 31, no. 6, 2008, pp. 1829-1832. - MA, X., B. DONG,
**Linear Physical Programming-Based Approach for Web Service Selection**. Proceedings of the International Conference on Information Management, Innovation Management and Industrial Engineering, Taipei, Taiwan, 2008, pp. 398-401. - MARIA, A., C. MATTSON, A. ISMAIL-YAHAYA, A. MESSAC,
**Linear Physical Programming for Production Planning Optimization**. Engineering Optimization, vol. 35, no. 1, 2003, pp. 19-37. - MARTINEZ, M., A. MESSAC, M. RAIS-ROHANI,
**Manufacturability-based Optimization of Aircraft Structures using Physical Programming**. AIAA Journal, vol. 39, no. 3, 2001, pp. 517-525. - MASSOUD, A., S. M. GUPTA,
**Linear Physical Programming for Solving the Multi-criteria Disassembly-to-Order Problem under Stochastic Yields, Limited Supply, and Quantity Discount**. Proceedings of 2010 Northeast Decision Sciences Institute Conference, Alexandria, Virginia, 2010, pp. 474-479. - MCALLISTER, C. D., T. W. SIMPSON, K. HACKER, K. LEWIS, A. MESSAC,
**Integrating Linear Physical Programming within Collaborative Optimization for Multiobjective Multidisciplinary Design Optimization**. Structural and Multidisciplinary Optimization, vol. 29, no. 3, 2005, pp. 178-189. - MELACHRINOUDIS, E., A. MESSAC, H. MIN,
**Consolidating a Warehouse Network: A Physical Programming Approach**. International Journal of Production Economics, vol. 97, no. 1, 2005, pp. 1-17. - MELACHRINOUDIS, E., H. MIN, A. MESSAC,
**The Relocation of a Manufacturing/Distribution Facility from Supply Chain Perspectives: A Physical Programming Approach**. Advances in Management Science, Multicriteria Applications. Kenneth Laurence, editor, Elsevier Science Inc., Vol. 10, 2000, pp.15-39. - MESSAC, A.,
**Physical Programming: Effective Optimization for Computational Design**. AIAA Journal, vol. 34, no. 1, 1996, pp. 149-158. - MESSAC, A.,
**Control-structure Integrated Design with Closed-form Design Metrics using Physical Programming**. AIAA Journal, vol. 36, no. 5, 1998, pp. 855-864. - MESSAC, A.,
**From Dubious Construction of Objective Functions to the Application of Physical Programming**. AIAA Journal, vol. 38, no. 1, 2000, pp. 155-163. - MESSAC, A., W. M. BATAYNEH, A. ISMAIL-YAHAYA,
**Production Planning Optimization with Physical Programming**. Engineering Optimization, vol. 34, no. 4, 2002a, pp. 323 – 340. - MESSAC, A., X. CHEN,
**Visualizing the Optimization Process in Real-time using Physical Programming**. Engineering Optimization, vol. 32, no. 6, 2000, pp. 721-747. - MESSAC, A., S. M. GUPTA, B. AKBULUT,
**Linear Physical Programming: A New Approach to Multiple Objective Optimization**. Transactions on Operational Research, vol. 8, no. 2, 1996, pp. 39-59. - MESSAC, A., P. HATTIS,
**Physical Programming Design Optimization for High Speed Civil Transport**. Journal of Aircraft, vol. 33, no. 2, 1996, pp. 446-449. - MESSAC, A., A. ISMAIL-YAHAYA,
**Multiobjective Robust Design using Physical Programming**. Structural and Multidisciplinary Optimization, vol. 23, no. 5, 2002, pp. 357-371. - MESSAC, A., M. P. MARTINEZ, T. W. SIMPSON,
**Effective Product Family Design Using Physical Programming**. Engineering Optimization, vol. 34, no. 3, 2002b, pp. 245 – 261. - MESSAC, A., M. P. MARTINEZ, T. W. SIMPSON,
**Introduction of a Product Family Penalty Function Using Physical Programming**. Journal of Mechanical Design, vol. 124, no. 2, 2002c, pp. 164-172. - MESSAC, A., C. MATTSON,
**Generating Well-distributed Sets of Pareto Points for Engineering Design using Physical Programming**. Optimization and Engineering, vol. 3, no. 