An inventory control model: Combining multi-objective programming and fuzzy-chance constrained programming

In this paper we developed an inventory model in mixed imprecise and uncertain environment. Presented model is the developed form of r,Q and is a multi-items model with two objectives as minimizing costs (holding and shortage) and risk level under constraints including available budgetary, the least service level, storage spaces and allowable quantities of shortage. Demand distribution functions are assumed to be exponential and extra demands are supposed in two situations as lost sales and backlogging. At first we develop crisp model then fuzzy stochastic model with fuzzy budgetary, allowable quantities of shortage and shortage spaces (that is stochastic with normal distribution function) parameter. All of fuzzy numbers are triangular type. In methodology of solution we change model to a crisp multi-objective by using difuzzification of fuzzy constraints and fuzzy chance-constrained programming methods, and then solve it by fuzzy logic method. Finally an illustrated example is taken and solved using LINGO package.


INTRODUCTION
After the publication of the EOQ model by Harris, some complexity and changes in their adjustment caused inefficiency, and some developments were in need (Naddor, 1986).The complexity arises in modeling a realistic decision-making inventory situation mainly due to the presence of some non-deterministic information; that is, that they are not able to encode precision and certainty of classical mathematical logic.Actually, a realistic situation is no longer realistic when the imprecise and uncertain information are neglected for the sake of mathematical model building (Das et al., 2004).One concern of studies on the inventory literature is (r,Q) model.This model have been popular in education and practiced at the beginning of infancy of inventory theory.In this regard, various computational procedures have been proposed in the *Corresponding author.E-mail: Amin.Nayebi@gmail.com.Tel: +989127831358.
textbooks and scientific studies for determining the control parameter "r" and "Q" (Yahua, 2005).Review (r,Q) model, among the stochastic demand, whenever inventory is the most r, Q would be established (Tersine, 1994).Optimization of an inventory system depends on finding a pair of "r" and "Q" parameters that minimize the long-run average of cost per period (Yahua, 2005), but in real-life situations decision-maker not only deals with minimizing costs but also concentrates on the other objectives which are limited.The concept of fuzzy sets theory helps us to solve the inventory management problems.
After the publication of classical lot-size formula by Harris in 1915, many researchers worked on the EOQ model whose results are currently available in reference books and survey papers (Clark, 1972;Hadely and Whitin, 1993;Raymond, 1931;Naddor, 1986).Other studies by Das et al. (2004), Goswami and Chaudhuri (1991), Bhunia and Maiti (1997) and Balkhi and Benkherof (1998) developed inventory models with variable replenishment production for multi-item inventory problems.According to other studies (Cheng, 1989;Cheng, 1991), demand and cost can be practically assumed to be inversely related to each other.Some inventory models have been developed and solved by geometric programming technique drawing on this assumption.Multi-item classical inventory models under resource constraints such as budgetary cost, limited storage area, number of orders, etc. are presented in well-known books (Silver and Peterson, 1985;Naddor, 1986;Hadely and Whitin, 1963;Lewise, 1970;Churchman et al., 1957).Ben-daya and Raouf (1993) discussed a multi-item inventory model with stochastic demand and Abou-el-ata and Kotb (1997) developed a crisp model inventory model under two restrictions.Initially, fuzzy sets theory presented by Zade in 1965 and Zimmerman used this theory to solve decision making problems (Das et al., 2004).Mandal et al. (2005) investigated a multi-objective fuzzy inventory model with three constraints using fuzzy geometric programming approach.They formulated the model with fuzzy parameters only in fuzzy environment and then solved it using modified geometric programming technique.Lodwick and Bachman (2005) used surprise function technique to convert a fuzzy possibility/necessity optimization problem into a deterministic one.Roy and Maiti (1997) used geometric programming technique to solve both single item and multi-item fuzzy inventory problems.Hadley and Whitin (1963) first extended the classical EOQ inventory model to the stochastic model.An application of fuzzy sets to inventory control models is presented by Yadavalli et al. (2005).In the survey by Das et al. (2004) named "Multi-item stochastic and fuzzy stochastic inventory models under two restrictions", the models are formulated under total budgetary and space constraints.Here the goal is to minimize the costs and all fuzzy models represented as stochastic and non-linear ones are solved by gradient technique and chance constrained programming.Yahua (2005) presented two procedures which determine optimal values for the control parameters (that is, reorder-point and order-quantity) when the holding costs are non-quasi-convex.In a study by Chu et al. (2001), a new form of partial backorder policy (PB2) with two-segment backorder control limits is introduced.Maiti and Maiti (2005) presented a multi-item inventory model with two-storage facilities under stochastic constraints that are solved by goal programming method (GPM) and fuzzy simulation-based single/multi -objective genetic algorithm (FSGA /FSMOGA) otherwise.In another study on stochastic inventory, Ouyang and Chang (2001) attempted to apply the fuzzy sets concepts to deal with the uncertain backorders and lost sales.Hariga (1999) incorporated a common stochastic continuous review inventory (r,Q) model and developed method for optimizing economic orders.Eynan and Kropp (2007) examined a periodic review system under stochastic demand with variable stock out costs by using a Taylor series expansion to approximate part of the cost function.
The other examples of developing stochastic inventory models are presented in studies by Wu (2001), Nielsen and Larsen (2005) and Ouyang et al. (2004).Nowadays, existence of a mixed environment or the coexistence of imprecision and uncertainly in an inventory is a realistic phenomenon (Das et al., 2004).Therefore, in this paper we developed inventory control model (r,Q) with two objectives as minimizing costs and risk level under limitations such as available budgetary, the performance level, number of shortages, and fuzzy-chance constrained of storage space.Shortages are assumed to be allowed and likely to face with a) lost sales and b) backlogging.Inventory costs depend on quantities.Available budgetary level is fuzzy factor and storage space is a stochastic parameter with fuzzy mean and fuzzy deviation.Storage space constraint is satisfied stochastically and the least allowed probability on this constraint is a fuzzy number which is definable.All random parameters are independent, storage spaces parameters follow normal distribution and demand follows distribution.In this model the amount of constant order is calculated based on the economical order amount.In this paper fuzzy constraints by difuzzification and fuzzy stochastic constraints of storage space by programming of fuzzy stochastic constraints change into crisp limitation and the output model which is a multi-objectives programming model can be solved by fuzzy logic technique.Finally an illustrated example is given and solved by using LINGO software package.
The rest of the paper is organized as follows: first, brief classification of ordering system is presented, then, in the second part we will do the same about symbols and signs.Models and assumption are discussed.The development of models and discussion of the methodologies are dealt with.A digital example is given and in final part conclusion and future research will follow.Also, we talked about developing the models, and solving methodology were discussed.Furthermore, a digital example was presented and a conclusion is drawn.Problem definition, models and their assumptions are also presented, developing the model is addressed.Solution methodology is proposed and Experimental results are presented.Finally, conclusion and future works.

