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An Economic order quantity model for Items with Three-parameter Weibull distribution Deterioration, Ramp-type Demand and Shortages
Applied Mathematical Modelling 37 (2013) 9698–9706
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An Economic order quantity model for Items with Three-parameter Weibull distribution Deterioration, Ramp-type Demand and Shortages S.S. Sanni ⇑, W.I.E. Chukwu Department of Statistics, Faculty of Physical Sciences, University of Nigeria, Nsukka, Enugu, Nigeria
a r t i c l e
i n f o
Article history: Received 30 May 2012 Received in revised form 16 April 2013 Accepted 21 May 2013 Available online 5 June 2013 Keywords: Inventory model Ramp type demand Differential equation Optimal policy Instantaneous rate function
a b s t r a c t In this paper, we develop an economic order quantity inventory model for items with three-parameter Weibull distribution deterioration and ramp-type demand. Shortages are allowed in the inventory system and are completely backlogged. The demand rate is deterministic and varies with time up to a certain point and eventually stabilized and becomes constant. The instantaneous rate of deterioration is an increasing function of time. We provide simple analytical tractable procedures for deriving the model and give numerical examples to illustrate the solution procedure. Our adoption of ramp-type demand reflects a real market demand for newly launched product. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Inventory control policy is concerned basically with two decisions ; ‘‘How much to order (produce or purchase) to replenish the inventory of an item’’ and ‘‘When to order so as to minimize the total cost’’ [1]. Several inventory models have been developed to answer the above questions. The economic-order-quantity model, hereafter EOQ, was originally developed by Ford W. Harris in 1913, but R. H. Wilson a consultant who applied it extensively, is given credit for his in-depth analysis [2]. The basic EOQ model assumes a constant demand rate and an infinite planning horizon, there is no deterioration of inventory and replenishment is instantaneous i.e. the lead time is zero. These assumptions restrict the applicability of the classical EOQ model. In order to make the basic EOQ model more realistic, many researchers have extended Wilson’s EOQ model by considering time/price varying demand pattern and deterioration rate. It ispertinent to note that the depletion of any inventory is due mainly to demand and partly to deterioration of the item. In what follows we give definition of terms central to the paper. In particular, we define deterioration, ramp-type function and Weibull function. Definition 1.1. Deterioration is defined as decay, damage, change or spoilage that prevents items from being used for its original purpose. Some examples of items that deteriorate are fashion goods, foods, mobile phones, chemicals, automobiles, drugs, etc.
Definition 1.2. A random variable Y (e.g. the time to deterioration of an item) is said to have a Weibull distribution if its b density is given, for some parameters a > 0; b > 0; and c, by f ðyÞ ¼ abðy cÞb1 eaðtcÞ ; y > 0. The parameters a; b, and c ⇑ Corresponding author. Tel.: +234 8035813754. E-mail addresses: [email protected] (S.S. Sanni), [email protected] (W.I.E. Chukwu). 0307-904X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.05.017
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are the the scale, shape and location parameters of the distribution, respectively. For time to deterioration data, the scale parameter, a represents the characteristic rate of deterioration of items, the shape parameter, b is a measure of the spread of time-to-deterioration and c is the minimum time such that y P c. A negative c may indicate that deterioration has occurred prior to the beginning of the inventory, namely during production, in transit, or prior to actual storage while c ¼ 0 reduces the density to the case of two-parameter Weibull distribution. The scale and the location parameters have the same unit as T (i.e. cycle length in time) while the shape parameter is dimensionless. Weibull model is widely used today for failure and survival analysis. hðxÞ; x < l Definition 1.3. Let R be the set of real numbers. A function g : R ! R defined by gðxÞ ¼ is called a ramp hðlÞ; x P l 0; x < l and a–0; l > 0 is a ramp-type function. function. In