A FUZZY INVENTORY MODEL (EOQ MODEL) WITH UNIT PRODUCTION COST,TIME DEPENDED HOLDING COST,WITH-OUT SHORTAGES UNDER A SPACE CONSTRAINT: A FUZZY PARAMETRIC GEOMETRIC PROGRAMMING (FPGP) APPROACH

 

Sahidul Islam

University of kalyani, India

E-mail: sahidul.math@gmail.com

 

Wasim Akram Mandal

University of kalyani, India

E-mail: wasim0018@gmail.com

 

Submission: 02/09/2016

Accept: 13/10/2016

 

ABSTRACT

In this paper, an Inventory model with unit production cost, time depended holding cost, with-out shortages is formulated and solved. We have considered here a single objective inventory model. In most real world situation, the objective and constraint function of the decision makers are imprecise in nature, hence the coefficients, indices, the objective function and constraint goals are imposed here in fuzzy environment. Geometric programming provides a powerful tool for solving a variety of imprecise optimization problem. Here we have used nearest interval approximation method to convert a triangular fuzzy number to an interval number then transform this interval number to a parametric interval-valued functional form and solve the parametric problem by geometric programming technique. Here two necessary theorems have been derived. Numerical example is given to illustrate the model through this Fuzzy Parametric Geometric-Programming (FPGP) method.

Keywords: Inventory model, Fuzzy number, Space constraint, Geometric Programming, Interval-valued function

1.     INTRODUCTION

            An inventory deals with decision that minimize the total average cost or maximize the total average profit. For this purpose the task is to construct a mathematical model of the real life Inventory system, such a mathematical model is based on various assumptions and approximations. In ordinary inventory model it consider all parameter like set-up cost, holding cost, interest cost   a fixed. But in real life situation it will have some little fluctuations. So consideration of fuzzy variables   is more realistic.

            Geometric Programming (GP) method is an effective method used to solve a non-linear programming problem like structural problem. It has certain advantages over the other optimization methods. Here, the advantage is that it is usually much simpler to work with the dual than the primal one. Solving a non-linear programming problem by GP method with degree of difficulty (DD) plays essential role. (It is defined as DD = total number of terms in objective function and constraints – total number of decision variables – 1). Since late 1960’s, Geometric Programming (GP) used in various field (like OR, Engineering science etc.). Geometric Programming (GP) is one of the effective methods to solve a particular type of Non linear programming problem. The theory of Geometric Programming (GP) first emerged in 1961 by Duffin and Zener. The first publication on GP was published by Duffin and Zener on (1967). There are many references on applications and methods of GP in the survey paper by Ecker. They describe GP with positive or zero degree of difficulty. But there may be some problems on GP with negative degree of difficulty. Sinha et al. proposed it theoretically. Abot-El-Ata and his group applied modified form of GP in inventory modelsPark and Wang studied shortages and partial backlogging of items. Friedman (1978) presented continuous time inventory model with time varying demand. Ritchie (1984) studied in inventory model with linear increasing demand. Goswami, Chaudhuri (1991) discussed an inventory model with shortage. Gen et. Al. (1997) considered classical inventory model with Triangular fuzzy number. Yao and Lee (1998) considered an economic production quantity model in the fuzzy sense. Kumar, Kundu and Goswami (2003) presented an economic production quantity inventory model involving fuzzy demand rate. Syde and Aziz (2007) applied sign distance method to fuzzy inventory model without shortage . D.Datta and Pravin Kumar  published several paper of fuzzy inventory with or without shortage. Islam and Roy (2006) presented a fuzzy EPQ model with flexibility and reliability consideration and demand depended unit  Production cost under a space constraint .A solution method of posynomial geometric programming with interval exponents and coefficients was developed by Liu (2008). Kotb, Fergancy (2011), presented Multi-item EOQ model with both demand-depended unit cost and varying  Lead time via Geometric Programming. Dey and Roy (2015) presented Optimum shape design of structural model with imprecise coefficient by parametric geometric programming.

In this paper we first considered crisp inventory model. There after it transformed to fuzzy inventory mode and developed. Here two necessary theorems have been derived.  At last it made an example and solved it by Parametric Geometric-Programming Technique.

