Novel Methods for Solving Multi-Objective Non-linear Inventory Model of Highly Deteriorating Items

In this study, multi-objective inventory model of deteriorating and perishable items is developed under space and budget constraints. Demand is stock dependent and power function of time. This model is completely a new model in the sense that the model is applied to those items whose deterioration rate is maximum. Shortages are allowed in each cycle. The main aim of this paper is to find different time points for each cycle where shortage occurs and inventory depletes, respectively, so that both total cost and shortage cost can be minimized simultaneously. The model is developed in both crisp and fuzzy environment. In fuzzy environment, the objectives are considered as fuzzy constraints. For this, the decision maker needs to establish an aspiration level for the objective functions which he wants to achieve as far as possible. This paper aims to use fuzzy non-linear programming (FNLP) and intuitionistic fuzzy optimization (IFO) techniques for the multi-objective inventory model. Comparison is based on different optimization techniques in different environment using numerical examples. A graph of the objective functions is provided. A system diagram of the model and an algorithm for solving the model are provided. Also, sensitivity analysis is made using different parameters of the model.


Introduction
Inventory is important since it affects our lives.It is everywhere as household inventory, social inventory, and business inventory.Inventory provides flexibility when it comes to cost.The classical pattern of something aims to minimize 56 Page 2 of 22 the total cost of inventory.The absence of inventories compels customers to wait until their orders are manufactured or filled from a source.Hence, a formal inventory policy or carrying of inventories is needed for almost all organizations that supply goods to customers.Generally, while considering an inventory policy, three things are focused: that is, customer demand, whether shortages are allowed or not, and finally, inventory system cost.Among these, demand is the most important factor of the inventory system.Researchers have been given their focus to different types of demand.Initially, Baker and Urban [1] studied a continuous deterministic case of inventory system where demand rate of an item is a polynomial functional form, dependent on the inventory level, while Goswami and Chaudhuri [2] discussed an economic order quantity (EOQ) model of deteriorating item including shortage, the linear trend in demand under finite time horizon.Datta and Pal [3] considered an EOQ model of a single item and linear time-dependent demand rate.Later, Goyal et al. [4] considered a single item EOQ model, linear trend in demand, no shortage, and no deterioration.Giri et al. [5] studied single item stock-depended demand rate with no shortage and no deterioration.Bhunia and Maiti [6] discussed a single item time-dependent EOQ model, no storage, and no deterioration.Then, Chakrabarti and Chaudhuri [7] considered the linear function in demand rate, with shortages and with no deterioration.Later, Chakrabarti and Chaudhuri [7] considered a model of deteriorating item, the linear trend in demand where shortages occurred in all cycles under finite time horizon.Mandal and Maiti [8] discussed the stock-dependent polynomial form demand pattern of a single item.
However, real-life inventory problem does not include single-item always.So, we studied some inventory problems related to multi-items.Hwang and Hahn [9] analyzed a single-objective multi-item deterministic EOQ model where demand rate is inventory level-dependent demand.Also, deterioration of items in inventory is very common.Items related to foods, pharmaceuticals, vegetables, fruits, etc. are few examples.So, we cannot neglect the loss due to deterioration.In recent years, deterioration in inventory models has been considered widely.Lee and Wu [10] studied an EOQ model for multi-items with weibull distribute deterioration including shortage where demand rate is a power function of times.Then, Wu [11] considered an EOQ inventory model for multi-items with weibull distribution deterioration where demand is time-varying and partial backlogging.Sicilia et al. [12] studied deterministic inventory systems with power demand patterns including shortages.Yang [13] analyzed an inventory model of different items with both stock-dependent demand rate and stock-dependent holding cost including shortages and no deterioration where stock-dependent demand is a power function.After then, Sicilia et al. [14] again considered an EOQ model for a power demand pattern system for deteriorating items where shortages are not allowed.Pando et al. [15] studied an inventory model for a single item in a finite time horizon where both demand rate and holding cost is a stock-dependent power function.Pando et al. [16] continuously reviewed an inventory policy for items in an infinite time horizon where both demand rate and holding cost is a potential function of both time and quantity having no shortages.