particular, gðxÞ ¼ a½x ðx lÞHðx lÞ where Hðx lÞ ¼ 1; x P l A variety of ramp-type functions have evolved from the above definition. Inventory models for deteriorating items have been widely studied by researchers. Inventory problem for deteriorating items was first studied by Whitin [3], he considered fashion items decaying at the end of the planning horizon. Wagner and Whitin[4] developed a dynamic version of the classical EOQ model. Thereafter, Ghare and Schrader[5] developed a model for exponentially deteriorating inventory. They proposed, explicitly for the first time, the differential equation governing the variation in the inventory system ; dIðtÞ=dt þ hIðtÞ ¼ DðtÞ. Donaldson[6] provided a somehow complicated analytical solution procedure for the basic inventory policy for the case of positive linear trend in demand. Silver[7] formulated a heuristic for deteriorating inventory model with time-dependent linear demand. Deb and Chuadhuri[8] extended the deteriorating inventory model with linear demand by incorporating shortages in the inventory. Covert and Philip [9] presented an inventory model where the time to deterioration is described with two-parameter Weibull distribution. Philip[10] generalized the model in [9] by considering a three-parameter Weibull distribution deterioration, no shortages and a constant demand. Chakrabarty et al. [11] proposed an EOQ model with three-parameter Weibull distribution deterioration, shortages and linear demand rate and obtained infinite series representation for the initial inventory level and the average total cost equation. Ghosh and Chaudhuri [12] developed an inventory model for two-parameter Weibull deteriorating items, with shortages and quadratic demand rate and gave infinite series representation for the initial inventory level and the total average variable cost equation. Sanni [13] developed an inventory model with three-parameter Weibull deterioration, shortages and quadratic demand rate. He derived explicit equations for the initial inventory level and the average total cost by approximating exponential functions by the first two terms of the Tailor series. An order-level inventory model for deteriorating items with ramp type demand rate was discussed by Mandal and Pal [14]. They considered an EOQ model for items with constant rate of deterioration, ramp-type demand rate and no shortage allowed in the system and obtained an approximate solution for the EOQ. This work was extended by Wu and Ouyang [15] by considering two types of shortages in the model: model that starts with stock and model that starts with shortages and obtained optimal replenishment policy for the different cases. This model was further generalized by Samanta and Bhowmick [16] by taking two parameter Weibull distribution to represent the time to deterioration and allowed shortages in the inventory.They studied two cases; where the inventory starts with shortages and the case where the system starts without shortages and derived the EOQ for the respective systems. For some literature on deteriorating inventory models, see [12,17,18]. In this paper we consider the problem of finding optimal replenishment policy for an inventory system which holds items with three parameter Weibull Distribution deterioration, ramp type demand rate and shortages are allowed and completely backlogged. The research focus of this paper is to develop a mathematical model for the system, provide an optimal replenishment policy for the model and establish the necessary and sufficient conditions for the optimal policy. The Weibull distribution is suitable for items whose rate of deterioration increases with time and the location parameter c, in the threeparameter Weibull distribution, is used here to depict the item shelf-life; an important feature of most deteriorating items. The ramp type demand rate describes the demand of products, such as fashion goods, electronics, automobiles, etc, for which the demand increases as they are launched into the market and after some time the demand stabilizes and becomes constant. 2. Assumptions and notation The mathematical model in this work is developed on the basis of the following assumptions and notation. Notation C1 C2 C3 C4 DðtÞ T Io
: : : : : : :
inventory holding cost per unit per unit time. shortage cost per unit per unit time. ordering cost per order. unit cost. demand rate at any time t P 0. cycle length. size of the initial inventory.