2.     MATHEMATICAL MODEL

            An Inventory model is developed under the following notations and assumptions.     

2.1.       Notations

I(t): Inventory level at any time, t≥0.

D: Demand per unit time, which is constant.

T: Cycle of length.

S: Set-up cost per unit time.

H: Holding cost per unit item, which is time depended.

P: Unit demand and set-up cost dependent production cost.

q: Production quantity per batch.

f(D,S): Unit production cost per cycle.

TAC(D,S,q): Total average cost per unit time.

w0: Space area per unit quantity.

W: Total storage space area.

2.2.       Assumptions

a) The inventory system involves only one item.

b) The replenishment occur instantaneously at infinite rate.

c) The lead time is negligible.

d) Demand rate is constant.

e) The unit production cost is continuous function of demand and Set-up cost and                  

     take the following form:

     P = θ,     θ,x (>0).

f) Holding cost is time depended, as “at”.

2.3.       Crisp model

Figure1: Inventory Model

The differential equation describing I(t) as follows

                ,                 0≤t≤T                                                  (2.3.1)

With the boundary condition I(0) = q, I(T) = 0.

The solution of (2.3.1) is obtained as

              I(t) = q – Dt                                                                             (2.3.2)

Also there are T= q/D.                                             

Here inventory holding cost = H = .                            (2.3.3)

Total inventory related cost per cycle = set-up cost + holding cost + production cost

                                        = S + +Pq                                              (2.3.4)

So total average cost per cycle is given by

                    TAC(D,S,q)

                                        + + θ.                               (2.3.5)                      

And storage area = w0q.

          So the inventory model can be written as

                       Min TAC(D,S,q)  + + θ                    (2.3.6)                       

                       subject to   w0q ≤ W, D,S,q > 0. 

2.4.       Fuzzy model

  When the objective and constraint goals, coefficients and exponents become fuzzy sets and fuzzy Numbers respectively, the crisp model (2.3.6) written to be a fuzzy model, as

                                 TAC(D,S,q)  =    +  +

                        subject to   q   ,        D,S,q > 0.                       (2.4.1)   

3.     GEOMETRIC PROGRAMMING (GP) PROBLEM

3.1.       Primal program

            Primal Geometric Programming (PGP) problem is:                  

            Minimize (t) =                                                 (3.1)

            subject to   (t) =     (r=1,2,…,l)              

                            tj> 0,    (j=1,2,….,m).

Where C0k(>0) (k=1,2,…..,T0), Crk(>0) and αrkj (k=1,2,….,1+Tr-1,…..,Tr;r = 0,1,2,……,l; j = 1,2,….,m) are real numbers. It is constrained polynomial PGP problem. The number of term each polynomial constrained functions varies and it is denoted by Tr for each r = 0,1,2,…,l.  Let T = T0+ T1+ T2+…. Tl be the total number of terms in the primal program. The Degree of Difficulty is (DD) = T – (m+1).

3.2.       Dual program:

            Dual programming (DP) problem of (3.1) is:

            Maximize   d(δ) =                       (3.2)         

            Subject to         = 1                              (Normality condition)

 = 0,              (Orthogonal conditions) 

                                        δrk > 0,          (k=1,2,……,Tr)        (Positivity constant) 

·        Case-1 :  For T0 ≥ M+1, the dual program presents a system of linear equations for the dual variables, where the number of linear equations is either less than or equal to dual variables.  More or unique solutions exist for the dual vectors. 

·        Case-2  :   For T0 < M+1, the dual program presents a system of linear equations for the dual variables, where the number of linear equations is greater than the number of  dual variables. In this case generally no solution vector exists for the dual variables. However one can get  an approximate solution vector for the system using either the Latest Square(SQ) or Max-Min(MN) method.

These are applied to solve such a system of linear equations. Ones optimal dual          variable vector  are known, the corresponding values of the primal variable vector x is found from the following relations:

             =     if i 

and    =       if i     (k=1,2, …….., ).

3.3.       Solution procedure of crisp model by Geometric Programming (G.P) technique:

           Here the primal problem is

           Min     TAC(D,S,q)  =  + + θ                                   (3.1.1) 

           subject to         w0q ≤ W,        D ,S, q > 0. 

