DECISION MAKING IN FUZZY REASONING TO SOLVE A BACKORDER ECONOMIC ORDER QUANTITY MODEL

. Fuzzy reasoning is the subject of fuzzy system where the fuzzy set is characterized by the randomization of the variable associated in the fuzzy set itself. It is the first-time application of fuzzy reasoning over the backorder economic order quantity (EOQ) inventory management problem. We first define the fuzzy reasoning membership function through the use of L-fuzzy number and possibility theory on fuzzy numbers. Considering the holding cost, set up cost, backordering cost and demand rate as reasoning based fuzzy number, we have constructed a dual fuzzy mathematical problem. Then this problem has been solved over the dual feasible space which is associated to the aspiration level and the fuzzy approximation constant. Numerical study reveals the superiority of the proposed method with respect to the crisp solution as well as general fuzzy solution. Sensitivity analysis and graphical illustrations have also been done to justify the novelty of this article.


Literature review
Traditionally, most of the inventory models are made in deterministic and stochastic environment.But if we wish to study the parameters associated to the inventory model assume fuzzy stochastic or the defined fuzzy membership function itself behave randomize number then it is noted that not a single article has been found in the literature.The related works of fuzzy reasoning is very much limited.Zadeh [1] invented fuzzy set theory to capture the problems of non-random uncertain world.Zadeh [2] applied the concept of Linguistic variable and its application to approximate reasoning.The role of fuzzy logic in management uncertainty was studied by Zadeh [3].Toga and Watanadc [4] discussed an expert system on a chip for real-time approximate reasoning.Pal and Mandal [5,6] studied a linguistic recognition system based on approximate reasoning.Gaines [7] invented the foundations of fuzzy reasoning.Buckley [8] solved a linear programming problem using possibilistic fuzzy sets.Negoita [9] discussed the current interest in fuzzy optimization.Carlsson [10] solved a multi criteria decision making problem using approximate reasoning.Delgado et al. [11,12] studied on imprecise costs in mathematical programming and new ranking method of optimization problems.Beyond the above methods, some remarkable solution methodologies of fuzzy (non)linear programming problems may also be considered over here [13][14][15][16][17][18].Lapuccl et al. [19] developed new optimality methods for cardinality constrained problems.However, in the literature very few articles over fuzzy optimization problem are available in which duality principle has been employed.Ma et al. [20] studied an interval type fuzzy linear programming problem where the matrix inversion method was utilized and found no duality gap.Authors were basically involved to focus no duality gap and several solution procedures with the help of Lagrangian of the primal (non)linear problems, strong-weak duality principles, pareto optimality considering the fuzzy parameters as Triangular and point to set fuzzy numbers [21][22][23][24][25][26].Fuller and Zimmerman [27] gave the concept of fuzzy reasoning in a theoretical manner but he did not apply this theory in any real-life problem.De [28] introduced a research article on inventory management problem using fuzzy reasoning and showed that a duality gap always exists.
Above discussion explores none of the research articles are made for solving the problems of backorder inventory models under fuzzy reasoning in which duality gap within the solution as well as in the feasible space exist.Thus, in this article we solve a backorder EOQ problem that definitely keeps a new footstep in the field of optimization itself.
The organization of this article as developed as follows: Section 2 is a preliminary section that discusses some useful concepts and definitions over fuzzy reasoning.Section 3 gives the formulation of the mathematical model, Section 4 explores the fuzzy mathematical model, Section 5 represents the numerical illustrations, Section 6 includes the sensitivity analysis of the proposed model, Section 7 covers the graphical illustrations and finally Section 8 gives a conclusion.

Preliminaries
Here we shall discuss some basic concepts on fuzzy sets, fuzzy reasoning, defuzzification methods and dual feasible space.