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hðtÞ ¼ abðt cÞb1 : instantaneous rate function for a three-parameter Weibull distribution; where a is the scale parameter, 0 < a 1; b is the shape parameter, b > 0 and c is the location parameter, c > 0. Also, t is time to deterioration, t P c t1 : time during which there is no shortage. T : optimal value of T. Io : optimal value of Io . t1 : optimal value of t 1 . Assumptions (i) The inventory system under consideration deals with single item. This assumption ensures that a single item is isolated from other items and thus preventing item interdependencies. (ii) The time horizon is infinite and a typical planning schedule of cycle of length T is considered. (iii) The demand rate is a ramp type function of time, i.e. DðtÞ ¼ a½t ðt lÞHðt lÞ where a and l are constants such 1; t P l that l > 0, and Hðt lÞ is a Heaviside unit function of time defined as follows: Hðt lÞ ¼ . Also, a stands 0; t < l for the initial demand rate and l is a fixed point in time. The implication of the ramp-type demand rate is that demand varies linearly with time up to some time l and then stays constant. (iv) Shortages in the inventory are allowed and completely backlogged so that at inventory level zero all arrived demand is permitted. (v) Replenishment is instantaneous and lead time is zero.The assumption of zero lead time is made so that the period of shortage is not affected. (vi) Deteriorated unit is not repaired or replaced during a given cycle. (vii) The holding cost, ordering cost, shortage cost and unit cost remain constant over time. (viii) There are no quantity discounts. (ix) The distribution of the time to deterioration of the items follows three-parameter Weibull distribution, i.e b f ðtÞ ¼ abðt cÞb1 eaðtcÞ ; t > 0. The instantaneous rate function is hðtÞ ¼ abðt cÞb1 . The implication of the threeparameter Weibull instantaneous rate is that the impact of the already deteriorated items that are received into the inventory as well as those items that may start deteriorating in future are accounted for. The graph of three-paprameter Weibull distribution- time relationship is given in Fig. 1 below. The essence of the assumptions is to make the complexity of the inventory system malleable to mathematical modelling. The assumptions are selected to give accurate approximation of real life inventory system for newly introduced product. Remark 1. The assumption of zero time is crucial to the mathematics of the model. If a positive lead time, L is assumed three possible situations may arise and the situations may be presented as follows: Let lL be a prescribed interval between orders, then situation 1. the prescribed time lL starts with inventory level say M 1 and ends with inventory level say M 2 , i.e. 0 < lL < l. situation 2. the prescribe time lL starts with inventory level say M 1 and ends with a shortage say M 2 , i.e. l < lL < t 1 . situation 3. the prescribed time lL starts with a shortage says M 1 and ends with a shortage say M 2 , i.e. t1 < lL < T. These three situations affect the shortage period and thus complicate the mathematics of the model if a positive lead time L > 0 is assumed. Motivated by the on-going research on inventory models for deteriorating items, it is our purpose in this paper to provide optimal inventory policy for the EOQ model with three-parameter Weibull distribution deterioration, ramp-type demand and shortages. And also establish the necessary and sufficient conditions for this optimal policy. Our model is a variant of Samanta and Bhowmick [16] and our results extend some results of [16] and many other recently known results in literature.
