A fuzzy replenishment policy for deteriorating items

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Abstract: Normally, the real-world inventory control problems are imprecisely defined .... the Triangular Fuzzy Number (TFN) ˜B = (a, b, c) is C( ˜B) = 1. 3 (a + b + c) and ..... and A. Goswami. A MONLP problem may be taken in the following form:.
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Int. J. Operational Research, Vol. 5, No. 3, 2009

A fuzzy replenishment policy for deteriorating items with ramp type demand rate under inflation G.C. Mahata Department of Mathematics, Sitananda College, Nandigram, Purba Medinipur, 721631 West Bengal, India E-mail: [email protected]

A. Goswami∗ Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India E-mail: [email protected] ∗ Corresponding author Abstract: Normally, the real-world inventory control problems are imprecisely defined and human interventions are often required to solve these decision-making problems. In this paper, a realistic inventory model with imprecise inventory costs have been formulated for deteriorating items under inflation. Shortages are allowed and the demand rate is taken as a ramp type function of time as well. For generality, we not only provide the solution for the inventory system, where the inventory starts with no shortages; but also enumerate two possible shortage models in the fuzzy sense. We have introduced fuzzy multi-objective mathematical programming with the triangular fuzzy number and it is applied as cost coefficients to the inventory problem. Pareto optimal solution of this multi-objective mathematical programming is established. Numerical example has been provided to illustrate the problem. Finally, sensitivity analysis is carried out to identify the most sensitive parameters in the system. Keywords: inventory; ramp type demand rate; deteriorating items; fuzzy cost coefficients; graded mean integration representation; fuzzy programming technique. Reference to this paper should be made as follows: Mahata, G.C. and Goswami, A. (2009) ‘A fuzzy replenishment policy for deteriorating items with ramp type demand rate under inflation’, Int. J. Operational Research, Vol. 5, No. 3, pp.328–348. Biographical notes: G.C. Mahata is a Lecturer in the Department of Mathematics, Sitananda College, Nandigram, Purba Medinipur, West Bengal, India. He received his BSc and MSc Degree in Mathematics on 1998 and 2000 respectively from Vidyasagar University, Midnapore, West Bengal, India. He received his PhD Degree in Inventory Control/ Management, Operations Research on 2007 from Indian Institute of Technology, Kharagpur, India. He is currently engaged in research in areas of fuzzy inventory control and optimisation theory. Copyright © 2009 Inderscience Enterprises Ltd.

A fuzzy replenishment policy for deteriorating items

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A. Goswami received his MSc and PhD Degree from Jadavpur University, India in 1985 and 1992, respectively. In 1992, he joined the Indian Institute of Technology, Kharagpur, India, where at present he is Professor in Mathematics Department. His research interests focus on fuzzy inventory control/optimisation, fuzzy database systems, distributed databases, objectoriented databases and data-mining techniques in fuzzy environments.

1 Introduction The inventory model with ramp type demand rate was proposed the first time by Hill (1995). Basically the ramp type of the demand pattern is generally seen in the case of any new brand of consumer goods coming to the market. The demand rate for such items increases with time up to certain time and then ultimately stabilises and becomes constant. For example, we can easily observe that some kind of Christmas season consumer products fits this kind of ramp type demand rate. In the beginning of Christmas season, about October or November, the sale increases linearly up to the beginning of December and then the sale reaches its climate and keep this climate sales situation until the end of Christmas season, 24th December. Hence, the ramp type demand rate is quite common to see and deserves comprehensive studies. Hill (1995) first considered the inventory models for increasing demand followed by a constant demand. He derived the exact solution to compare with the Silver-Meal heuristic. Mandal and Pal (1998) extended the inventory model with ramp type demand for deteriorating items and allowing shortage. However, they only derived an approximated solution for the inventory model starting with no shortage. Recently Giri et al. (2003) extended the ramp type demand inventory model with a more generalised Weibull deterioration distribution. Besides, Manna and Chaudhuri (2006) developed an EOQ model with ramp type demand rate, time dependent deterioration rate, unit production cost and shortages. The effect of deterioration is very important in many inventory systems. Deterioration is defined as decay or damage such that the item cannot be used for its original purpose. Most of the physical goods undergo decay or deterioration over time. Commodities such as fruits, vegetables, foodstuffs, etc., suffer from depletion by direct spoilage while kept in store. Highly volatile liquids such as gasoline, alcohol, turpentine, etc., undergo physical depletion over time through the process of evaporation. Electronic goods, radioactive substances, photographic film, grain, etc. deteriorate through a gradual loss of potential or utility with the passage of time. Thus decay or deterioration of physical goods in stock is a very realistic feature and inventory modellers felt the need to take this factor into consideration. Ghare and Schrader (1963), first, proposed an inventory model having a constant rate of deterioration and a constant rate of demand over a finite-planning horizon. Shah and Jaiswal (1978) presented an order-level inventory model for deteriorating items with a constant rate of deterioration. Aggarwal (1978) developed an order-level inventory model by correcting and modifying the error in Shah and Jaiwal’s (1978) analysis in calculating the average inventory holding cost. Covert and Philip (1973) extended Ghare and Schrader’s (1963) model by considering variable rate of deterioration. Shah (1977) suggested a further generalisation of

