An Approach for Solving Goal Programming Problems using Interval Type-2 Fuzzy Goals

where ^∼B_k ∈ F₁ the fuzzy aspiration level, dk1,dk2 ∈ R are negative and positive deviations from the goal b_k, A_k is the set of n constraints related to fuzzy goals, A'_k is a set of crisp constraints of the problem, B'_k is its set of boundaries, and x ∈ R^m is the set of decisionvariables of the problem.

Finally, the proposal of Yang [20] is based on a simpler LP model in which G(x) ≡ A_kx, as follows:

where λ ∈ [0,1] is the global satisfaction degree of all goals.

This approach uses λ as a global variable that represents the satisfaction of every fuzzy goal μ_Bk. The first constraint represents the satisfaction degree for b_k ≤ G_k(x) ≤ b_k+b_k2 (see Eq. (3)), and the second constraint represents the satisfaction degree for b_k -b_k1 ≤ G_k(x) ≤ b_k (see Eq. (3)). As x is a free variable in this model, it operates over A_kx and finally moves λ to its maximum value.

5 Goal programming with Interval Type-2 fuzzy aspiration levels

Disagreement among people who are involved into decision making is a common issue in real scenarios. Some people is pessimistic while others are optimistic about different goals. This leads to have different perceptions coming from different experts, so we handle those perceptions using two functions LMF and UMF which are defined as follows:

where ¯μ_^∼bk defines the UMF of the k_th goal, and μ_^∼bk defines the LMF of the k_th goal. A graphical display of a Interval Type-2 fuzzy goal is shown in Figure 1.  

Based on (3) and (5), we extend its results to a Interval Type-2 fuzzy environment. Thus, we define the Interval Type-2 fuzzy aspiration level as ˜b_k which leads to the following LP model:

where ^∼B_k ∈ F₂ is the Interval Type-2 fuzzy aspiration level, dk1,dk2 ∈ R are negative and positive deviations from the goal B_k, A_k is the set of n constraints, and x ∈ R^m is the set of decision variables of the problem.  

Therefore, we extend the proposal of Yang [20] to a Interval Type-2 fuzzy model using a two-step method that finds two different λ values, one for ¯μ_˜b and one for μ_˜b. To do so, we have to solve the following two LPs:

where ¯λ is the overall upper satisfaction degree of the goals, and λ is the overall lower satisfaction degree of the goals. A_kx is the k_th technological constraint, and ¯b_k1,b_k1,¯b_k2,b_k2 are the admissible deviations from b_k.

Our approach finds two values:min{λ} = λ* and max{¯λ} = ¯λ* that represent pesimistic and optimistic perceptions about b_k, and also compose the interval [λ*,¯λ*] = {λ* ∈ [0,1]|λ* ≤ λ* ≤ ¯λ*}of satisfaction of all experts.

6 Experimentation and Results

6.1 Interval Type-2 fuzzy Goals

As application example we use the proposed by Narasimhan [15] and extended by Chen & Tsai [3] which is composed by three fuzzy goals, as shown as follows:

where x₁ and x₂ are the manufacturing quantities of two products which regard to three goals:G₁ is a profitability goal, and G₂ , G₃ are the expected selling quantities per product. The maximum deviations from G_k = {630, 7, 4} are symmetrically handled where b_k1 = b_k2 = {10, 2, 2}.  

We use those values as the LMF of an extended problem e.g b_k1 = b_k2 = {10,2,2}, and the UMF is defined using ¯b_k1 = ¯b_k2 = {15,3,3}. Now, their LPs are based on Eqs. (9) and (10):

The solution of the model (12) is ¯λ* = 0.76 reached by (x1,x2) = (6.28, 3.28) and the solution of (13) is λ* = 0.64 with (x1,x2) = (6.28, 3.28). A graphical description of the results can be seen in Figures 3, 4 and 5 respectively (see Appendix 1). The optimal value of the goals 1, 2 and 3 are 633.6, 6.28 and 3.28 respectively for both ¯λ* and λ*.  

Both models reach the same values of the decision variables x1 and x2 which is a logical solution since all deviations are defined as L1 distances using ¯λ and λ as linear functions of d_k1 and d_k2. The optimal values ¯λ* = 0.76 and λ* = 0.64 are global satisfaction degrees, which means that all three goals are satisfied at the same level.

Now, we solve another example to illustrate how an embedded Type-1 fuzzy set (Be) into FOU (^∼b_k ) works. To do so, we have selected the deviations for goals 1, 2 and 3 as 12.5, 2.5, 2.5 which corresponds to the middle point of the support of FOU(^∼b_k) as shown in Figure 2:

The LP formulation based on Eq. (5) is:

The crisp solution to the problem (14) given Type-1 fuzzy goals is λ*=0.712 which fits into the obtained range [ λ*= 0.64, ¯λ* = 0.76] obtained through (9) and (10). Also note that any  B_e ∈ F₁ embedded into supp(^∼b_k) leads to an optimal satisfaction degree λ* that fits into the range [ λ, ¯λ] as described in the Appendix 2. By continuity of LP models (see Dantzig [5], Hladík [8], and Fiedler et al [6]), if ^∼b_k ∈ F₂ is continuous then its UMF leads to an optimalsolution namely ¯λ* and every B_e ∈FOU(^∼b_k) leads to an optimal satisfaction degree, namely λ* e ∈ [λ, ¯λ].