Rate of deterioration
Increasing rate ( 2 > β > 1)
γ <0 γ =0 γ >0
Decreasing rate, γ < 0 ( β < 1)
Time Fig. 1. Rate of deterioration-time relationship for three-parameter Weibull distribution.
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3. Mathematical model and analysis At the start of the cycle, the inventory level reaches its maximum I0 units of item at time t ¼ 0. During the time interval ½0; t1 , the inventory depletes due mainly to demand and partly to deterioration. At time t ¼ l < t1 , the inventory level depletes to S units and at t 1 , the inventory level is zero and all the demand hereafter (i.e. T t1 Þ is completely backlogged. The total number of backordered items is replaced by the next replenishment. The demand varies with time up to a certain time and become constant. The deterioration rate is described by an increasing function of time hðtÞ ¼ abðt cÞb1 . A Graphical representation of the considered inventory system is given below (Fig. 2). The changes in the inventory at any time t are governed by the differential equations:
dIðtÞ þ IðtÞabðt cÞb1 ¼ at; dt
0 6 t 6 l;
ð3:1Þ
dIðtÞ þ IðtÞabðt cÞb1 ¼ al; dt
l 6 t 6 t1 ;
ð3:2Þ
dIðtÞ ¼ al; dt
t1 6 t 6 T;
ð3:3Þ
with boundary conditions Ið0Þ ¼ I0 ; IðlÞ ¼ S; Iðt1 Þ ¼ 0 and IðTÞ ¼ 0. Using the assumptions and the boundary conditions in Eq. 3.1, Eq. 3.2 and Eq. 3.3, we have
"
" # # t 2 atðt cÞbþ1 aððt cÞbþ2 ðcÞbþ2 Þ þ I0 expfaðcÞb g expfaðt cÞb g; þ ðb þ 1Þðb þ 2Þ 2 bþ1 " " ## bþ1 aððt1 cÞ ðt cÞbþ1 Þ expfaðt cÞb g; l 6 t 6 t 1 ; IðtÞ ¼ al ðt1 tÞ þ ðb þ 1Þ
IðtÞ ¼ a
IðtÞ ¼ alðt t 1 Þ;
0 6 t 6 l;
t 1 6 t 6 T:
ð3:4Þ ð3:5Þ ð3:6Þ
Applying the condition IðlÞ ¼ S in Eqs. 3.4 and 3.5, we get
"
"
S ¼ a "
l2 2
þ
alðl cÞbþ1 bþ1
"
aððl cÞbþ2 ðcÞbþ2 Þ
#
# b
þ I0 expfaðcÞ g expfaðl cÞb g;
ðb þ 1Þðb þ 2Þ
ð3:7Þ
##
S ¼ al ðt1 lÞ þ
aððt1 cÞbþ1 ðl cÞbþ1 Þ ðb þ 1Þ
expfaðl cÞb g:
ð3:8Þ
Eliminate S by equating Eq. 3.7 and 3.8, we obtain the initial inventory level
"
# l alðt1 cÞbþ1 aððl cÞbþ2 ðcÞbþ2 Þ I0 ¼ a l t 1 þ expfaðcÞb g ðb þ 1Þðb þ 2Þ 2 bþ1 " # l alðt1 cÞbþ1 aððl cÞbþ2 ðcÞbþ2 Þ l : aðcÞb l t 1 þ ¼ a l t1 ðb þ 1Þðb þ 2Þ 2 bþ1 2
ð3:9Þ
Inventory level
Depletion curve with detarioration
I0 Without detarioration
S I
t1 t=μ
0
( I − I0 ) T Fig. 2. Inventory system for deteriorating items.
Time
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The total inventory cost per unit time consists of the the following components: Deterioration cost (DC) in the cycle ½0; t1 ;
" !# Z l Z t1 C4 C4 h li ¼ DC ¼ at dt þ al dt I0 I0 al t1 T T 2 0 l " # bþ1 bþ2 bþ2 C 4 a alðt1 cÞ aððl cÞ ðcÞ Þ l ; aðcÞb l t 1 ¼ T ðb þ 1Þðb þ 2Þ bþ1 2
ð3:10Þ
Shortage cost (SC) in the interval ½t1 ; T;
SC ¼
C2 T
Z
T
IðtÞ dt ¼
t1
C 2 al ðT t1 Þ2 : : T 2
ð3:11Þ
Ordering cost(OC);
OC ¼
C3 T
ð3:12Þ
and the inventory holding cost (HC) in ½0; t1 ;
C1 HC ¼ T
Z l
IðtÞ dt þ
0
Z
!
t1
IðtÞ dt :
ð3:13Þ
l
To make the derivation of the cost function simple, we approximate the inventory depletion curve with a straight line. Similar treatment of the inventory depletion curve with linear approximation can be found in [9,11,12]. Thus, the inventory holding cost per unit time is approximately;