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all these models by allowing shortages and using a general distribution for the deterioration rate. On the other hand, several studies have examined the inflationary effect on an inventory policy. Buzacott (1975) and Misra (1975) simultaneously developed EOQ models with constant demand and a single inflation rate for all associated costs. Bierman and Thomas (1977) then proposed an inflation model for the EOQ which also incorporated the time value of money. Again, under the assumption of constant demand, Misra (1979) first provided different inflation rates for various costs associated with an inventory system. Bose et al. (1995) developed the EOQ inventory model under inflation and time discounting. Recently, Yang et al. (2001) generalised the inventory model under inflation for fluctuating demand. In the development of inventory models, the researchers have assumed that the set up cost, unit holding cost, purchasing cost and shortage cost are constants. These kind of assumptions are not always true. It may not be possible to specify the values of these cost parameters precisely but they may contain some uncertain values. For example, “unit holding cost is about Ch ”, or “unit shortage cost is approximately Cs or more”, etc. In other sense, these parameters may contain some uncertain values. On this view, several researchers developed fuzzy inventory models in situations where these parameters are described imprecisely. Several authors namely Chang et al. (1998), Lee and Yao (1998), Lin and Yao (2000), Yao et al. (2000), Mahata et al. (2005) and Mahata and Goswami (2006) developed inventory models in fuzzy sense by considering different parameters as fuzzy parameters. In this paper, an effort has been made to analyse an EOQ model for deteriorating items under inflation assuming the demand rate to be a ramp type function of time. Such type of the demand pattern is generally seen in the case of any new brand of consumer goods coming to the market. The demand rate for such items increases with time up to certain time and then ultimately stabilises and becomes constant. It is believed that such type of demand rate is quite realistic. In this paper, we present two possible shortage models: •

model starting with no shortage



model starting with shortage.

In both cases fuzzy total relevant costs per unit time have been derived by considering holding cost, shortage cost, purchase cost and set up cost as triangular fuzzy numbers. We use Graded Mean Integration Representation method for defuzzifying fuzzy total relevant cost per unit time. For determination of the optimal ordering policies, we have used fuzzy programming technique. Pareto optimal solution of this problem is established. Numerical example has been provided to illustrate the problem. Finally, sensitivity analysis is carried out to identify the most sensitive parameters in the system.

2 Preliminaries In Chen and Hsieh (1999), introduced Graded Mean Integration Representation method based on the integral value of graded mean h-level of generalised fuzzy number for defuzzifying generalised fuzzy number. Here, we first describe generalised fuzzy number as follows:

A fuzzy replenishment policy for deteriorating items

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 is described as any fuzzy subset of the real line R, A generalised fuzzy number A whose membership function µA(x) satisfies the following conditions: 1

µA(x) is continuous mapping from R to the closed interval [0, 1],

2

µA(x) = 0; −∞ < x ≤ a,

3

µA(x) = L(x) is strictly increasing on [a, b],

4

µA(x) = w; b ≤ x ≤ c, where 0 < w ≤ 1,

5

µA(x) = R(x) is strictly decreasing on [c, d],

6

µA(x) = 0; d ≤ x < ∞.

Here a, b, c and d are real numbers. We denote this type of generalised fuzzy number  = (a, b, c, d)LR .  = (a, b, c, d; w)LR . When w = 1, it can be simplified as A as A Second, by Graded Mean Integration Representation method L−1 and R−1 are the inverse functions of L and R, respectively, and the graded mean h-level value of  = (a, b, c, d; w)LR is h(L−1 (h) + R−1 (h))/2 (see Figure 1). generalised fuzzy number A  is P (A)  with grade w, where Then the Graded Mean Integration Representation of A  = P (A)



w

h 0

(L−1 (h) + R−1 (h)) dh 2



w

hdh

(1)

0

with 0 < h ≤ w and 0 < w ≤ 1.  is a special case of The Generalised Triangular Fuzzy Number (GTFN) B  generalised fuzzy number, and be denoted as B = (a, b, c; w). It’s corresponding graded mean integration representation is  = P (B)

w 0

h{a + (b − a)h/w + c − (c − b)h/w}/2dh a + 4b + c w . = 6 hdh 0

(2)

 = (a1 , a2 , a3 , a4 ; wA ) Figure 1 The graded mean h-level value of generalised fuzzy number A

Remark: By formula (2), it is easy to observe that the graded mean integration  = (a, b, c; w) is independent of w. representation of the GTFN B

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 = (a, b, c). The defuzzification of B  = (a, b, c) When w = 1, it can be simplified as B can be found by centroid or graded mean integration method. The centroid of  = (a, b, c) is C(B)  = 1 (a + b + c) and the the Triangular Fuzzy Number (TFN) B 3  = (a, b, c) is P (B  = 1 (a + 4b + c). graded mean integration representation of B 6 a+c The mid-point of the interval [a, c] is M = 2 . Thus,  − P (B)  = 1 (M − b) C(B) 3 1  − b = (M − b) P (B) 3  = 1 (M − b). M − C(B) 3 i

 < C(B)  0, =  ∈ FN . kA kA (kc, kb, ka) if k < 0.