7 Concluding Remarks

We have presented and solved an extension of the fuzzy goal programming basic model proposed by Narasimhan [15], Yang [20] ,and Chen & Tsai [3] to a Interval Type-2 fuzzy environment, which includes linguistic uncertainty and numerical imprecision coming from multiple experts opinions and perceptions.

Our approach gives the model flexibility to find other kind of solutions in cases where the system has no the ability to fulfill all goals. As higher λ* as closer to the goal the model is. ¯λ and λ describe overall optimistic and pessimistic satisfaction degrees regarding different experts of the system.

There is a relationship among λ*, d_k1 and d_k2 since as wider ˜b_k as higher λ* is, which means higher satisfaction values. In the first example if d_k1  and d_k2  are increased in a 50% then λ* is increased only in 12%, and if d_k1  and d_k2  are decreased in a 50% then α* is decreased in 36% (LP formulations for 50% decreased d_k1 , d_k2 ,¯λ and λ are shown in Appendix 2). Finally, all experts are satisfied into the range [ λ = 0.64, ¯λ = 0.76].

The second example shows an embedded set A_e into FOU(˜b_k) whose optimal satisfaction degree λ* fits into the range [λ, ¯λ] as described in the Appendix 2. This helps decision making when having multiple experts and helps to see how different selections of A_e affect the problem.

References

1. A. Charnes and W. W. Cooper. Management models and industrial applications of linear programming, vol. i, 1961.

2. Abraham Charnes and William Wagner Cooper. Goal programming and multiple objective optimizations:Part 1. European Journal of Operational Research, 1(1):39-54, 1977.

3. Liang-Hsuan Chen and Feng-Chou Tsai. fuzzy goal programming with different importance and priorities. European Journal of Operational Research, 133(3):548-556, 2001.

4. Ting-Yu Chen. An electre-based out ranking method for multiple criteria group decision making using interval type-2 fuzzy sets. Information Sciences, 263:1-21, 2014.

5. George Dantzig. Linear Programming and Extensions. Princeton University Press, 1963.

6. M. Fiedler, J. Nedoma, J. Ramík, J. Rohn, and K. Zimmermann. Linear Optimization Problems with Inexact Data. Springer-Verlag, 2006.

7. Edward L Hannan. On fuzzy goal programming*. Decision Sciences, 12(3):522-531, 1981.

8. Milan Hladík.Weak and strong solvability of interval linear systems of equations and inequalities. Linear Algebra and its Applications, 438(11):4156-4165, 2013.

9. Chaofang Hu, Shaokang Zhang, and Na Wang. Enhanced interactive satisficing method via alternative toler- ance for fuzzy goal programming with progressive preference. Applied Mathematical Modelling, 38(19):4673- 4685, 2014.

10. Kaveh Khalili-Damghani, Soheil Sadi-Nezhad, and Madjid Tavana. Solving multi-period project selection problems with fuzzy goal programming based on topsis and a fuzzy preference relation. Information Sciences, 252:42-61, 2013.

11. George Klir and Bo Yuan. fuzzy sets and fuzzy logic, volume 4. Prentice Hall New Jersey, 1995.

12. Gang Li. fuzzy goal programming-a parametric approach. Information Sciences, 195:287-295, 2012.

13. Jerry M Mendel. Uncertain rule-based fuzzy logic system:introduction and new directions. Prentice Hall PTR, 2001.

14. Jerry M. Mendel. On centroid computations for Type-2 fuzzy sets. Appl. Comput. Math., 10(1):88-96, 2011.

15. Ram Narasimhan. Goal programming in a fuzzy environment. Decision sciences, 11(2):325-336, 1980.

16. M. Pagola, C. Lopez-Molina, J. Fernandez, E. Barrenechea, and H. Bustince. Interval type-2 fuzzy sets con- structed from several membership functions. application to the fuzzy thresholding algorithm. IEEE Transac- tions on fuzzy Systems, 21(2):230-244, 2013.

17. Jindong Qin and Xinwang Liu. Multi-attribute group decision making using combined ranking value under interval type-2 fuzzy environment. Information Sciences, 297:293-315, 2015.

18. Turgay Safiye and Taskin Harun. fuzzy goal programming for health-care organization. Comp. & Ind. Eng., 2014.

19. I.B. Türksen. Interval valued fuzzy sets based on normal forms. fuzzy Sets and Systems, 20(2):191-210, 1986.

20. Taeyong Yang, James P Ignizio, and Hyun-Joon Kim. fuzzy programming with nonlinear membership functions:piecewise linear approximation. fuzzy sets and systems, 41(1):39-53, 1991.

21. Zhiming Zhang and Shouhua Zhang. A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets. Applied Mathematical Modelling, 37(7):4948-4971, 2013.

A Appendix 1

This appendix contains the results of the optimization process for the first Interval Type-2 fuzzy goals example.

B Appendix2

This appendix shows the model of the Type-1 example shown in Section 6 where b_k1 = b_k2 = {5, 1, 1} and ¯b_k1 = ¯b_k2 = {10, 2, 2}: