Fuzzy Goal Programming

Fuzzy goal programming uses the ideal values as targets and minimizes the maximum normalized distance from the ideal solution for each objective. An ideal solution is the vector of best values of each criterion obtained by optimizing each criterion independently ignoring other criteria. In this example, ideal solution is obtained by minimizing price, lead-time, and quality independently. In most situations, the ideal solution is an infeasible solution since the criteria conflict with one another. 

If M equals the maximum deviation from the ideal solution, then the fuzzy goal programming model is as follows  

Constraints (6.28 through 6.37) stated earlier will also be included in this model, except that the target for Equations 6.28 through 6.30 are set to their respective ideal values. In this model Xi, and 3 are scaling constants to be set by the user. A common practice is to set the values X2, equal to the respective ideal values. The advantage of Fuzzy GP is that no target values have to be specified by the DM. 

For additional readings on the variants of fuzzy GP models, the reader is referred to Ignizio and Cavalier (1994), Tiwari et al. (1986), Tiwari et al. (1987), Mohammed (1997), and Hu et al. (2007). An excellent source of reference for goal programming methods and applications is the textbook by Schniederjans (1995). 

We shall now illustrate the four goal programming methods using a supplier order allocation case study. The data used in all four methods is presented next. 

---

Recall that in Fuzzy GP, the ideal values are used as targets for the different goals. The solution obtained using fuzzy goal programming is shown in Table 6.37. The scaling constants are calculated such that all the objectives have similar values. 

Hu, C. R, C. J. Teng, and S. Y. Li. 2007. A fuzzy goal programming approach to multi objective optimization problem with priorities. European Journal of Operational Research. 176 1319-1333. 

Tiwari, R. N., S. Dharmar, and J. R. Rao. 1986. Priority structure in Fuzzy goal programming. Fuzzy Sets and Systems. 19 251-259. 

In this section, we propose a weighted fuzzy goal programming (GP) model to solve the bi-criteria problem. In fuzzy GP, ideal values can be used as targets for the objectives (Masud and Ravindran 2008 Subramanian et al. 2013). In the proposed model, we minimize the sum of the weighted satisfaction degrees of the objectives. The proposed weighted fuzzy GP model is as follows ... 

In this section, we solve the bi-criteria model using fuzzy goal programming approach. The results confirm that the two objective functions, maximization of the supply chain profit and maximization of the supply chain density, are conflicting objectives. The increase in supply chain density leads to a decrease in supply chain profit and vice versa. The graphical representation of an efficient frontier is in Fig. 5. 

Mehrbod et al. (2012) considered a multi-product, multi-time period CLSC network with manufacturing plants, distributors, retailers, return product collection centers, and recycling centers. The authors developed a bi-criteria mixed integer nonlinear program for the problem to minimize supply chain costs, delivery time of new products, and collection time of returned products. An interactive fuzzy goal programming method was used to solve the problem. 

Mehrbod et al. (2012) Minimize cost, nimimize delivery and collection time MILP, solved using interactive fuzzy goal programming Strategic... 

Mehrbod, M., Tu, N., Miao, L., and Wenjing, D. Interactive fuzzy goal programming for a multi-objective closed-loop logistics network. Annals of Operations Research 201, no. 1 (2012) 367-381. 

Ravi, V. and Reddy, P.J. (1998) Fuzzy linear fractional goal programming applied to refinery operations planning. Fuzzy Sets and Systems, 96, 173. 

All four goal programming models (Preemptive, Non-Preemptive, Tchebycheff, and Fuzzy) are used to solve the supplier order allocation problem. Each model produces a different optimal solution. They are discussed next. 

Mohammed, R. H. 1997. The relationship between goal programming and fuzzy programming. Fuzzy Sets and Systems. 89 215-22Z... 

Deporter, E.L., Ellis, K.P., 1990. Optimization of project networks with goal programming and fuzzy linear programming. Comput. Ind Eng. 19,500-504. 

---