(3)

A fuzzy replenishment policy for deteriorating items

333

3 Assumptions and notation The mathematical model is developed on the basis of the following assumptions and notation: 1

The replenishment rate is infinite; replenishments are instantaneous.

2

The lead time is zero.

3

T is the fixed length of each ordering cycle.

4

S is the maximum inventory level for each ordering cycle.

5

Shortage is allowed and fully backordered.

6

A constant fraction θ, assumed to be small, of the on-hand inventory gets deteriorated per unit time.

7

Ch : the inventory holding cost per unit per unit of time; Cs : the shortage cost per unit per unit of time; Cp : the cost of each unit which deteriorates and C0 : the s , C p and h , C set up cost per order, are known and constant in the crisp model. C  C0 are the fuzzy holding cost, fuzzy shortage cost, fuzzy cost of each unit and fuzzy set up cost respectively in the fuzzy model.

8

I(t) is the on-hand inventory at time t over [0, T ].

9

The demand rate R(t) is assumed to be a ramp type function of time: R(t) = D0 [t − (t − µ)H(t − µ)], D0 > 0 (see Figure 3), where H(t − µ) is the well known Heaviside’s function defined as follows  1; if t ≥ µ, H(t − µ) = 0; if t < µ.

Figure 3 A ramp type function of the demand rate

4 Crisp mathematical models The objective of the inventory problem here is to determine the optimal order quantity so as to keep the annual total relevant cost as low as possible. Based on whether the inventory starts with shortages or not, two different models have been discussed.

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4.1 Model-I: The inventory model starts without shortages In this subsection, we will discuss the deterministic inventory model for deteriorating items, where the inventory starts with no shortages. The behaviour of the inventory at any time during a given cycle is depicted in Figure 4. Replenishment is made at time t = 0 when the inventory level is at its maximum, S. From t = 0 to t = t1 time units, the inventory level decreases due to both demand and deterioration. At time t1 , the inventory level reaches zero; thereafter, shortages are allowed to occur during the time interval (t1 , T ), and all of the demand during period (t1 , T ) is backlogged. The total number of backlogged items is replaced by the next replenishment. Figure 4 An illustration of an order-level cycle

The inventory level of the system at time t over period [0, T ) can be described by the following equations: dI(t) + θI(t) = −R(t); 0 ≤ t ≤ t1 , dt dI(t) = −R(t); t1 ≤ t ≤ T. dt

(4) (5)

In this model, it is assumed that µ < t1 , therefore, the above two governing equations becomes dI(t) + θI(t) = −D0 t; 0 ≤ t ≤ µ, dt dI(t) + θI(t) = −D0 µ; µ ≤ t ≤ t1 , dt dI(t) = −D0 µ; t1 ≤ t ≤ T. dt

(6) (7) (8)

The solutions of the differential equations (6)–(8), with the boundary conditions I(0) = S and I(t1 ) = 0 are   t 1 (9) − 2 (1 − e−θt ) ; 0 ≤ t ≤ µ, I(t) = Se−θt − D0 θ θ

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A fuzzy replenishment policy for deteriorating items D0 µ θ(t1 −t) − 1}; µ ≤ t ≤ t1 , {e θ I(t) = −D0 µ(t − t1 ); t1 ≤ t ≤ T.

(10)

I(t) =

(11)

Note that, in equations (9) and (10), the values of I(t) at t = µ should coincide, which implies that S=

D0 µ θt1 D0 θµ e − 2 (e − 1). θ θ

(12)

t Thus, the cumulative inventory during (0, t1 ) is 0 1 I(t)dt and the cumulative shortages T during (t1 , T ) is t1 I(t)dt. As a result, the present value of the inventory holding cost is  µ  t1  t1 −rt −rt −rt e I(t)dt = Ch e I(t)dt + e I(t)dt Ch 0

0

µ



1 − e−(θ+r)µ D0 = Ch S − 2 (1 − e−µr − µre−µr ) θ+r θr

−µr −(θ+r)µ D0 1 − e −1 e + 2 + θ r θ+r

 D0 µ θt1 e−(θ+r)µ − e−(θ+r)t1 e−rt1 − e−rµ + e + θ θ+r r

and the present value of the shortage cost is  T Cs D0 µ −rt1 e−rt I(t)dt = [e − e−rT − (T − t1 )re−rT ]. Cs r2 t1

(13)

(14)

t In addition, the amounts of deteriorated items during (0, t1 ) is θ 0 1 I(t)dt. Therefore, the present value of the cost for the deteriorated items is  t1 Cp θ e−rt I(t)dt 0 