When objectives of the inventory model are more than one, i.e., maximizing profit, minimizing wastage cost, minimize total production, maximizes return on inventory investment, minimizing expected annual cost for different demand, in these situations, the decision-maker (DM) requires a compromise solution, i.e., extreme solution.Since these objectives are confusing with each other, the decision maker finds a compromise solution that is extreme with respect to one objective.In most situations of inventory management, uncertainty plays an important role.In the inventory model, generally, demand for items is uncertain.Lack of information, conflict evidence, complexity, etc. are the cause of uncertainty.Fuzziness describes the ambiguity of an event.Roy and Maiti [17] developed a multi-objective inventory model of deteriorating items with stock-dependent demand.Inventory costs, total average cost, warehouse space, and purchasing and selling prices are assumed to be imprecise and vague.Inventory costs and prices are expressed by a triangular fuzzy number (TFN), and different objectives and constraints goals are expressed by a fuzzy linear membership function.Then, this model is solved by the fuzzy non-linear programming (FNLP) method.Also, taking into consideration the cost parameters, the objective functions and constraints become imprecise in nature.Mandal et al. [18] discussed an inventory model with shortages and demanddependent unit cost.Three objectives are considered here.Garai et al. [19] presented a multi-objective, multi-item inventory model in which demand is stock-dependent within exponential fuzzy environment.The solution has been made using chanceoperator techniques.Soni and Suthar [20] studied an EOQ model of deteriorating items using demand as fuzzy in nature.The fuzzy objective function is converted to a crisp model by using the centroid method.Results are compared in three different environment: crips, fuzzy, and fuzzy learning environments.Also, Kar et al. [21] discussed three objectives (maximization of profit, minimization of total production cost, minimization of wastage cost) in inventory models of deteriorating items where vagueness and impreciseness arise due to the availability of objective goals and different parameters.The model is solved using fuzzy non-linear goal programming techniques.Asma and Amirtharaj [22] presented an inventory model of deteriorating items.A fuzzy programming technique is used to solve the model.Impreciseness and fuzziness appeared in a fuzzy objective and various cost parameters.
In real-life multi-objective problems, objectives are conflicting in nature.So, to get a compromise solution of multi-objective problems, the IFO technique is another interesting technique.When available information is not sufficient to define impreciseness by means of a conventional fuzzy set, an intuitionistic fuzzy set (IFS) can be viewed as an alternative approach.Many researchers like Mahapatra and Maiti [23], Banerjee and Roy [24], Chakrabortty et al. [25], Garai and Roy [26], Bharati and Singh [27], Jafarian et al. [28], Garai et al. [19], and Pawar et al. [29] have been given their contribution to solve multi-objective inventory problem using intuitionistic fuzzy optimization (IFO) technique.
In this paper, we discussed the multi-objective inventory model of stock-dependent demand of deteriorating items including shortages under space constraints and budget constraints.When objective functions are considered as fuzzy constraints, the proposed model is solved using FNLP and IFO techniques.The advantage of IFO is that both degrees of acceptance and degrees of rejection of objectives are taken into consideration.The notations and assumptions are discussed in Section 2. In Section 3, a model is formulated.In Section 4, algorithms for solving the model in different environments are provided.In Section 5, methodology for solving the multi-objective nonlinear programming (MONLP) problem is discussed.In Section 6, validity of the model is discussed through numerical examples in crisp and fuzzy environments.Sensitive analysis and conclusions are discussed from the results obtained in Sections 7 and 8.The article comes to an end with a scope and future work.

Model and Assumptions
The model uses the following parameters and assumptions.

