Application of fuzzy goal programming approach to multi-objective linear fractional inventory model

In this paper, we propose a model and solution approach for a multi-item inventory problem without shortages. The proposed model is formulated as a fractional multi-objective optimisation problem along with three constraints: budget constraint, space constraint and budgetary constraint on ordering cost of each item. The proposed inventory model becomes a multiple criteria decision-making (MCDM) problem in fuzzy environment. This model is solved by multi-objective fuzzy goal programming (MOFGP) approach. A numerical example is given to illustrate the proposed model.


Introduction
published their classical paper in which they show that a linear fractional programme with one ratio can be reduced to a linear programme using a nonlinear variable transformation. Brief definitions of Multi-Objective Linear Fractional Programming (MOLFP) problem and Fuzzy Linear Programming (FLP) are presented in Appendix. Zimmermann (1976) first extended the FLP approach to a conventional multi-objective linear programming (MOLP) problem. For each of the objective functions in this problem, the DM was assumed to have a fuzzy goal, such as 'the objective function should be substantially less than or equal to some value'. Later, Zimmermann (1978) described the fuzzy programming and linear programming with several objective functions. The study of fractional programmes with only one ratio has largely dominated the literature in this field until about 1980. Kornbluth and Steuer (1981) proposed the goal programming with linear fractional criteria. Luhandjula (1984) tried to apply the variable transformation to solve the MOLFP problem. Since then two other monographs solely devoted to fractional programming appeared, one authored by Craven (1988). Membership functions, such as linear, piecewise linear, exponential and hyperbolic functions, were used in different analysis. In general, the non-increasing and non-decreasing linear membership functions are frequently applied for the inequalities with less than or equal to and greater than or equal to relationships, respectively. Dutta, Tiwari, and Rao (1992) developed a set theoretic approach to solve a multiple objective linear fractional programming. Furthermore, Dutta, Tiwari, and Rao * Corresponding author. Email: pavankmaths@gmail.com (1993a) commented over the fuzzy approaches for multiple criteria linear fractional optimisation. Subsequently, Dutta, Tiwari, and Rao (1993b) studied the effect of tolerance in fuzzy linear fractional programming. A Goal Programming Method for Solving Fractional Programming Problems via Dynamic Programming was proposed by Pal and Basu (1995). At the same time, Barros and Frenk (1995) studied the generalised fractional programming and cutting plane algorithms. Hariri and Abou-El-Ata (1997) proposed a multi-item production lot-size inventory model with varying order cost under a restriction. Geometric programming approach was applied to solve this inventory model. Some researchers considered multi-objective approach to supply chain management. Sabri and Beamon (2000) proposed a multi-objective approach to simultaneous strategic and operational planning in supply chain design. Later, Maulik, Pal, and Moitra (2003) developed a goal programming procedure for fuzzy MOLFP problem. At the same time, Fung, Tang, and Wang (2003) published a paper on multi-product aggregate production planning with fuzzy demands and fuzzy capacities. Rafiei, Mohammadi, and Torabi (2013) presented a multi-segment multi-product multi-period supply chain network design model, which minimises the expected cost. They proposed an algorithm based on genetic algorithm (GA)-priority to solve this model. Moreover, Ramana, Rao, and Kumar (2013) developed a generic hierarchy model for decision-makers who can prioritise the supply chain metrics under performance dimensions of leagile supply chain.
Different models arise naturally in decision-making when several rates are to be optimised simultaneously and a compromise is sought which optimises a weighted sum of these rates. In light of the applications of single-ratio fractional programming numerators and denominators may be representing output, input, profit, cost, capital, risk or time, for example. A multitude of applications of the sumof-ratios problem can be found in this way. Included is the case where some of the ratios are not proper quotients. This describes situations where a compromise is sought between absolute and relative terms like profit and return on investment (profit/capital) or return and return/risk, for example. A fuzzy approach to solve a multi-objective linear fractional inventory model was used by Sadjadi, Aryanezhad, and Sarfaraj (2005). They considered two goals as fractional under two constraints: space capacity constraint and budget constraint. Their aim was simultaneously to maximise the profit ratio to holding cost and to minimise the back orders ratio to total ordered quantities. Chen (2005) used fractional programming approach to stochastic inventory problems. Some applications of MOLFP in inventory were proposed by Toksarı (2008). An application of fuzzy goal programming approach with different importance and priorities to aggregate production planning was introduced by Belmokaddem, Mekidiche, and Sahed (2009). They made an attempt to minimise total production and work force costs, inventory carrying costs and rates of changes in work force.
Very recently, Dutta and Kumar (2012) proposed a goal programming approach using trapezoidal membership function for fuzzy multi-objective linear fractional optimisation. Multi-objective optimisation models were proposed by several researches in fuzzy environment as well as in stochastic environment. Most of researchers assume that the product quality is perfect. However, practically, it can often be observed that the product quality is not always perfect. Gani and Maheswari (2013) presented a model on inspection cost and imperfect quality items with multiple imprecise goals in supply chains in an uncertain environment. Chakraborty and Chatterjee (2013) considered the material selection problem as multi-criteria decisionmaking (MCDM). They solved five material selection problems by using different MCDM techniques. Finally, they observed that choices of the best suited materials solely depend on the criterion having the maximum priority value. Chakrabortty and Hasin (2013) attempted to solve an aggregate production planning problem by fuzzy-based genetic algorithm (FBGA) approach. Sadi-Nezhad and Shahnazari-Shahrezaei (2013) investigated the ranking fuzzy numbers using the concept of preference ration. They introduced the weakness of this method. Then, they proposed a new approach based on the concept of utility function which takes the opinion of decision-maker (DM) for ranking fuzzy numbers into account. This paper presents an application of fuzzy goal programming approach to solve the multi-objective fractional inventory model. Many realistic inventory problems deal with more than one objective function which may be in conflict with each other. Our model refers to a multi-item inventory problem, with limited capacity of warehouse, constraints on investment in inventory and budgetary constraint on ordering cost of each item. Our aim is to simultaneously maximise the profit ratio to backordered quantity and to minimise the holding cost ratio to total ordered quantities.
The paper is outlined as follows. The notations and assumptions are defined in Section 2. The proposed inventory model is presented in Section 3. In Section 4, we describe the solution procedure. A numerical example is solved in Section 5. In Section 6, we analyse the results. Finally, we conclude in Section 7.