1 − e−(θ+r)µ D0 = Cp θ S − 2 (1 − e−µr − µre−µr ) θ+r θr

D0 1 − e−µr e−(θ+r)µ − 1 + 2 + θ r θ+r

 e−rt1 − e−rµ D0 µ θt1 e−(θ+r)µ − e−(θ+r)t1 + + e . (15) θ θ+r r Consequently, the present value of the total relevant cost per unit time for Model I during the cycle (0, T ) is    t1  t1  T e−rt I(t)dt + Cp θ I(t)dt + Cs e−rt I(t)dt T T C1 (t1 ) = C0 + Ch 0

0

= X11 C0 + X12 Ch + X13 Cd + X14 Cs ,

t1

(16)

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G.C. Mahata and A. Goswami

where 1 , T 

1 − e−(θ+r)µ D0 1 = S − 2 (1 − e−µr − µre−µr ) T θ+r θr

e−(θ+r)µ − 1 D0 1 − e−µr + 2 + θ r θ+r

 D0 µ θt1 e−(θ+r)µ − e−(θ+r)t1 e−rt1 − e−rµ + e + , θ θ+r r 

1 − e−(θ+r)µ θ D0 S − 2 (1 − e−µr − µre−µr ) = T θ+r θr

D0 1 − e−µr e−(θ+r)µ − 1 + 2 + θ r θ+r

 D0 µ θt1 e−(θ+r)µ − e−(θ+r)t1 e−rt1 − e−rµ + + e and θ θ+r r D0 µ = 2 {e−rt1 − e−rT − (T − t1 )re−rT }. r T

X11 = X12

X13

X14

Furthermore, the total backorder amount at the end of the cycle is D0 µ(T − t1 ); therefore, the order quantity, denoted by Q, is Q = S + D0 µ(T − t1 ). Also, t1 = T − =

Q−S D0 µ

2{θ2 (Q − D0 µT ) + D0 (eθµ − θµ − 1)} D0 θµ

(17)

neglecting powers of θ higher than the first and using equation (12).

4.2 Model-II: The inventory model starts with shortages In this subsection, we consider the deterministic inventory model for deteriorating items, where the cycle starts with shortages. The behaviour of the inventory system at any time during a given cycle is depicted in Figure 5. Depending on the procurement time t2 , two different situations may arise: •

µ < t2



µ > t2 .

The inventory system starts with zero inventory at t = 0, and shortages are allowed to accumulate up to t2 . Procurement is done at time t2 . The quantity received at t2 is used partly to make up for the shortages which accumulated in the pervious cycle from

A fuzzy replenishment policy for deteriorating items

337

time 0 to t2 . The rest of the procurement accounts for the demand and deterioration in [t2 , T ]. The inventory level gradually falls to zero at time T . The inventory level of the system at time t over the period [0, T ] can be described by the following equations: dI(t) = −R(t); 0 ≤ t ≤ t2 , dt dI(t) + θI(t) = −R(t); t2 ≤ t ≤ T. dt

(18) (19)

Figure 5 An illustration of an order-level cycle, where the inventory is permitted to start with shortages: (a) µ < t2 and (b) µ > t2

4.2.1 Situation 1 (µ < t2 ) When µ < t2 , the above two governing equations become dI(t) = −D0 t; 0 ≤ t ≤ µ, dt dI(t) = −D0 µ; µ ≤ t ≤ t2 , and dt dI(t) + θI(t) = −D0 µ; t2 ≤ t ≤ T. dt

(20) (21) (22)

The solution of the differential equations (20)–(22) with the boundary conditions I(0) = 0 and I(T ) = 0 are D0 2 t ; 0 ≤ t ≤ µ, 2

µ I(t) = D0 µ − t ; µ ≤ t ≤ t2 , and 2

I(t) = −

I(t) =

D0 µ θ(T −t) [e − 1]; t2 ≤ t ≤ T. θ

(23) (24) (25)

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T Thus, the cumulative inventory during (t2 , T ) is t2 I(t)dt and the cumulative t shortages during (0, t2 ) is 0 2 I(t)dt. As a result, the present value of the holding cost is    T e−rT − e−rt2 D0 µ θT e−(θ+r)t2 − e−(θ+r) T −rt + e I(t)dt = Ch Ch e (26) θ θ+r r t2 and the present value of the shortage cost is  µ   t2 −rt −rt Cs e I(t)dt = Cs e I(t)dt + Cs 0

0

 = Cs

t2

e−rt I(t)dt

µ

D0 (2 − 2e−rµ − 2rµe−µr − µ2 r2 e−rµ ) 2r3

D0 µ −rt2 (e − e−rµ ) 2r  D0 µ + 2 ((1 + µr)e−µr − (1 + rt2 )e−rt2 ) . r +

(27)