4, 2002, pp. 431-450. - MESSAC, A., C. SUKAM, E. MELACHRINOUDIS,
**Mathematical and Pragmatic Perspectives of Physical Programming**. AIAA Journal, vol. 39, no. 5, 2001, pp. 885-893. - MESSAC, A., S. VAN DESSEL, A. A. MULLUR, A. MARIA,
**Optimization of Large-scale Rigidified Inflatable Structures for Housing using Physical Programming**. Structural and Multidisciplinary Optimization, vol. 26, no. 1, 2004, pp. 139-151. - MESSAC, A., B. WILSON,
**Physical Programming for Computational Control**. AIAA Journal, vol. 36, no. 2, 1998, pp. 219-226. - MIRAKHORLI, A., M. H. FARAHANI, F. RAMTIN,
**New Approach in Supplier Selection using Linear Physical Programming**. Proceedings of IEEE/INFORMS International Conference on Service Operations, Logistics and Informatics, Chicago, IL, 2009, pp. 47-51. - MULLUR, A., C. A. MATTSON, A. MESSAC,
**New Decision Matrix based Approach for Concept Selection using Linear Physical Programming**. Proceedings of the AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Conference, Norfolk, VA, 2003, Paper No: AIAA 2003-1446. - NAGRATH, D., M. AVILA-ELCHIVER, F. BERTHIAUME, A. W. TILLES, A. MESSAC, M. L. YARMUSH,
**Soft Constraints-based Multiobjective Framework for Flux Balance Analysis**. Metabolic Engineering, vol. 12, no. 5, 2010, pp. 429-445. - NAGRATH, D., B. W. BEQUETTE, S. M. CRAMER, A. MESSAC,
**Multiobjective Optimization Strategies for Linear gradient Chromatography**. AIChE Journal, vol. 51, no. 2, 2005, pp. 511-525. - NAGRATH, D., C. CANEBA, T. KAREDATH, N. BELLANCE,
**Metabolomics for Mitochondrial and Cancer Studies**. Biochimica et Biophysica Acta (BBA) – Bioenergetics, vol. 1807, no. 6, 2011, pp. 650-663. - NUKALA, S., S. M. GUPTA,
**Strategic and Tactical Planning of a Closed-Loop Supply Chain Network: A Linear Physical Programming Approach**. Proceedings of the 2006 POMS Meeting, Boston, MA 2006, Paper No: 004-0210. - ONDEMIR, O., S. M. GUPTA,
**Order-driven Component and Product Recovery for Sensor-embedded Products (SEPS) Using Linear Physical Programming**. Proceedings of the 41st International Conference on Computers & Industrial Engineering, Los Angeles, CA 2011, pp. 714-719. - ONUT, S., U. R. TUZKAYA, G. TUZKAYA, B. GULSUN,
**A Multi-Objective Energy Resource Allocation Model for Turkish Manufacturing Industry using Linear Physical Programming**. International Journal of Innovative Computing, Information and Control, vol. 7, no. 6, 2011, pp. 3147-3169. - PATEL, M., K. LEWIS, A. MARIA, A. MESSAC,
**System Design through Subsystem Selection using Physical Programming**. AIAA Journal, vol. 41, no. 6, 2003, pp. 1089-1096. - POCHAMPALLY, K. K., S. GUPTA, S. KAMARTHI,
**Identification of Potential Recovery Facilities for Designing a Reverse Supply Chain Network using Physical Programming**. Proceedings of the SPIE International Conference on Environmentally Conscious Manufacturing III, Providence, Rhode Island, 2003, pp. 139-146. - POCHAMPALLY, K. K., S. M. GUPTA,
**A Linear Physical Programming Approach for Designing a Reverse Supply Chain**. Proceedings of the Fifth International Conference on Operations and Quantitative Management Seoul, South Korea, 2004, pp. 261-269. - POCHAMPALLY, K. K., S. M. GUPTA,
**Use of Linear Physical Programming and Bayesian Updating for Design Issues in Reverse Logistics**. International Journal of Production Research, vol. 50, no. 5, 2012, pp. 1349-1359. - POCHAMPALLY, K. K., S. M. GUPTA, K. GOVINDAN,
**Metrics for Performance Measurement of a Reverse/Closed-loop Supply Chain**. International Journal of Business Performance and Supply Chain