Notations
In our proposed inventory model, we adopt the following notations: n = Number of items k = Fixed cost per order B = Maximum available budget for all items F = Maximum available space for all items For ith item: (i = 1, 2,. . . n), we define the following: Q i = Ordering quantity of item i (a decision variable) h i = Holding cost per item per unit time for ith item P i = Purchasing price of ith item S i = Selling price of ith item D i = Demand quantity per unit time of ith item f i = Space required per unit for ith item OC i = Ordering cost of ith item.

Assumptions
The following assumptions are being made in developing the mathematical model: (1) A multi-item inventory model is considered.
(2) Time horizon is infinite, and there is only one period in the cycle time. i.e. no discount is available. (7) Shortages are not allowed. (8) No deterioration allowed.

Proposed inventory model
A multi-item inventory system under resources constraints is introduced as a linear fraction programme. This model refers to a multi-item inventory problem, with limited capacity of warehouse and constraints on investment in inventories. For each item, we also imposed the constraint on ordering cost. Demand for each item is known and constant and it must be met over an infinite horizon without shortages or backlogging. Replenishments are instantaneous and we assume a zero lead time. In real life, we observe inventory problems deal with more than one objective function. These objectives may be in conflict with each other, or may not be. In such type of inventory models, the decision-maker is interested to maximise or minimise two or more objectives simultaneously over a given set of decision variables. We call this model inventory as linear fractional inventory model.
We propose a multi-item inventory system with fractional objective functions. Without loss of generality, we assume there is only one period in the cycle time. Then Constraints: We consider the following constraints in this modelling formulation.
Upper limit of the total amount investment: Limitation on the available warehouse floor space in the store: Budgetary constraints on ordering cost as follows.
For an effective inventory management, the control over the ordering cost is also one of the priorities of the decisionmaker (DM). The low ordering cost plays an important role to minimise the total inventory cost, and to optimise value of ordering quantity. We are interested to impose the upper limit of ordering cost as a constraint.
Since OC 1 , OC 2 , . . ., OC n are the ordering cost of first item, second item,. . ., nth item, therefore, we can express the concerned constrained as follows: For optimal policy of inventory, the decision-maker desires to generate an optimal policy so that the profit can be maximised, and the holding cost can be minimised.
With the maximisation of profit, if the backordered quantity is minimised, then this policy will be most preferred, and economical. Likewise, with the minimisation of holding cost, the policy is such that the total order quantity is optimised. So, it is practical and wise to consider the objective functions in the form of fractions and automatically trade-offs between the above objectives will be considered. Such objectives are called fractional objectives.
Hence, the multi-objective fractional programming model is: Mathematically, we can rewrite above optimisation problem as follows: kD n − (OC n )Q n ≤ 0 (for each n), and all Q n ≥ 0, OC n > 0.
Obviously, this model is multi-objective linear fractional programming problem (MOLFPP), which can be easily solved by using fuzzy goal programming (FGP) approach.