T In addition, the amounts of deteriorated items during (t2 , T ) is θ t2 I(t)dt. Therefore, the present value of the deterioration cost is    T e−(θ+r)t2 − e−(θ+r)T e−rT − e−rt2 Cp θ + . (28) e−rt I(t)dt = Cp D0 µ eθT θ+r r t2 For this model, the present value of the ordering cost is C0 e−rt2 . Consequently, in situation 1 the present value of the total relevant cost per unit time for Model II during the cycle (0, T ) is   −rt2 + Ch T C21 (t2 ) = C0 e 

T

−rt

e

t2 T

+ Cp θ

e−rt I(t)dt

 I(t)dt + Cs



t2

e−rt I(t)dt

0

T

t2

= X21 C0 + X22 Ch + X23 Cp + X24 Cs , where e−rt2 , T   e−rT − e−rt2 D0 µ θT e−(θ+r)t2 − e−(θ+r)T e + , = θT θ+r r   e−rT − e−rt2 D0 µ θT e−(θ+r)t2 − e−(θ+r)T + = e and T θ+r r  1 D0 D0 µ −rt2 (e = (2 − 2e−rµ − 2rµe−µr − µ2 r2 e−rµ ) + − e−rµ ) 3 T 2r 2r  D0 µ + 2 ((1 + µr)e−µr − (1 + rt2 )e−rt2 ) . r

X21 = X22 X23 X24

(29)

A fuzzy replenishment policy for deteriorating items

339

Furthermore, the backorder amount for the entire cycle is 12 D0 µ2 + D0 µ(t2 − µ) and the order quantity, Q, is 1 Q = I(t2 ) + D0 µ2 + D0 µ(t2 − µ) 2

µ D0 µ θ(T −t2 ) [e . − 1] + D0 µ t2 − = θ 2 2Q − 2D0 µT + D0 µ2 , Also, t2 = T − 2D0 θµ

(30)

neglecting powers of θ higher than the first.

4.2.2 Situation 2 (µ > t2 ) If µ > t2 , the above two governing equations (18) and (19) become dI(t) = −D0 t; 0 ≤ t ≤ t2 , dt dI(t) + θI(t) = −D0 t; t2 ≤ t ≤ µ, and dt dI(t) + θI(t) = −D0 µ; µ ≤ t ≤ T. dt

(31) (32) (33)

The solution of the differential equations (31)–(33) with the boundary conditions I(0) = 0 and I(T ) = 0 are 1 I(t) = − D0 t2 ; 0 ≤ t ≤ t2 , 2 D0 I(t) = Seθ(t2 −t) − 2 [(θt − 1) − (θt2 − 1)eθ(t2 −t) ]; t2 ≤ t ≤ µ, and θ D0 µ θ(T −t) I(t) = [e − 1]; µ ≤ t ≤ T. θ

(34) (35) (36)

Note that, in equations (35) and (36), the values of I(t) at t = µ should coincide, which implies that S=

D0 µ (T −t2 ) D0 θ(µ−t2 ) e − 2 (e + θt2 − 1). θ θ

(37)

By using a similar argument as in Model-II of situation 1, the present value of the total relevant cost per unit time for Model-II of situation 2 is obtained as follows: T C22 (t2 ) = X31 C0 + X32 Ch + X33 Cp + X34 Cs , where X31 =

e−rt2 , T

(38)

340 X32

X33 X34

G.C. Mahata and A. Goswami −(θ+r)t2

 − e−(θ+r)µ e D0 1 θt2 = S + 2 (θt2 − 1) e T θ θ+r D0 + 2 2 ((θ − r + θrt2 )e−rt2 − (r − θ − θrµr)e−µr ) θ r

 e−rT − e−rµ D0 µ θT e−(θ+r)µ − e−(θ+r)T + + e , θ θ+r r = θX32 and D0 = 3 (2 − 2e−rt2 − 2rt2 e−rt2 − r2 t22 e−rt2 ). 2r T

Furthermore, the total backorder amount for the entire cycle is 12 D0 t22 , therefore, the order quantity, Q, is 1 Q = S + D0 t22 . 2 2D0 µT + D0 µθT 2 − D0 µ2 − 2Q Also, t2 = 1 − 1 − D0 µθ

(39)

neglecting powers of θ higher than the first and using equation (37). From the above arguments, the present value of the total relevant cost per unit time for Model II can be expressed as T C2 (t2 ) =

 T C21 (t2 ); if µ < t2 ,

(40)

T C22 (t2 ); if µ > t2 ,

where T C21 (t2 ) and T C22 (t2 ) are given by equations (29) and (38). Hence the problem is Minimise T C1 (t1 ) Minimise T C2 (t2 )

(41)

subject to the constraints T > t1 , T > t2 , t2 > 0.

