Application of fuzzy random-based multi-objective linear fractional programming to inventory management problem

This research article aims to study a multi-objective linear fractional programming (FMOLFP) problem having fuzzy random coefficients as well as fuzzy pseudorandom decision variables. Initially, the FMOLFP model is converted to a single objective fuzzy linear programming (FLP) model. Secondly, we show that a fuzzy random optimal solution of an FLP problem is resolved into a class of random optimal solution of relative pseudorandom linear programming (LP) model. As a result, some of theorems show that a fuzzy random optimal solution of a fuzzy pseudorandom LP problem is combined with a series of random optimal solutions of relative pseudorandom LP problems. As an application, the developed approach is implemented to an inventory management problem by taking the parameters as trapezoidal fuzzy numbers, ultimately resulting in a new initiative for modelling real-world problems for optimization. In the last, some numerical examples are introduced to clarify the obtained results and their applicability.


Introduction
Fractional programming, i.e. the optimization of a fraction of two functions subject to some prescribed conditions, plays important role in modelling and optimization in the field of management, engineering, finance, economics and science. Recently, huge developments have taken place in this area. Charnes and Cooper (1962) proposed the programming with linear fractional functions, termed as fractional programming problem (FPP). Normally, FPP is a decision making model that aims to optimize the ratio subject to some constraints. In real-life situations, the decision maker (DM) sometimes may face to compute the ratio between stock of goods and sales, output and employee etc., with both denominator and numerator are linear. When one ratio is considered as an objective function under linear constraints, the problem is referred as linear FPP. As per applications scenario, FPP is used in the fields of traffic planning (Dantzig et al., 1966), and many more. In the meantime, some applications of FPP and the algorithms to solve this kind of problems were presented by Dinkelbach (1967). Luhandjula (1984) developed some fuzzy approaches to solve the multi-objective linear FPP. Sakawa and Yano (1988) proposed an approach for multi-objective linear FPPs. Guzel LP problems with fuzzy random variable coefficient were presented and their applications in the area of distribution problems by Zhong (1993a, 1993b). Tanaka and Asai (1984) presented the fuzzy LP problem with fuzzy numbers. Dutta et al. (1993) investigated the effect of tolerance in fuzzy linear FPP. Many authors have investigated the fuzzy random variables with various types of membership functions. Zhong et al. (1994) studied the fuzzy random LP problem with fuzzy as well as random nature, called fuzzy random LP problems. Liu and Liu (2002) derived a model for expected value of fuzzy variables. Pop and Stancu (2008) presented a method to solve fully fuzzified linear FPPs.
Recently, various applications of FPP were proposed by researchers. Atanassov (1986) developed the idea of intuitionistic fuzzy sets in many applications. Banerjee and Roy (2010) derived the solution methodology for single as well as multi-objective stochastic inventory models with fuzzy cost components. Das and Maiti (2013) studied the fuzzy stochastic inequality and equality possibility constraints and the applications in a production-inventory model using optimal control method. Chakraborty et al. (2013Chakraborty et al. ( , 2016 presented an intuitionistic fuzzy method for pareto optimal solution of manufacturing problem. Singh and Yadav (2016) and Ali et al. (2018) derived a fuzzy programming method to solve an intuitionistic fuzzy linear FPP.
In literature, several authors considered fuzzy as well as random (probabilistic) nature of parameters of the problem. EI-Asharm and Girgis (1996) presented their research work on linear multi-objective programming in random and fuzzy environments. Chen (2005) developed an FPP method to solve the stochastic inventory control model. Nasseri and Bavandi (2019) proposed a study on fuzzy stochastic linear FPP based on fuzzy mathematical programming. Very recently, Yang et al. (2020) applied fractional programming to solve an agricultural planting problem. Valipour and Yaghoobi (2021) investigated some fuzzy linearization methods to solve FMOLFP problem. Table 1 illustrates the description of related work by different authors.
In this research work, a fuzzy random multi-objective linear FPP approach is proposed. The proposed approach is demonstrated with an application to inventory management problem.
The cardinal contributions of the current work are demonstrated below: • Introducing the concept of fuzziness and randomness in FPP. • Applying FPP approach to solve the multi-item inventory management model. • Probabilistic constraints are converted to the corresponding crisp constraints. • Demonstrating the problem and algorithm illustrating a numerical application of two-item inventory management problem. • A comparative study is performed between the existing methods and the suggested approach.
The remainder of the paper is designed as follows: Section 2 demonstrates some preliminaries required in this paper. Section 3 introduces fuzzy random multiobjective linear FPP formulation. In Section 4, some of basic theorems with proof are presented. In Section 5, an inventory model is given as an application. In the end, some conclusions along with upcoming research ideas are reported in the last Section.