Fuzzy membership function under fuzzy reasoning
We consider the L-fuzzy number ̃︀  with the membership grade where  0 refers initial value of the fuzzy number ̃︀  and  gives tolerance level with the graphical representation defined at Figure 1.Now as per De [28] the approximated fuzzy membership of ̃︀  can be stated in Figure 2 which is the better realization of Figure 3 obtained in (5) that came from the simplification of (2) using ( 3), (4) exclusively.We have where and which gives after simplification Now (5) can be simplified as where 2.2.Choice of the probability density function of the random variable For the random variable  we assume the probability density function ( 8) that satisfies 2.3.Constructions of expected value through -cuts of   () and the values of () and  (︀  2 )︀ can be obtained from (11) and (12). and After substitutions (10) becomes Also, from (6) we get That appears as 2.4.Selection of dual feasible space under -dual  cuts (Extension of De [28]) Here we shall set a dual feasible space as 0 <  < 1 defined in Figure 4 where  and  are complementary to each other.Now, according to the above feasible space, we may approximate the bounds of the crisp number  0 by utilizing (15) and get (16) and the schematic diagram of model optimization is given in Figure 5.

Problem definition under fuzzy duality principle
Let us take Primal fuzzy problem as Minimize ̃︂ () where ̃︂ () and ̃︂  () are both differentiable fuzzy objective and fuzzy constraint functions respectively defined on Ω ⊆ R  .To construct a saddle point problem, we may use the fuzzy Lagrangian and get a fuzzy dual problem for the divergence operator ∇.

Problems for defuzzification
Since  and  are complementary to each other so the above problem ( 9) can be viewed for a maximization and minimization problem under strong-weak cuts of fuzzy numbers simultaneously and they can be put in (19) and ( 20) respectively.
Note.For solving (19) and (20), we may go for an efficient solution algorithm otherwise some software like LINGO 8.0 may be employed.

Formulation of backorder inventory model
In this section we use some notations and assumptions (Sect.3.1) and the development of crisp inventory model with backorder (Sect.3.2).

The mathematical model
Let us consider the classical backorder inventory model (shown in Fig. 6) where the parameter  stands for the time horizon (months), ℎ represents the holding cost per unit quantity per month.Similarly, the parameters , , ,   ,  and  refer to the unit backorder cost, ordering cost, demand rate, inventory cycle time (months), the order quantity and the shortage quantity per cycle respectively.Our aim is to minimize the objective function for the optimum values of the decision variables  and  respectively.
Then we have The average total inventory cost in the planning period [0,  ] is given by The crisp optimal solutions are

Representing mathematical model of fuzzy reasoning
Let all the cost components and the demand rate assume fuzzy flexibilities via fuzzy reasoning.Then the fuzzy problem of (10) can be expressed as Here the tilt bar (∼) denotes the fuzzification of the parameters.Moreover to construct the membership functions of the corresponding fuzzy parameters we may initialize each parameter along with their fuzzy deviation in the following: holding cost assumes value within ( 0 ,  1 ), that for unit backordering cost per cycle is ( 0 ,  2 ), the ordering cost per order is ( 0 ,  3 ), the demand rate is ( 0 ,  4 ) and for the objective function is ( 0 ,  0 ) respectively.Hence the corresponding membership functions are given by: For the parameter , let the membership function be For , let the membership function be For , let the membership function be For , let the membership function be And for the fuzzy objective function, we have Now to carry out the fuzzy problem under fuzzy reasoning, we shall utilize the estimated values of the approximated fuzzy parameters developed at (13) and ( 16) the following relationship would be arrived