            Corresponding dual form of (3.1.1) is given by

            Max d() =

            subject to    = 1                                                          (3.1.2)           

                                = 0

                                + (1-x)= 0

                            + 2= 0

                                0.      

From (3.1.2) we get    .

Putting the values in (3.1.2) we get the optimal solution of dual problem. The values of D, S, q is obtained by using the primal dual relation as follows.

From primal dual relation we get

                       = (,

                      = (,

                      = (,

                      = 1.

The optimal solution of the model  through the parametric approach is given by

  (                                            

  and             S* ,      

                     D* ,

                      q*  .                                             

4.     FUZZY NUMBER AND ITS NEAREST INTERVAL APPROXIMATION:

4.1.       Fuzzy number: 

A real number  described as fuzzy subset on the real line  whose membership function  has the following characteristics with

                       =  

Where

 is continuous and strictly increasing and  is continuous and strictly decreasing.

α-cut of The α-cut of , is defined by Aα={x: μA(x)≥α.}                     

Figure2: Trapezoidal fuzzy number of   with α-cut.

Aα is a non-empty bounded closed interval in X and it can be denoted by Aα = [AL(α), AR(α)]. Where AL(α) and AR(α) are the lower and upper bounds of the closed interval respectively.

Figure 2 shows a fuzzy number  with α-cuts Aα1 = [AL(α1), AR(α1)], Aα2 = [AL(α2), AR(α2)]. It Seen that if α2 ≥ α1 then AL(α2) ≥ AL(α1) and AR(α1) ≥ AR(α2).

4.2.       Interval number

An interval number A is defined by an ordered pair of real numbers as follows A = [ where andare the left and right bounds of interval A, respectively. The interval A, is also defined by center () and half-width () as follows

A = ( = {x: where  =  is the center and  =  is the half-width of A.

4.3.       Nearest interval approximation

Here we want to approximate a fuzzy number by a crisp model. Suppose  and are two fuzzy numbers with α-cuts are [AL(α), AR(α)] and[BL(α), BR(α)]  respectively. Then the distance between  and  is

                     d() = .

Given is a fuzzy number. We have to find a closed interval , which is the nearest to with respect to metric. We can do it since each interval is also a fuzzy number with constant α-cut for all α ∈ [0,1]. Hence (,.  Now we have to minimize

                  d(

with respect to .

In order to minimize d(, it is sufficient to minimize the function

            D(, = ()).

The first partial derivatives are

          

And

           

Solving   and  we get

CL =  and CR = .

Again since   (D(,)) =2 > 0,   (D(,)) =2 > 0 and

H(,) =  (D(,)). (D(,)) –  = 4 > 0.

So D(,) i.e. d( is global minimum. Therefore, the interval

Cd( = [ ] is the nearest interval approximation of fuzzy number  with respect to the metric d.

Let  = (a1,a2,a3) be a triangular fuzzy number. The α-cut interval of  is defined as

            Aα = [,] where  = a1+α(a2 - a1) and  = a3 - α(a3 – a2). By nearest interval approximation method,

the lower limit of the interval is

           CL =  =  = ,

and the upper limit of the interval is

           CR =  =   =  .

Therefore, the interval number corresponding to a given fuzzy number  is [. In the centre and half–width form the interval number of  is defined as     .

4.4.       Parametric Interval-valued function:

Let [m, n] be an interval, where m > 0, n > 0.. From analytical geometry point of view, any real number can be represented on a line. Similarly; we can express an interval by a function. The parametric interval-valued function for the interval [m, n] can be taken as g(s) =  for s ∈ [0,1], which is strictly monotone, continuous function and its inverse exits. Let be the inverse of g(s), then

               s.

5.     GEOMETRIC PROGRAMMING WITH FUZZY COEFFICIENT:

When all coefficients of Eq. (6) are triangular fuzzy number, then the geometric programming problem is of the form

        Min           (x)                                                                                 (5.1)

        subject to  (x)       (1≤i≤n)  

                         x>0.

Its objective functions

        (x) =      

and constraints of the form

            (x) =       (0≤i≤n)

are all posynomials of x in which coefficients  and indexes  are fuzzy numbers.

