Mathematical Programming Approaches

Mathematical programming models are used to optimize decisions concerning execution of certain activities subject to resource constraints. Mathematical programming models have a well-defined structure. They consist of mathematical expressions representing objective function and constraints. The expressions involve parameters and decision variables. The parameters are input data, while the decision variables represent the optimization outcome. The objective function represents modeling objectives and makes some decisions more preferable than others. The constraints limit the values that decision variables can assume. 

There are also some important limitations. Mathematical programming models have a lower level of validity compared to some other typies of models—particularly, simulation. In the supply chain configuration context, mathematical programming models have difficulties representing the dynamic and stochastic aspects of the problem. Additionally, solving of many supply chain configuration problems is computationally challenging. 

Following the supply chain configuration scope, mathematical programming models are suited to answer the following supply chain configuration questions  

How do specific parameters influence supply chain performance  

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S JiihrMis of Mass-Exch fnge JVel nrks A Mathematical Programming Approach... 

Grossmann, I. E. J. A. Caballero and H. Yeomans. Mathematical Programming Approaches to the Synthesis of Chemical Process Systems. Korean J Chem Eng 16(4) 407-426 (1999). 

Martel A (2005) The design of production-distribution networks a mathematical programming approach. In Geunes J, Pardalos PM (eds) Supply Chain Optimization. Springer, Berlin et al., pp 265-305... 

Grossmann I.E., Caballero J.P. and Yeomans H. 1999. Mathematical programming approaches to the synthesis of chemical process systems, Korean J. Chem. Eng., 16(4), 407-426. 

Chu, Sydney C. K. (1995), A mathematical programming approach towards optimized master production scheduling, International Journal of Production Economics, 38, 269-279. 

Degraeve, Z. and R Roodhooft. 2000. A mathematical programming approach for procurement using activity based costing. Journal of Business Finance and Accounting. 27 69-98. 

In terms of analytical methods, Weber et al. (1991) present an important review of the most relevant articles in supplier selection. Out of these articles, only 10 of them used mathematical programming. Ghodsypour and O Brien (2001) also present a comprehensive review of the supplier selection methods. These methods vary in their level of complexity, from simple sourcing and matrix methods to advanced mathematical programming approaches. 

Talluri S, Baker RC (2002) A multi-phase mathematical programming approach for effective supply chain design. Eur J Oper Res 141 544—558 Truong TH, Azadivar F (2005) Optimal design methodologies for configuration of supply chains. Int J Prod Res 43 2217-2236... 

We would like to highlight that in the mathematical programming approach with algebraic modeling systems, the declaration of models is expressed with an index notation, and not matrix notation, as seen in Table 10.4, where both notations are shown for an LP problem. 

Zsitkai, Z., Rev, E., Fonyo, Z., Msiza, A.K. and Fraser, D.M. (2002). Comparison of different mathematical programming approaches for mass exchange network synthesis. Proceedings ESCAPE-12 Conference, Eds Grievink, J., Schijndel, J. van. The Hague, The Netherlands, 361-366, Elsevier, Amsterdam. 

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