5 Fuzzy model In the fuzzy model, we have considered that the inventory cost coefficients are fuzzy h , C d and C s . Then the cost functions (16), (29) 0 , C numbers and are denoted by C and (38) reduce to 0 + X12 C h + X13 C d + X14 C s ,  T C 1 (t1 ) = X11 C  0 + X22 C h + X23 C p + X24 C s , T C 21 (t2 ) = X21 C

(42)

0 + X32 C h + X33 C p + X34 C s .  T C 22 (t2 ) = X31 C

(44)

(43)

The Triangular Fuzzy Numbers (TFN) here represent the holding cost Ch , shortage cost Cs , deterioration cost Cd and set-up cost C0 per order. These are

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h = (Ch1 , Ch2 , Ch3 ), C s = (Cs1 , Cs2 , Cs3 ), C d = (Cd1 , Cd2 , Cd3 ), and C 0 = (C01 , C  C02 , C03 ). Then we get T C 1 (t1 ) as  T C 1 (t1 ) = [X11 C01 + X12 Ch1 + X13 Cd1 + X14 Cs1 , X11 C02 + X12 Ch2 + X13 Cd2 + X14 Cs2 , X11 C03 + X12 Ch3 + X13 Cd3 + X14 Cs3 ].

(45)

 We defuzzify T C 1 (t1 ) by formula (2) and the graded mean integration representation  of T C 1 (t1 ) is 1  P (T C 1 (t1 )) = (X11 C01 + X12 Ch1 + X13 Cd1 + X14 Cs1 ) 6 + 4(X11 C02 + X12 Ch2 + X13 Cd2 + X14 Cs2 )  + (X11 C03 + X12 Ch3 + X13 Cd3 + X14 Cs3 ) = F1 X11 + F2 X12 + F3 X13 + F4 X14 , where C01 + 4C02 + C03 Ch1 + 4Ch2 + Ch3 , F2 = , 6 6 Cd1 + 4Cd2 + Cd3 Cs1 + 4Cs2 + Cs3 and F4 = . F3 = 6 6 F1 =

  C 22 (t2 ) are Similarly, the graded mean integration representation of T C 21 (t2 ) and T respectively,  P (T C 21 (t2 )) = F1 X21 + F2 X22 + F3 X23 + F4 X24 ,

(46)

 P (T C 22 (t2 )) = F1 X31 + F2 X32 + F3 X33 + F4 X34 ,

(47)

where t1 and t2 are functions of Q, the order quantity, and are given by equations (17) and (30) or (39). Hence problem (41) becomes  Minimise P (T C 1 (Q))  Minimise P (T C 2 (Q))

(48)

subject to the same constraints as in (41), where   P (T C 21 (Q)); if µ < t2 ,  P (T C 2 (Q)) =  P (T C 22 (Q)); if µ > t2 .

(49)

6 Mathematical analysis General Fuzzy Non-Linear Programming (GFNLP) technique Multi-Objective Non-Linear Programming (MONLP) problem.

to

solve

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G.C. Mahata and A. Goswami A MONLP problem may be taken in the following form:

Maximise/Minimise

f (x) = [f1 (x), f2 (x), . . ., fk (x)]T

subject to :

x ∈ X = {x ∈ Rn : gj (x) ≤ or = or ≥ bj

(50)

for j = 1, . . ., m; li ≤ xi ≤ ui for i = 1, 2, . . ., n}. Some basic definitions on Pareto optimal solutions are introduce below: Definition 1 (Complete optimal solution): x∗ is said to be a complete optimal solution to the MONLP (50) if and only if there exits x∗ ∈ X such that fr (x∗ ) ≤ fr (x), for r = 1, 2, . . . , k and for all x ∈ X. However, when the objective functions of the MONLP conflict with each other, a complete optimal solution does not always exist and hence the Pareto optimality concept arises and it is defined as follows. Definition 2 (Pareto optimal solution): x∗ is said to be a Pareto optimal solution to the MONLP (50) if and only if there does not exit another x ∈ X such that fr (x∗ ) ≤ fr (x), for r = 1, 2, . . . , k and fr (x∗ ) = fr (x) for at least one j, j ∈ {1, 2, . . . , k}. Zimmermann (1978) showed that fuzzy programming technique can be used nicely to solve the multi-objective programming problem. To solve the MONLP (50) problem, the following steps are used: Step 1: Solve the MONLP (50) as a single objective non-linear programming problem using only one objective at a time and ignoring the others. These solutions are known as ideal solutions. Step 2: From the results of Step 1, determine the corresponding values for every objective for each derived solution. With the values of all objectives at each ideal solution, pay-off matrix can be formulated as follows: f1 (x) 1

x x2 ... xk

f1∗ (x1 ) f1 (x2 ) ... f1 (xk )

f2 (x) 1

f2 (x ) f2∗ (x2 ) ... f2∗ (xk )

...

fk (x)

... ... ... ...

fk (x1 ) fk (x2 ) ... fk∗ (xk )

Here x1 , x2 , . . ., xk are the ideal solutions of the objectives f1 (x), f2 (x), . . ., fk (x) respectively. So Ur = max{fr (x1 ), fr (x2 ), . . ., fr (xk )}, and Lr = min{fr (x1 ), fr (x2 ), . . ., fr (xk )}.

(51) (52)

[Lr and Ur are lower and upper bounds of the rth objective function fr (x) for r = 1, 2, . . . , k.]