Preliminaries
This Section presents some of basic concepts.
Definition 2.2: (Kaufmann & Gupta, 1988). A fuzzy num-berÃ(a 1 , a 2 , a 3 , a 4 ) is a trapezoidal fuzzy number if its membership function The interval confidence ofÃ(a 1 , a 2 , a 3 , a 4 ) is defined as follows: Let F 0 (R) be the set of all compact trapezoidal fuzzy numbers on R. For anyÃ ∈ F 0 (R),Ã satisfies the following conditions: for any α ∈ (0, 1], and a L α ≤ a U α .
More definitions related to the intervals are all discussed in the Appendix.

Basics theorems
In this Section, some theorems with proofs point out the relations of optimal solution of problem (6) and the corresponding relative random programming problems (7)-(12).

Theorem 4.2: Assume thatÃ
. Also, letX • be a fuzzy pseudorandom optimal solution of problem (6). Then, we have α is a pseudorandom optimal solution of problem (9), 2. X •U α is a pseudorandom optimal solution of problem (10), and Proof: LetX • be a fuzzy random optimal solution of problem (6). Theñ Using Lemma 3.3 and sinceÃ ≤ 0, we have Based on the factC ≥ 0,D ≤ 0, and from Definition A3 and Lemma A1, we have On the other hand, by using Definitions A3, A4, and A5 and Lemma A1, we have Thus for any α ∈ (0, 1], and from problems (15) and (16) we have the following expression

Proof: LetX
Based on Lemma 3.4 and the theorem conditions, it is clear that X •U α ∈S (L,U) α and also, since On the other hand, we obtain the following expression Thus, for any α ∈ (0, 1], and from problems (19), (20) and (21), we observe that X •U α is a pseudorandom optimal solution of problem (11).

Proof:
The proof is similarly to Theorem 4.

Case study (Multi-item inventory problem)
We consider the multi-objective linear fractional inventory model as Kumar and Dutta (2015):

Nomenclature
The following nomenclature is adopted to deal the suggested approach. Ordering cost of ith item.

Assumptions
The following assumptions are considered to formulate the suggested model: i. Multi-item inventory model without shortages is assumed. ii. Infinite time horizon considered. iii. Zero lead time considered. iv. Holding cost is constant for each item. v. Demand is inversely related to the selling price: where β > 0 is a scaling constant, and γ > 1 is the coefficient of price-elasticity. For notational simplicity, D i and D i (S i ) may be used interchangeably in this research work.
vi. There is no discount, i.e. the purchase price is constant for each item.
Following the above assumptions, this inventory model is formulated as multi-objective linear fractional inventory model as follows : Subject to: where, H iQi 2 represents the holding cost, n i=1Q i represents the total ordered quantity, Constraint I represents the restriction on total budget, Constraint II represents the restriction on storage space, Constraint III represents the upper limit on number of orders, Constraint IV represents the budgetary constraint on ordering cost. Constraint V represents the non-negative restrictions.
We illustrate the following numerical example for the proposed model. Example 5.1: (Two-item inventory problem) (Model in Crisp Environment) Consider inventory model of two items with the following input data (in proper units) as presented in the following Table 2: At α = 0.6, the input data are presented in the following Table 3, as follows: Based on the data in as shown in Table 3, the multiobjective inventory problem can be expressed as: Problem (24) according to problem (7) can be rewritten as follows: 1.6(Q 1 ) L α + 2.6(Q 2 ) L α ≤ 300, 718.213(Q 1 ) L α + 675.339(Q 2 ) L α ≥ 16, 000, The solution of problem (25) is given by Also, problem (24) according to problem (8) can be rewritten as follows The solution is given by Z 2 = .10780.9700, (Q 1 ) L α = 86.0708, (Q 2 ) L α = 13.4095.

Conclusions
In this work, the solution method of fuzzy random linear multi-objective fractional programming (FRLMOFP) problem has investigated. With α-level set concept, the FRLMOFP can be classified into six subproblems according different criteria. The relations of these models as well as the FRLMOFP have mathematically formulated. An inventory problem with price-dependant has introduced as an application to support the suggested method. This approach comprises of the deterministic nature of demand. This research work assumes the demand function which depends exclusively on selling price. Further research directions can be explored for different type demand functions, variable deterioration and use of preservation technology to reduce the deterioration. One can also introduce the impact of lead time as random variable that may occur in supply chain during the present COVID 19 pandemic outbreak. Another direction for further research is to use the proposed approach to model different dynamics of decision sciences, such as multi-criterion decision making, networking, scheduling problem, etc.
where I is the index set, then = i∈IÃ i (u) = = i∈IÃ i (u), for any u ∈ .
(ii) If E(a L α ) and E(a U α ) are the expectation of a L α and a U α , respectively, then the expectation ofÃ is defined as follows: Definition A.6: (Zhong et al., 1994):X ∈S(Ã,B,X) is said to be fuzzy pseudorandom efficient solution of problem (3) if and only ifZ k (C,D,C 0 ,D 0 ,X) ≤Z k (C,D,C 0 ,D 0 ,X) withZ k (C,D,C 0 ,D 0 ,X) <Z k (C,D,C 0 ,D 0 ,X) holds for at least one k = 1, 2, . . . , K.