Fuzzy goal programming (FGP) approach
In our approach, first the objectives are transformed into fuzzy goals by means of assigning an aspiration level to each of them. Then achievement of the highest membership value (unity) to the extent possible of each of the fuzzy goals is considered.
Let be the aspiration level assigned to the kth objective (X). Then, the fuzzy goals appear as: and Z k (X) g k (for minimising Z k (X)), where , indicate the fuzziness of the aspiration levels, and are to be understood as 'essentially more than' and 'essentially less than' in the sense of Zimmermann [2]. Hence, the Fuzzy Linear Fractional Goal Programming (FLFGP) can be stated as follows: Find X Satisfying Z k (X) g k , k = 1, 2, . . . , k 1 (17) Subject to AX ≤, =, or ≥ b, X ≥ 0.
Now, in case of fuzzy programming, the fuzzy goals are characterised by their associated membership functions. For kth fuzzy goal Z k (X) g k , the membership function μ k can be written as: Also, for kth fuzzy goal Z k (X) W g k , the membership functionμ k is: where u k is the upper tolerance limit, and l k ≤ g k ≤ h k ≤ u k = real numbers.
In fuzzy programming, the highest degree of membership function is 1. Hence, for the above-defined membership functions, the flexible membership goals with the aspiration level 1 can be written as: and where d − k (≥ 0) and d + k (≥ 0) with d − k d + k = 0 are the underdeviation and over-deviation, respectively, from the aspiration levels. We see that the membership goals in Equations (22) and (23) are inherently nonlinear in nature and this may create computational difficulties in the solution process. To avoid such problems, the linearisation of membership goals must be done. So, we express the kth membership goal in Equation (22) as follows: (1) and letting L k =1 + L k l k , where C k = L k c k − L k d k and G k = L k β k − L k β k . (25) Similar goal expressions for the membership goal in Equation (23) can also be obtained. Now, using the method of variable change, the goal expression in Equation (24) can be linearised as follows.
Taking (24) is obtained as

the linear form of the expression in Equation
Now, to minimise d − k means to minimise D − k /(d k X + β k ) which is also a nonlinear expression. Also, when a membership goal is fully achieved, d − k = 0 and when its achievement is zero, d − k = 1 are found in the solution. So, to involve d − k ≤ 1 in the solution, we need to impose the following constraint to the model of the problem: It may be noted that any such constraint corresponding to does not arise in the model formulation. Now, if the simplest version of GP is introduced to formulate the model of the problem under consideration, then the GP model formulation becomes: Find X so as to where Z is the fuzzy achievement function consisting of the weighted under-deviational variables, in which the numerical weights w − k (≥ 0), k = 1, 2, . . ., K are the relative importance of achieving the aspiration levels of the respective fuzzy goals subject to the constraints set. Now, for the relative importance of the fuzzy goals properly, the weighting scheme can be used to assign the values to w − k (k = 1, 2, . . ., K). In our formulation, w − k can be determined as The mathematical programming problem in Equation (29) can be solved by using min-sum method. The final goal expression is: which is actually the linearised form of the original goal expression where L k = 1/ (g k − l k ) , L k = 1 + L k l k , and where (g k − l k ) denotes the admissible violation constant.
Its generalised fractional form is given by Now, the conventional form of a goal in GP is where Z k (X) may be linear/nonlinear including fractional, where n k and p k represent the under and over-deviational variables, respectively. Comparing the above two forms of goals, we find that Now, due to the minimising the weighted goal deviational variables in GP, we observe that the GP formulations for the two forms of goals are equivalent with d − k = n k / (g k − l k ). Thus, the weights of the goals in the proposed model are reciprocal to the admissible violation constants.