Where  = ( and  = (. Using nearest interval approximation method, we transform all triangular fuzzy number into interval number i.e. [and [. The geometric programming problem with imprecise parameters is of the following form

          Min          (x)                                                                              (5.2)

        subject to  (x)  1               (1≤i≤n)  

                         x>0,

Its objective function is

        (x) = ,

and constraints of the form is

         (x) = ,      (1≤i≤n).

Where  denote the interval counterparts i.e. ∈ [ and∈ [   for all i and k. Using parametric interval-valued functional form, the problem (5.2) reduces to

        Min (x,s) =                            (5.3)

        Subject to   (x,s) =  

                           xj > 0 for i = 1,2,…….n,   j = 1,2,……..m.

This is a parametric geometric programming problem. We get different solutions of this problem for different value of the parameter s.

the dual programming of (5.3) is as follows:

       Max d(δ,s)  =        (5.4)

       Subject to

            = 1,

             = 0,

             > 0.

5.1.       Theorem 5.1.

If x is a feasible vector for the constraints PGP and if δ is a feasible vector for the corresponding DP, then (x,s) ≥ d(δ,s) (Primal- Dual Inequality).

5.2.       Proof.           

The expression for (x,s) can be written as

(x,s) = .

Here the weights are  and positive terms are      , ……… ,   .

Now applying A.M.-.G.M inequality, we get

       ()

Or    [

 Or   

 Or                                                (5.1.1)

Again (x,s) can be written as

(x,s) = .

Now applying A.M.-.G.M inequality, we get

 Or    

              (5.1.2)

Using   1,                                                 [as (x,s)  1]

We have                                           

1                                        (5.1.3)

Multiplying (5.1.1) and (5.1.3) we get

               (5.1.4)

Using orthogonal condition the inequality (5.1.4) becomes

    = d(δ,s)       (5.1.5)

i.e., (x,s) ≥ d(δ,s) . (Proof).

5.3.       Theorem 5.2.

δ is a feasible vector for the dual programming (DP) problem , then d(δ,1)  ≥ d(δ,0).

5.4.       Proof:

We have   ,    for all k, (k=1,2,…….,). 

Or

Or

Or

Or  

Or

 

i.e., d(δ,1)  ≥ d(δ,0). (proof)

5.5.       Solution procedure of fuzzy model by Geometric Programming (G.P) technique:

When  = (),  = (),  = () and  = () are triangular fuzzy number .then the fuzzy model is

           TAC(D,S,q)  =    +  +                                 (5.3.1)

          subject to  q  , D, S, q > 0.

Using nearest interval approximation method, the interval number corresponding triangular number  = () is [] = []. Similarly interval number corresponding  and are [] = [], [] = [] and [] = [] respectively. The problem (5.3.1) reduces to

         Min TAC(D,S,q)  =  + +            (5.3.2)

         subject to w0q ≤ [],D, S, q > 0.

Which is equivalent to

          Min TAC(D,S,q)  =    +  +                                 (5.3.3)

          subject to  q  , D, S, q > 0,

where   ∈ [],  ∈ [],  ∈ [] and  ∈ [].

According to section 4.4, the fuzzy model (5.2.3) reduces to a parametric programming by replacing

  where s ∈ [0,1].

The model takes the reduces form as follows

         Min TAC(D,S,q)  =    +  +    (5.3.4)

         subject to  q   , D,S,q > 0

Corresponding dual form of (5.3.4) is given by

         Max d() =

         subject to    = 1                                                        (5.3.5)

                            = 0

                            + (1-x)= 0

                         + 2= 0

                           ≥ 0.      

From (5.3.5) we get  .

Putting the values in (5.3.5) we get the optimal solution of dual problem. The values of D, S, q is obtained by using the primal dual relation as follows:

From primal dual relation we get

                  = (,

                 = (,

                = (,

                 = 1.

The optimal solution of the model  through the parametric approach is given by

      (

       and

                     S* = ,     

                     D* = ,

                     q* =  .     

6.     NUMERICAL EXAMPLE AND SOLUTION:

A manufacturing company produces a machine. It is given that the inventory carrying cost of the machine is $15 per unit per year. The production cost of the machine varies inversely with the demand and set-up cost. From the past experience, the production cost of the machine is 120 where D is the demand rate and S is set-up cost. Storage space area per unit time () and total storage space area (W) are 100 sq. ft. and 2000 sq. ft. respectively. Determine the demand rate (D), set-up cost (S), production quantity (q), and optimum total average cost (TAC) of the production system.