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Step 3: Using aspiration levels of each objective of the MONLP (50) may be written as follows: Find x so as to satisfy ≤



fr (x) ∼ Lr /fr (x) ∼ Ur (r = 1, 2, . . ., k),

(53)

x ∈ X. Here objective functions of equation (50) are considered as fuzzy constraints. Step 4: Define a membership function is defined by   if  0; µr (fr (x)) = dr (x); if   1; if

function for each objective function. A membership fr (x) ≥ Ur , Lr ≤ fr (x) ≤ Ur , fr (x) ≤

(54)

Lr ,

where Lr = Lr + r and 0 ≤ r ≤ Ur − Lr . Here dr (x) is a strictly monotonic decreasing function with respect to fr (x). Having elicited the membership functions (as in (54)) µr (fr (x)) for r = 1, 2, . . . , k, a general aggregation function µD (x) = µD (µ1 (f1 (x)), µ2 (f2 (x)), . . . , µk (fk (x))) is introduced. So a multi-objective decision making problem can be defined as maximise µD (x)

(55)

subject to x ∈ X.

Here one can adopt the well-known Bellman and Zadeh’s (1970) fuzzy decision to solve (55). According to fuzzy decision based on minimum operator (like Zimmermann’s approach (1978)), the problem (55) is reduced to maximise α, subject to wr µr (fr (x)) ≥ α; for r = 1, 2, . . ., k,

(56)

x ∈ X, 0 ≤ α ≤ 1. Step 5: Solve (56) to get Pareto optimal solution of (50).

7 General Fuzzy Non-Linear Programming Technique to solve the inventory model To solve the multi-objective inventory problem (48), Step 1 is used. according to Step 2 pay-off matrix is formulated as follows:

Q1 Q2

After that

 P (T C 1 (Q))

 P (T C 2 (Q))

∗  P (T C 1 (Q1))  P (T C 1 (Q2))

 P (T C 2 (Q1))  P (T C 2 (Q2))

  From this matrix U1 , L1 , U2 , L2 (where L1 ≤ P (T C 1 (Q)) ≤ U1 and L2 ≤ P (T C 2 (Q)) ≤ U2 are identified. Here, for simplicity, fuzzy linear membership functions

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   µT (P (T C 1 (Q))) and µT (P (T C 2 (Q))) for the objective functions P (T C 1 (Q)) C1 C2  and P (T C 2 (Q)) respectively are defined as follows:    0; for P (T C 1 (Q)) ≤ L1 ,      P (T C 1 (Q)) − L1   µT (P (T C 1 (Q))) = ; for L1 ≤ P (T C 1 (Q)) ≤ U1 , C1  −L  U  1 1     1; for P (T C 1 (Q)) ≥ U1 ,

where U1 = U1 − 1 and 0 < 1 < U1 − L1 ,    0; for P (T C 2 (Q)) ≤ L2 ,      P (T C 2 (Q)) − L2   µT (P (T C 2 (Q))) = ; for L2 ≤ P (T C 2 (Q)) ≤ U2 , C2  −L  U  2 2     1; for P (T C 2 (Q)) ≥ U2 ,

(57)

(58)

where U2 = U2 − 2 and 0 < 2 < U2 − L2 . According to Step 4, having elicited the above membership functions non-linear programming problem is formulated as follows: Maximise

α

subject to

 P (T C 1 (Q)) −

α (U  − L1 ) ≥ L1 , w1 1 α  P (T C 2 (Q)) − (U  − L2 ) ≥ L2 , w2 2 0 ≤ α ≤ 1,

(59)

and the same restrictions of equation (41), where w1 and w2 are two normalised positive weights of two objectives. As Step 5, solving the above non-linear programming problem, one can get optimal objective functions and variables of the inventory problem (48).

8 Numerical example To illustrate the results of the proposed method, we solve the following numerical h = Rs. (2, 3, 4) per unit per year, 0 = Rs. (75, 80, 85) per order, C examples. C s = Rs. (13, 15, 17) per unit per year, C p = Rs. (4, 5, 6) per unit, D0 = 100 units, C µ = 0.12 year, T = 1 year, θ = 0.001, r = 0.002. Considering equivalent importance on two objectives (i.e., w1 = w2 = 0.5), the Pareto optimal solutions of equation (48) by GFNLP method are derived. If we let Model I denote that the inventory model starts without shortages and let Model II denote that the inventory model starts with shortages, then applying the procedure described in previous section, the optimal solutions for Model I and Model II are those given in Table 1. From Tables 1 and 2, it is clear that the ordering strategy for Model II is better than that for Model I because the former has a smaller minimum average total cost per unit of time, which confirms that the inventory model starting with shortages always produces better optimal solution.