Parameters
The various parameters for the model are given below.q i (t) : Inventory level at time t for ith item.Q i : Order quantity per cycle for ith item.D i (t) : Demand rate at time t for ith item.c i : Purchasing cost per unit for ith item.oc i : Ordering cost per order for ith item.h i : Holding cost per unit time for ith item.t i : Length of each cycle for ith item.t i1 : Time at which inventory level reaches zero for each cycle for ith item.i : Scale parameter,  i > 0. : Shape parameter, 0 ≤ ≤ 1 s i : Shortage cost per quan- tity for ith item.S i : Shortage amount per order for ith item.i : rate of deterioration, 0 ≤ i ≤ 1 B: Budget available for replenishment.F: Available storage space in the inventory system.f i : Storage space per unit quantity for the ith item.
Decision variables t i =Time at which replenishment occurs.t i1 =Time at which shortage occurs.

Assumption
-The inventory system involves multi-item.
-Items are deteriorating.
-Lead time is zero.
-Replenishment rate is instantaneous.-Time horizon of the inventory system is infinite.
-Demand is deterministic and is a power function of instantaneous stock level q i which is expressed as follows: are allowed, and unsatisfied demand is back-ordered.

Model Formulation
Figure 1 shows the inventory policy of the model in one cycle.The said model is developed in crisp and fuzzy environment.

Crisp Inventory Model
Here, a crisp inventory model is developed over infinite time horizon.
Case-i (When shortages do not occur) q i (t) = Inventory level for ith item for each cycle.This decreases slowly partly due to match demands and partly due to match deterioration of item.Based upon above ideas and using Fig. 2, the differential equation over the time interval (0 ≤ t ≤ t i ) is as follows.

Case-ii (When shortages occur)
Shortages occur in t i1 ≤ t ≤ t i (Fig. 3).So the differential equation (D.E) satisfying the above conditions is given below: with boundary condition q i (t i1 ) = 0. Solving Eq. ( 3) for  i > 0 (1) (2)  Then, the total cost (TC) of the inventory model for each replenishment for ith item is TC(t i1 , t i ) (Fig. 4).
Total cost is a function of two variables.Using partial derivatives and hessian matrix, the model is a minimization problem.
The multi-objective inventory problem using the objectives as minimization of TC function and SC function under two constraints, budget constraint and space constraint, is as follows: (6) ) Equation ( 10) is a multi-objective non-linear inventory model.

Algorithms for Solving MONLP in Different Environments
Followings are the steps to solve the MONLP: Step 1: The MONLP is solved in crisp environment using LINGO software.
Step 2: The MONLP is solved in fuzzy environment considering each objective function as fuzzy constraints.For this, the decision maker can establish an aspiration level of the objective function which he wants to achieve as far as possible.
Step 3: The MONLP is solved in fuzzy environment using two methods: the FNLP method and the IFO method.
Step 4: In the FNLP method, the aspiration level is tried to achieve considering linear membership function for each objective function.In the IFO method, linear membership and non-membership function for the objective functions are considered in order to maximize degree of acceptance and minimize degree of rejection.
Step 5: Results are compared in fuzzy environment using FNLP and IFO methods.Then, results of the model in crisp and fuzzy environments are compared.
Step 6: Results of the proposed method are compared with the latest method.

Methodology for Solving MONLP Problem
A general MONLP problem takes the following form: (10) min TC(t i1 , t i ) min SC(t i1 , t i ) The objective functions f j (x) and/(or) constraints g k (x) are non-linear in "n" decision variables x 1 , x 2 , ...x n .In the formulation, there are J objective functions f (x) = (f 1 (x), f 2 (x), ...f J (x)) T .Each objective function can be either maximized or minimized.There are K constraints g 1 (x), g 2 (x), ...g K (x).