Illustrative numerical example
To perform a numerical study, we shall consider the parametric values in the following: Let,  = $10 per unit,  = $20 per unit,  = $200 per cycle,  = 2000 units per months,  = 12 months, then using LINGO 8.0 software we get the optimal solution as  * = 100,  * = 33.3333and the minimal total cost  ( * ,  * ) = 8000 for the crisp model.However, for fuzzy model we may take the initial values and their corresponding tolerances as ( 0 ,  0 ) = (8000, 750), ( 0 ,  1 ) = (10, 2), ( 0 ,  2 ) = (20, 5), ( 0 ,  3 ) = (200, 40) and ( 0 ,  4 ) = (2000, 250) respectively.The optimal results are put in Table 1. the Table 1 it is seen that; the crisp solution is maximum with respect to the general fuzzy and dual space fuzzy reasoning problems.In crisp model, the order quantity and the backorder quantity are 100 units and 33.333 units respectively but for the case of fuzzy reasoning, the case of highest aspiration level they are 135.959units and 45.747 units respectively and that of dual aspiration level they are 110.136units and 36.650units alone.The model minimum becomes $7400.0.
Table 2 shows the optimum value of the decision variable whenever the higher aspiration level is considered.For the ranges of 0.19-0.91 of the aspiration levels the objective value assumes the range $7393.541-$7931.818 with respect to the range of order quantity 109.7847-135.9589units and that for backorder quantity 36.5262-45.9589units exclusively.The lower aspiration level giving lower minimum inventory cost.
Table 3 reveals the values of the objective function under optimum dual aspiration level giving the minimum value $7400.00.The higher dual aspiration level gives the lower average inventory cost.The range of the order quantity and backorder quantity are respectively obtained as 110-138 units and 37-47 units respectively.

Sensitivity of the model
Here we shall perform the sensitivity analysis of the proposed model for each of the fuzzy deviation parameters as well as their initial approximations of the fuzzy cost components and fuzzy demand parameters on and from (−20% to +20%), the outputs are put in Table 4.
Table 4 gives the values of the objective functions that have low positive deviation respect to crisp solution.All the initial values of the cost parameters and the demand parameters including their fuzzy deviations are poorly sensitive.The cost function assumes the range on and from 0.50% to +2.96% but for the change of the initial value (approximation) objective function this range shifted to −19.57% to +22.99% exclusively.However, the amount of order quantity and the backorder quantity keep values around 70 units and 23 units respectively in most of the cases.Another study reveals that, the values of the aspiration levels (in dual space) which are frequently arrived belongs to the weaker family of decision maker's (DM's) satisfactions.Moreover, some cases have arrived where the strong fuzzy/cut sets are very much supportive to have minimum objective value.

Graphical illustrations
Here we shall draw several graphs of the proposed model using the data set obtained from sensitivity analysis Table 4.  Figure 7 reveals the average inventory cost variation with respect to the several aspiration and dual aspiration levels of the fuzzy reasoning model.At the value 0.4, the average inventory costs are equal for both the cases (dual spaces) of fuzzy reasoning model.For increasing aspiration level causes the incremental values of the objective function but for dual aspiration level the results become reverse.
Figure 8 states the optimum order quantity curves due to the variation of dual aspiration levels.The order quantity becomes fixed at 117 units for the fixed dual space point 0.4 alone.Beyond this fixed point the values of the order quantity getting diffracted.
Figure 9 indicates the optimum shortage quantity curve due to the changes of the dual spaces.The backorder quantity remains fixed at 39 units with respect to the aspiration level 0.4, but beyond this value the values of the backorder quantity are getting scattered.
Figure 10 showing the nature of dual axis of the dual space  < 1, 0 <  < 1, 0 <  < 1.The aspiration level curve assumes a concave like shape but its dual assumes almost a straight line.

Conclusion
This study has developed for classical backorder inventory model of fuzzy approximate reasoning.The most fundamental concept of fuzzy reasoning is that the fuzzy membership functions are assumed to be randomized.By this way we are able to get finer optimal solution than the traditional fuzzy alpha cut approach.Here we have introduced a new solution space named dual aspiration level space.In this new feasible region, all the decision variables become finer than the values so obtained from some other existing methods.The managerial insights are (a) The decision maker can order/back order more quantities than (s)he wishes.(b) By this approach the DM can look the model for strong and weak aspiration level simultaneously.(c) The DM has many options to set several strategic plans once at a time.(d) By this method the DM might be able to know the upper limit of the inventory cost that would incur in the inventory process itself.

Figure 7 .
Figure 7. Variation of cost function under fuzzy reasoning.

Figure 8 .
Figure 8. Variation of order quantity over dual space.

Figure 9 .
Figure 9. Variation of optimum shortage quantity over dual space.

Table 2 .
Optimal solution for maximum .

Table 3 .
Optimal solution for maximum .

Table 4 .
Sensitivity analysis of the fuzzy reasoning backorder EOQ Model.