MODEL FORMULATION
The following notations and assumption are used to develop the models.Allowed minimum fuzzy probability for the warehouse restriction Model and assumptions (Tersine, 1994) r,Q ordering system with a cost functions is assumed as follows:

: n Number of items
and assumptions of these models are: (i) Demand functions are assumed to be exponential: D~ Exp (  ) (ii) r and Q is independence (iii) Extra demand consider as: 1) lost sales; 2) backlogging.

METHODOLOGY
In this paper, methodology has two sections: section one is to develop the model and section two is to solve the model.In the first section, original model with the objectives and constraints is converted to crisp multi objectives model and fuzzy stochastic multi objectives model by fuzzy and stochastic variables and fuzzy and stochastic parameters.In solving section, first we will apply difuzzification on constraints and then convert the model to a crisp model using fuzzy chance constrained programming.Finally, resulted model will be solve through fuzzy logic method.The motioned methodology is illustrated in Figure 1.

Developing the model
To develop the (r,Q) model, we present crisp model and then fuzzystochastic model.Afterwards we propose two objectives and six constrains.Model objectives consist of;

Minimizing costs
We use the following formula for this objective where:

Minimizing risk level
Risk level gets over zero demand which can be more than the order point "r".We used the following relation for minimizing the risk level: Because stochastic demand is possible, risk level is directly related to the shortage and changes as shortage changes.So when it increases, shortage increases too in the same way.To reach this purpose we have:

Original model
The Objectives

Multiple Crisp Model
The Constraints

Fuzzy Stochastic Model
Developing the Model

Difuzzification of fuzzy constraints Crisp Multi-objective Model
Fuzzy -logic method shows the use of various constraints.In this study we propose 6 constraints as follows: 1. Constraint of budgetary: 3. Constraint of storage spaces: where: 4-Constraint of service level: for this purpose we have: Thus we applied the following equation for this:

5-In this paper we assumed
, thus we used EOQ formula as a constraint in this model: Where: 6-Finally, because probability rate is a number in the range [0,1], we considered the following constraint: Thus when inventory policy follows lost sales the developed crisp model of (r, Q) is:

Fuzzy stochastic model
and allowed minimum probability is a fuzzy number, As a result, fuzzy-stochastic model is presented as follows: When inventory policy is backlogging we use the following relation instead of equation (2) for minimizing costs: And then have:

Solution methodology
Developed fuzzy-stochastic model can be solved if the model changes to a crisp model and by using one of multi-objective programming techniques, it can assume the following steps:

First step: Difuzzification of fuzzy constraints
If the programming model is like this (Kilir and Bo, 2001): Supposed the fuzzy number from the left side of the first number of mean number has one degree   and second number is the distance between the first number, and the left bound and third number is the distance between the first number and right bound.Solving the model and difuzzification constraints, we obtain:

Converting the Fuzzy-stochastic constraint to a crisp constraint
Fuzzy-chance constrained programming: To this end, we used fuzzy-chance constrained programming (FCCP) (Nanda et al., 2006).A crisp CCP problem is a particular type of stochastic programming which is of the form: The crisp conversion of CCP for each α [0, 1] is: This problem can be solved by using crisp stochastic programming method.