Solution Of MONLP Problem in Fuzzy Environment
Two methods have taken into consideration to solve the MONLP problem:

Fuzzy Programming Techniques
To solve crisp MONLP problems, the following are the steps of the method: Step 1: Solve the MONLP as a single objective non-linear programming considering one of the objectives at a time and ignoring all others.Repeat the process 'J' times for 'J' different objective functions.The ideal solutions are obtained for the respective objective functions.Let(x 1 , x 2 … , x J ) be the ideal solutions for the respective j th objective functions wherex r = (x r 1 , x r 2 , … x r n ).
Step 2: Construct a pay-off matrix of size (J × J) using ideal solutions obtained in Step 1.
Lower bound L r and upper bound U r for rth objective are estimated from the pay- off matrix as L r ≤ f r ≤ U r , (r = 1, 2, ...j) where L r = min[f r (x 1 ), f r (x 2 ), ..., f r (x j )] andU r = max[f r (x 1 ), f r (x 2 ), ..., f r (x j )] Step 3: Objective functions of (4.1) are considered as fuzzy constraints as below: Corresponding to each objective function f r (x) , define a fuzzy linear or non-lin- ear membership function f r (x) .For simplicity, for rth objective function, a linear membership function is defined as Step 4: A fuzzy nonlinear programming (FNLP) method.According to Zimmerman [30], the MONLP Eq. ( 11) can be solved by the FNLP method as is the aspiration level of each objective function of the MONLP Eq. ( 11) Step 5: Stop.

An Intuitionistic Fuzzy Approach for Solving Multi-Objective Inventory Problem (MOIP)
According to intuitionistic fuzzy optimization (IFO) theory, we are to minimize the degree of rejection of intuitionistic fuzzy (IF) objectives and constraints and simultaneously maximize the degree of acceptance of the IF objectives and constraints.denotes the minimal acceptable degree of objectives and constraints.denotes maximal degree of rejection of objectives and constraints.
Step 1: Solve the MONLP as a single objective non-linear programming considering one of the objectives at a time and ignoring all others.Repeat the process "J" times for "J" different objective functions.The ideal solutions are obtained for the respective objective functions.Let (x 1 , x 2 , ...x J ) be the ideal solutions for the respective j th objective functions where x r = (x r 1 , x r 2 , ...x r n ).
Step 2: Construct a pay-off matrix of size (J × J) using ideal solutions obtained in Step 1.
Then, let L acc r and U acc r be the lower and upper bounds of acceptance for rth objective function.Similarly, let L Step 3: From step 2, best L r and worst U r values for each objective correspond- ing to the set of solutions are obtained.To find the solution X for each objective function in terms of aspiration levels such that f r (x) ≤L r (r = 1, 2, .., j) subject to the non-negativity conditions.Then, the linear membership function for the objective function f r is defined as follows: and a linear non-membership function for the objective function f r is defined a From Banerjee and Roy [24], the new lower and upper bounds for the nonmembership function are as follows L rej r = L acc r + t(U acc r − L acc r ), where 0 < t < 1 U rej r = U acc r + t(U acc r − L acc r ), for t = 0.
Step 4: Considering together the above linear and non-linear membership function, an intuitionistic fuzzy optimization model of MOIM problem following Bellman-Zadeh [31] can be written as Following Angelov [32], the MONLP Eq. ( 11) reduced to the following form Step 5:

Numerical Examples
The input data for different parameters taken from Mishra and Umap [33] is given in Table 1.

Result of the MONLP Problem in Crisp Environment
The result of the MONLP problem Eq. ( 10) using input data of Table 1 in crisp environment by LINGO software is shown in Table 2.