Numerical example
We consider a three item inventory problem. All input parameter values are given in Table 1. Using this data, the multi-objective inventory problem can be expressed as: Subject to 625Q 1 +730Q 2 + 440Q 3 ≤ 900000, (38) We find the individual best solution: Z 1 ≡ 11.56 at (1363.71, 40, 42) and Z 2 ≡ 6.14 at (1363.71, 40, 42). Let the fuzzy aspiration levels of the two objectives be g Z 1 = 13 and g Z 2 = 5, respectively. Then the problem can be designed as: Find Q (Q 1 , Q 2 , Q 3 ) so as to satisfy the following two fuzzy goals: Subject to constraints (38)-(43). For the fuzzy goal Z 1 13, let the lower tolerance limit be l z 1 = 8. For the fuzzy goal Z 2 5, let the upper tolerance limit be u z 2 = 10, as shown in Figures 1 and 2.  Analytically, the linear membership functions are defined as follows: For Z 1 13, and for Z 2 5, Then the membership goals can be expressed as where From Equations (48) and (49), we obtain and D + 2 = 5d + 2 (Q 1 + Q 2 + Q 3 ).
Now, from the restrictions d − 1 ≤ 1 and d − 2 ≤ 1, we obtain ≤ 1, and Thus the equivalent GP formulation is obtained as: Find Q (Q 1 , Q 2 , Q 3 ) so as to The LINGO computer software package is operated to run the linear programming model. The optimal solution is obtained as: Q 1 = 1363.712, Q 2 = 40, Q 3 = 42. The achieved objective function values are Z 1 = 11.5617, Z 2 = 6.1424. The resulting membership values are μ z 1 = 0.7123, μ z 2 = 0.7715. The optimal results obtained for the different values of demand quantity are shown in Table 2-4.

Analysis of results
From above Table 2-4, we observe the following points: (1) The optimal order quantity is Q 1 = 1363.712 units, Q 2 = 40, Q 3 = 42 and achieved objective function values are Z 1 = 11.5617, Z 2 = 6.1424. The resulting membership values are: μ z 1 = 0.7123, μ z 2 = 0.7715. This means the resulting achievement degrees for the two fuzzy goals are achieved up to the level of 71.23%, and 77.15% respectively, all of which satisfy the requirements of decisionmakers.
(2) The maximum achievement degrees for the two fuzzy goals are 0.7145 and 0.7732 corresponding to the demand values (900,1900,1400), as displayed in Table 2.  (3) In Table 3, the maximum achievement degrees for the two fuzzy goals are 1.0 and 0.7715, which corresponding to different set of purchasing price values. (4) In Table 4, the maximum achievement degrees for the two fuzzy goals are 1.0 and 0.7715, which corresponding to different set of selling price values. (5) Using fuzzy goal programming approach to simultaneously optimise two objectives, yields optimal order quantity over a wide range of demand values. The demand variation, up to a finite lower and upper limits, gives the global optimal and feasible results. Beyond these limits, the solution becomes non-optimal and infeasible. This is because of the model assumptions.
In both examples, the demand quantity values vary but the selling price and purchasing price do not vary. If we impose another assumption: dependency of demand quantity values with selling price and the purchasing price, then the global optimal results will be obtained over a wide range of variation in demand quantity values. Then, the model may be more realistic.
The proposed model is solved by using LINGO computer package and getting optimal order quantity plan. The proposed model gives an efficient compromise solution and the overall levels of decision-making (DM) satisfaction with the multiple fuzzy goal values.

Conclusion
In this paper, we proposed an application of a fuzzy goal programming approach with trapezoidal fuzzy number developed by us (Dutta & Kumar, 2012) to multi-objective linear fractional inventory problem in a simple way. The proposed approach attempts simultaneously to maximise the profit ratio to backordered quantity and to minimise the holding cost ratio to total ordered quantities, so that in the end, the proposed model is solved by using LINGO programme to get optimal order quantities. This model gives optimal results over a wide range of demand values. However, the major limitations of the proposed model concern the assumptions made in determining each of the decision parameters, with reference to forecasted demand, selling price and purchasing.