Then the input value of the model (2.3.6) is 

                          Table-1(Input values)

    a

      H

       x

      θ

     

        W

    7

     15

    1.75

    120

     100

     2000

Then the model is of the form

              Min TAC(D,S,q)  =  + +

             subject to 100q ≤ 2000, D, S, q > 0.                                       (6.1)

                           Table 2: Optimal solution of (2.3.6) for crisp model

 Crisp model

       S*

      D*

   q*

TAC*(S*,D*,q*)$

      G.P

    0.684

   4048

  20

    140.517

    N.L.P

    0.685

   4047

  20

    140.685

                                   

When the input data of inventory model is taken as triangular fuzzy number i.e..  = (5,7,9),  = (13,15,17), = (116,120,124) and = (1800,2000,2200). Using nearest interval approximation method, we get the corresponding interval number and interval-valued function i.e. 

             ≈ [6,8],     ∈ [6,8],

             ≈ [14,16],     ∈ [14,16],

            ≈ [118,122],     ∈ [118,122],

 ≈ [1900,2100],     ∈ [1900,2100],  where s ∈ [0,1].

The optimal solution of the fuzzy model by interval-valued parametric geometric programming is presented in Table

         Table 3:  Optimal Solution for Fuzzy Inventory Model

          s

           S*

           D*

          q*

   TAC*(S*,D*,q*)$  

        0.0

        0.820

       2983.86

        21.00

          119.801

        0.1

        0.786

       3175.16

        20.79

          123.060

        0.2

        0.753

       3378.71

        20.58

          126.396

        0.3

        0.722

       3595.32

        20.38

          129.924

        0.4

        0.693

       3825.81

        20.18

          133.735

        0.5

        0.664

       4071.08

        19.97

          137.533

        0.6

        0.637

       4332.07

        19.78

          141.519

        0.7

        0.610

       4609.80

        19.58

          145.473

        0.8

        0.585

       4905.33

        19.38

          149.793

        0.9

        0.561

       5219.81

        19.19

          154.196

        1.0

        0.538

       5554.45

        19.00

          158.767

                    

For s=0, the lower bound of the interval value of the parameter is used to find the optimal solution. For s=1, the upper bound of interval value of the parameter is used for the optimal solution. These results yield the lower and upper bounds of the optimal solution. The main advantage of the proposed technique is that one can get the intermediate optimal result using proper value s.

Here we have given a rough graph, which shown how change the value of TAC*(S*,D*,q*)  for difference values of s.

Figure 3:   Change of the value of objective function for change of s, by Fuzzy Geometric Programming Technique.

7.     SENSITIVITY ANALYSIS:

 Effect, for increment the parameter “s”.

(1)  For increasing of “s”, set-up cost S* is decreasing.

(2)  For increasing  of “s”, demand rate D* is increasing.

(3)  For increasing of “s , Production quantity q* is decreasing. 

(4)  For increasing of “s”, Total average cost TAC*(S*,D*,q*) is increasing. 

8.     CONCLUSION:

            In this paper, we have proposed a real life inventory problem in a fuzzy environment and presented solution along with sensitivity analysis approach. The inventory model developed with unit production cost, time depended holding cost, with-out shortages. This model has been developed for single item.

            In this paper, we first create a crisp model then it transformed to fuzzy model and solved by parametric Geometric-Programming technique. Here decision maker may obtain the optimal results according to his expectation .In fuzzy we have considered triangular fuzzy number(T.F.N) In future, the other type of membership functions such as piecewise linear hyperbolic, L-R fuzzy number, Trapezoidal  Fuzzy Number (TrFN), Parabolic flat Fuzzy Number (PfFN), Parabolic Fuzzy Number (pFN), pentagonal fuzzy number etc can be considered to construct the membership function and then model can be easily solved.

ACKNOWLEDGEMENTS

The authors are thankful to University of Kalyani for providing financial assistance through DST-PURSE Programme. The authors are grateful to the reviewers for their comments and suggestions.

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