Second step: Problem solving of crisp multi-objective programming
The fuzzy-stochastic model in this step is modified to a crisp multiobjective problem by using one of the multi-objective programming methods, which can be approach by using the fuzzy-logic method.
Fuzzy-logic method for problem solving of multi-objective decisionmaking (MODM) (Zimmerman, 1996): Assuming this multi-objective decision-making is presented as follows: Where "n" is the number of variables, " m " is the number of constraints and " k " is the number of object functions.The result obtained by calculating the objective functions, quantities and its variables are as shown in Table 1 Nayebi and Tavasolian 11937 If  is not of the same value, we have:

NUMERICAL EXAMPLE
This information includes two items, if decision-maker uses fixed order-quantity (r,Q).How are the reorder-points level ( r ), among minimizing cost and risk level when policy inventory are 1) lost sales and 2) backlogging?
When policy inventory about extra demand is lost sales, we have Considering the above mentioned methodology, the crisp multiobjective is as follows.
Consequently, we have a crisp multi-objective programming that can be solved by fuzzy logic method and its results are presented in Table 2. Considering Table 2 and defining membership functions as follow:

RESULT AND DISCUSSION
Results of this paper are developed models in fuzzy-Stochastic environment and presented methodology for solving them.But to illustrate the developed models in use we tested them in numerical example.Therefore in previous section, the developed models are run.A numerical example using the models solved that models' outputs illustrated as Lingo's outputs in Tables 3 and 5. Tables 3 and 5 show quantities of order points in two inventory policies: backlogging and lost sales.
Important points in this paper were solving methodology and quantity of α.This variable considered all objectives together and there was no point combining the result of each objective that did not exist in relative literature separately.In this paper, we apply two objectives with six constraints for numerical example because developed models became large but the models can be applied to actual environment with more objectives and constraints.By considering the results of solving two models, management can use various ( 1 r , 2 r ) without any changes in optimizing level.For this, we applied Lingo to a sensitivity analysis illustrated in Table 6.In the case where inventory policy follows the lost sale, decision makers can change 1 r quantity in range of 24.06 to 28.06059 and when without changing There is a similar analysis on backlogging policy as seen in Table 6.

Conclusion
In this paper, a traditional inventory control model in two policies; lost sales and backlogging with two objectives of minimizing costs and risk level and budgetary limitation

Inventory policy r1 r2
Lost sale [24.06,28.06059] 24.999 28.605 [24.38, 25] Backlogging [24, 33.699] 24.99 26.00375 [24.387, 25] and also warehouse, number of allowed shortages and minimum level of allowed performance were developed and discussed.Developed models are nonlinear and some of its parameters are considered as fuzzy ones.
Warehouse limitation which changed to a crisp constraint using fuzzy chance constraint programming in fuzzychance environment was discussed and finally a crisp multi objectives model was solved by fuzzy logic method.
In future research we can use some assumptions about demand distribution and warehouse space like Erlang.Also we can consider some objectives and other limitations.Instead of triangular fuzzy, we can use trapezoid type and also use optimization function method and lexicography from other solving method of multi objectives programming like geometric programming.We can apply Grey theory and develop the models with grey numbers.The obtained results also can be compared with each other.
number.Regarding the special state of i th constraints, CCP problem is: is a normally distributed FRV (in fact we may consider any type of FRV), whose mean and variance are fuzzy numbers denoted by bi m ~ and bi


When inventory policy follows backlogging about extra demand, we have:


Considering the mentioned methodology, the crisp multi-objective is presented as follows: , the obtained model should be like subsequent model solved by Lingo package and its results are illustrated in

:
This section is presented to develop model in fuzzy environment, A m ~ and fuzzy variance 2 ~A  : while the degree of membership is tolerance of per unit of i function.
i 

Table 2 .
Results of solving the functions in lost sales model

Table 3 .
Lingo output the lost sales model.

Table 4 .
Results of solving the functions in backlogging model.

Table 4 .
Hence from Table 4, membership functions can be defined as follows:

Table 5 .
Lingo output the lost sales model

Table 6 .
Results on sensitivity analysis.