Modelling, vol. 1, no. 1, 2009a, pp. 8-32. - POCHAMPALLY, K. K., S. NUKALA, S. M. GUPTA,
**Quantitative Decision-Making Techniques for Reverse/Closed-Loop Supply Chain Design**. Environment Conscious Manufacturing, S. M. Gupta and A. J. D. Lambert (Editors), Boca Raton, Florida, USA, CRC Press, 2008, pp. 105-214. - POCHAMPALLY, K. K., S. NUKALA, S. M. GUPTA,
**Strategic Planning Models for Reverse and Closed-Loop Supply Chains**. Boca Raton, Florida, USA, CRC Press, 2009b. - SANCHIS, J., M. A. MARTÍNEZ, X. BLASCO, G. REYNOSO-MEZA,
**Modelling Preferences in Multi-objective Engineering Design**. Engineering Applications of Artificial Intelligence, vol. 23, no. 8, 2010, pp. 1255-1264. - SIMPSON, T. W., J. R. A. MAIER, F. MISTREE,
**Product Platform Design: Method and Application**. Research in engineering Design, vol. 13, no. 1, 2001, pp. 2-22. - SROKA, M., D. LONG,
**Exploring Metric Sensitivity of Planners for Generation of Pareto Frontiers**. Proceedings of the Sixth Starting AI Researchers’ Symposium, Montpellier, France, 2012, pp. 306-317. - SULEMAN, A., M. GONICALVES,
**Multi-objective Optimization of an Adaptive Composite Beam using the Physical Programming Approach**. Journal of Intelligent Material Systems and Structures, vol. 10, no. 1, 1999, pp. 56-70. - TAPPETA, R., J. RENAUD, A. MESSAC, G. SUNDARARAJ,
**Interactive Physical Programming: Tradeoff Analysis and Decision Making in Multicriteria Optimization**. AIAA Journal, vol. 38, no. 5, 2000, pp. 917-926. - TIAN, Z.-G., H.-Z. HUANG, L.-W. GUAN,
**Fuzzy Physical Programming and Its Application in Optimization of Through Passenger Train Plan**, Proceedings of the International Conference on Traffic and Transportation Studies, Guilin, China, 2002, pp. 498-503. - TIAN, Z., D. LIN, B. WU,
**Condition based Maintenance Optimization Considering Multiple Objectives**. Journal of Intelligent Manufacturing, vol. 23, no. 2, 2012, pp. 333-340. - TIAN, Z., M. ZUO,
**Redundancy Allocation for Multi-state Systems using Physical Programming and Genetic Algorithms**. Reliability Engineering & System Safety, vol. 91, no. 9, 2006, pp. 1049-1056. - TIAN, Z., M. J. ZUO, H. HUANG,
**Reliability-redundancy Allocation for Multi-state Series-Parallel Systems**. IEEE Transactions on Reliability vol. 57, no. 2, 2008, pp. 303-310. - TONG, X.-Y., G.-B. CAI, Y.-T. ZHENG, J. FANG,
**Optimization of System Parameters for Gas-generator Engines**. Acta Astronautica, vol. 59, no. 1-5, 2006, pp. 246-252. - WANG, T.-T., X.-K. CHEN, Y. LIN,
**Multi-objective Optimization Based on the Integration of Linear Physical Programming within Analytical Target Cascading**. Proceedings of the 4th International Conference on Biomedical Engineering and Informatics, Shanghai, China 2011, pp. 2286-2289. - WILSON, B. H., C. ERIN, A. MESSAC,
**Optimal Design of a Vibration Isolation Mount using Physical Programming**. Journal of Dynamic Systems, Measurement and Control, vol. 121, no. 2, 1999, pp. 171-178. - ZHANG, N.,
**2D Turbine Airfoil Optimization using Physical Programming**. Proceedings of the International Conference on Mechatronics and Automation, Xi’an, China 2010, pp. 852-856. - ZHANG, N.,
**Physical Programming based Multidisciplinary Optimization for Aircraft Conceptual Parameter Design**. Proceedings of the Chinese Control and Decision Conference, Mianyang, China 2011, pp. 2387-2392. - ZHANG, X., H.-Z. HUANG, L. YU,
**Fuzzy preference based Interactive Fuzzy Physical Programming and its application in multi-objective optimization**. Journal of Mechanical Science and Technology, vol. 20, no. 6, 2006, pp. 731-737.