Result of the MONLP Problem in Fuzzy Environment
The solution of model Eq. ( 10) using input data of Table 1 taken from Mishra and Umap [33] in fuzzy environment is as follows: Considering one objective at a time and ignoring other objective function of Eq. ( 10), the result is as follows: Putting the values of the different parameters in Eq. ( 18) and using step 1 of Section 5.1.1,the following results are obtained: (18) min TC(t i1 , t i ) Then, SC = 176.7471.
Putting the values of the different parameters in Eq. ( 19) and using step 1 of Section 5.1.1,the following results are obtained: Then, TC = 150.0182.Pay-off matrix is Then, lower bound and upper bound for rth objective function are as follows: Considering objective functions of Eq. ( 10) as fuzzy constraints as below.
Using step 3, the linear membership function for each objective function for r = 1,2 is as follows: Then, by step 4, for a desired aspiration level and for r = 1,2 (a) By the FNLP method 56 Page 14 of 22

(b) By the IFO method
Steps 1 and 2 are the same as discussed above.Using step 2 of Section 5.1.2,lower and upper bounds of acceptance and rejection for rth (r = 1,2) objective function are as follows: By step 3, the linear membership and non-membership function for rth (r = 1,2) objective function are as follows: By step 4, for maximum degree of acceptance and minimum degree of rejection , the intuitionistic fuzzy optimization model of model Eq. ( 10) is as follows: Solving the MONLP model Eq. ( 25) by LINGO software, the result is shown in Table 3.

Sensitivity Analysis
Sensitive analysis is made on the basis of the effect of different parameters.Here, the effect of the parameters s i , i , and i has taken into consideration.Tables 4, 5, and 6 show the results.( 24)   -From Table 4, the result is as follows: As the parameter s i per unit item is increasing, SC of the model Eq. ( 10) is fluctuating in both the FNLP and IFO methods, whereas TC is fluctuating in the FNLP method and remains constant in the IFO method.-From Table 5, the result is as follows: As the parameter i increases, TC of the model Eq. ( 10) increases for the FNLP method and decreases for the IFO method.At the same time, SC of the model Eq. ( 10) increases for the FNLP method and fluctuates for the IFO method.-From Table 6, the result is as follows: As the parameter i increases, the TC of the model Eq. ( 10) remains constant for both the FNLP and IFO methods, whereas SC of the model Eq. ( 10) for both the FNLP and IFO methods is fluctuating.

Conclusion
In this study, we have presented a multi-objective inventory model of deteriorating items under two constraints: availability of budget and space constraint.Demand pattern is the power function of instantaneous stock level.Parameters are not taken as fuzzy in nature, rather the goal of total cost and shortage cost is considered as fuzzy constraints.The main objective of the model is to calculate two time points for each item and for each cycle in order to minimize two objective functions TC and SC simultaneously.The model is solved in crisp environment using LINGO software.In fuzzy environment, two different fuzzy techniques (FNLP and IFO) are used to solve the MONLP problem.The result is shown in Table 3.A comparison table for the MONLP is shown in Table 7.In the FNLP method, the aspiration level for objective functions for the proposed method obtained is = 90%, whereas aspiration level for objective functions for the previous paper obtained is = 76%.In the IFO method, degree of acceptance of objectives and degree of rejection of objectives are obtained.It has been shown that in the IFO method, a TC of 149.99 is obtained which is less in comparison to the TC obtained in the FNLP method.Also, the shortage cost of the proposed method is 1.9830 which is less in comparison to the shortage cost obtained in the previous previous as 18.4467 using the IFO technique.Sensitivity analysis has been made considering the effects of different parameters.In each case, better results are obtained in the IFO method.Considering different demand patterns based on real-life inventory problems, the parameters of the problem as imprecise in nature, models can be framed and solved using different fuzzy techniques which can be our future research work.
The following are the limitations of the proposed method.
The auxiliary parameter = 1-is introduced in order to solve the differential Eq. (1).Integration of the term ∫ t i1 0 q i (t)dt is too difficult to compute while finding the expression for HC, DC, and SC.

Fig. 1
Fig. 1 System diagram of the model

Fig. 4
Fig. 4 Graphical representation of total cost function lower and upper bounds of rejection for rth objective function where L acc r ≤ L rej r ≤ U rej r ≤ U acc r , r = 1, 2, ..., J.

Table 1
Input data

Table 3
Optimal solutions of different methods

Table 5
Effect of the parameter i Parameters

Table 6
Effect of the parameter i Parameters