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Hence, the optimal ordering strategy is as follows. The inventory starts with shortages, procurement is done at time t2 = 0.2509 year, the optimal order quantity, Q∗ , for each ordering cycle is 18.0056 units, the maximum inventory level, S ∗ , is 14.9886 units, and the minimum average total cost per unit of time, T C ∗ , is Rs. 100.7239. Table 1

Pareto optimal solutions for Model I and II

Model

t∗1 or t∗2

S∗

Q∗

T C∗

Model I Model II

0.8329 0.2509

14.6650 14.9886

18.0068 18.0056

104.6204 100.7239

Table 2

Sensitivity analysis for Model I and II Optimal value of

Parameter Model I D0

µ

θ

r

Model II D0

µ

θ

r





% Change in

Q

T C1∗

S

% change

S

+50 +25 −25 −50 +50 +25 −25 −50 +50 +25 −25 −50 +50 +25 −25 −50

21.9975 18.3318 10.9989 7.3325 21.0433 17.9344 11.2374 7.6509 14.6665 14.6659 14.6644 14.6635 14.7937 14.7559 14.5783 14.5375

27.0102 22.5085 13.5051 9.0034 25.5104 21.8836 13.8800 9.5033 18.0113 18.0087 18.0050 18.0031 18.0069 18.0069 18.0067 18.0066

113.1601 108.8903 100.3505 96.0807 106.1771 105.9292 102.1890 98.5731 104.6338 104.6271 104.6137 104.6070 104.6529 104.6271 104.6137 104.5868

+50 +25 −25 −50 +50 +25 −25 −50 +50 +25 −25 −50 +50 +25 −25 −50

22.4829 18.7363 11.2416 7.4943 21.0072 18.2212 11.5572 7.9065 14.9897 14.9890 14.9883 14.9873 14.9935 14.9889 18.9883 14.9833

27.0084 22.5070 13.5042 9.0028 25.5073 21.8816 13.8794 10.0433 18.0084 18.0070 18.0041 18.0034 18.0069 18.0060 18.0053 18.0044

108.9456 104.8346 96.6130 96.5312 108.7666 104.8745 96.2289 91.3190 100.7401 100.7271 100.7208 100.7079 100.7531 100.7318 100.7174 100.7118



Q∗

T C1∗

50 25.0036 −24.9987 −50.0002 43.4933 22.2942 −23.3729 −47.8290 0.0103 0.0058 −0.0042 −0.0101 0.8776 0.6198 −0.5912 −0.8691

50 25 −24.9999 −49.9999 41.6712 21.5297 −22.9179 −47.2239 0.0205 0.0103 −0.0102 −0.0204 0.0007 0.0005 −0.0005 −0.0010

8.1626 4.0813 −4.0813 −8.1626 1.4880 1.2510 −2.3240 −5.7802 0.0128 0.0064 −0.0064 −0.0128 0.0311 0.0064 −0.0064 −0.0321

50 25.0036 −24.9987 −50.0001 40.1545 21.5668 −22.8937 −47.2496 0.0071 0.0026 −0.0019 −0.0087 0.0326 0.0018 −0.0018 −0.0356

50 25 −24.9999 −49.9999 41.6634 21.5268 −22.9160 −44.2213 0.0156 0.0078 −.0082 −0.0121 0.0073 0.0023 −0.0019 −0.0069

8.1626 4.0813 −4.0813 −8.1626 7.9849 4.1208 −4.4627 −9.3373 0.0161 0.0032 −0.0031 −0.0159 −0.0290 0.0078 −0.0065 −0.0120

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9 Sensitivity analysis We now study the sensitivity of the optimal solution to changes in the values of the different parameters associated with the inventory system. Sensitivity analysis is performed by changing (increasing or decreasing) the parameters by 25% and 50%, and taking one parameter at a time, keeping the remaining parameters at their original values. The results are given in Table 2. A careful study of Table 2 reveals the following points. i

It is seen that the percentage change in the optimal cost is almost equal for both positive and negative changes of all the parameters.

ii

The optimal total cost increases (decreases) and decreases (increases) with the increase (decrease) and decrease (increase) in the value of the parameters D0 , µ, θ and r.

iii

It can be seen that S ∗ is insensitive to changes in the values of the parameters θ and r and highly sensitive to changes in the values of parameters D0 and µ.

iv

The optimal order quantity Q∗ is insensitive to changes in the values of the parameters θ and r. It is highly sensitive to changes in the values of parameters D0 and µ.

v

The optimal total cost is insensitive to changes in the values of the parameters θ and r. It is moderately sensitive to changes in the values of the parameters D0 and µ.

10 Conclusions We have presented an order-level inventory system for deteriorating items with a ramp type demand function of time under inflation. The problem is then solved by Fuzzy Non-Linear Programming (FNLP) method. Here inventory costs namely holding cost, shortage cost, purchase cost, set up cost, are taken as triangular fuzzy numbers. We have not only provided the solution procedure for the problem, but also presented two possible shortage models. Numerical example and sensitivity analysis have been implemented to study the theory. It is observed that the ordering strategy for Model II is better than that for Model I. Thus it may be concluded that the inventory model starts with shortages always produces better results.

Acknowledgements The authors are grateful to the editor and the anonymous referee for his helpful comments and suggestions on improving this paper.

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