HC ¼
1 C1 : I0 t 1 : 2 T
ð3:14Þ
See Appendix A for ‘exact’ inventory holding cost from Eq. 3.13. Hence, the total inventory cost per unit time, uðT; t1 Þ is given by:
uðT; t1 Þ ¼
C4 l 1 C 1 C 2 alðT t 1 Þ2 C 3 þ : þ : I0 t 1 þ I0 al t 1 2 T T 2 2T T
ð3:15Þ
We assume t 1 ¼ KT; 0 < K < 1. This assumption seems reasonable since the length of the shortage interval is part of the cycle length. In addition, this restriction on values of t1 enhances the convexity of the inventory cost function in Eq. 3.15. Substituting t 1 ¼ KT in (3.15), we get
C4 C1K C 4 al l C 2 alðT KTÞ2 C 3 I0 þ þ ; þ KT 2 T T 2 2T T " # C 4 C 1 K alðKT cÞbþ1 aððl cÞbþ2 ðcÞbþ2 Þ l b l KT ðaðcÞ 1Þ ; þ ¼a 2 ðb þ 1Þðb þ 2Þ T bþ1 2
uðT; KÞ ¼
ð3:16Þ
C 4 al l C 2 alð1 KÞ2 T C 3 þ : KT þ 2 T 2 T
We now proceed to determine T and K optimally by treating them as decision variables. The Inventory cost per unit time,
uðT; KÞ being a function of two variables T and K has to be partially differentiated with respect to T and K separately and then put equal to zero. This gives the necessary condition for minimizing the total inventory cost per time uðT; KÞ. That is; " # @ u C 4 a alðKTðb þ 1ÞðKT cÞb ðKT cÞbþ1 Þ aððl cÞbþ2 ðcÞbþ2 Þ l2 b þ ðaðcÞ 1Þ ¼ 2 ðb þ 1Þ ðb þ 1Þðb þ 2Þ @T 2 T þ
C 1 alK 2 C 4 al2 C 2 alð1 KÞ2 C 3 ½aðKT cÞb ðaðcÞb 1ÞÞ þ þ 2 ¼ 0; 2 2 2T 2 T
ð3:17Þ
@u ¼ C 4 al½aðKT cÞb ðaðcÞb 1Þ @K " # C 1 a alðKTðb þ 1ÞðKT cÞb þ ðKT cÞbþ1 Þ aððl cÞbþ2 ðcÞbþ2 Þ l b þ l 2KT ðaðcÞ 1Þ þ 2 ðb þ 1Þ ðb þ 1Þðb þ 2Þ 2 C 4 alT C 2 alð1 KÞT ¼ 0:
ð3:18Þ
The solutions of Eq. 3.17 and Eq. 3.18, solving simultaneously, give the optimal values T and K which minimize the total inventory cost per unit uðT; KÞ provided they satisfy the sufficient condition below.
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" l1 ¼ C 4 a
al
ðb þ 1ÞT
4
h b1 b b T 2 ðK ðb þ 1Þ½K T bðK T cÞ þ ðK T cÞ K ðb þ 1ÞðK T cÞ Þ
b
2TðK T ðb þ 1ÞðK T cÞ ðK T cÞ þ
bþ1
# i aððl cÞbþ2 ðcÞbþ2 Þ l2 ðaðcÞb 1Þ Þ þ ðb þ 1Þðb þ 2ÞT 3 2T 3
C 1 aK 3 l C al2 C abðK T cÞb1 4 3 þ 33 ; 2 2T T
l2 ¼
C 4 aal T 2 ðb þ 1Þ
½T ðb þ 1ÞðK T bðK T cÞ
b1
b
b
þ ðK T cÞ Þ ðb þ 1ÞðK T cÞ þ
C 1 al b1 ½abK 2 T ðK T cÞ 2
b
þ 2K ðK T cÞ 2K ðaðcÞb 1Þ þ C 2 alð1 K Þ; l3 ¼ C 4 aK labðK T cb1 Þ " # b1 b b C 1 a al½K ðb þ 1ÞðK T bðK T cÞ þ ðK T cÞ Þ þ K ðb þ 1ÞðK T cÞ b þ 2K lðaðcÞ 1Þ C 4 al 2 bþ1 C 2 alð1 K Þ; b1
l4 ¼ C 4 aT labðK T cÞ " # b1 b b C 1 a al½T ðb þ 1ÞðT K bðK T cÞ þ ðK T cÞ Þ þ T ðb þ 1ÞðK T cÞ b þ 2T lðaðcÞ 1Þ þ C 2 aT l; 2 bþ1 0
where li s; i ¼ 1; 2 . . . 4, are: @ 2 u=@T 2 ; @ 2 u=@T@K; @ 2 u=@K@T and @ 2 u=@K 2 respectively. The sufficient condition is:
l1 > 0;
l4 > 0;
l1 :l4 l2 :l3 > 0:
ð3:19Þ
If l2 ¼ l3 , the condition reduces to:
l1 > 0;
l4 > 0;
2
l1 :l4 ðl2 Þ > 0:
Remark 2. We can put (3.19) in equivalent form as; the total cost per unit time uðT; KÞ is minimized if its Hessian matrix evaluated at ðT ; K Þ is positive definite. The total back-order quantity for the cycle is alðT t1 Þ . Thus, the optimal order quantity , I is:
I ¼ I0 þ alðT t 1 Þ:
ð3:20Þ
Corollary 3.1. A change in the inventory ordering cost C 3 by the amount qT, where q is the ordering cost per unit of item produced, will leave the optimal ordering quantity unchanged. Proof. Replacing C 3 in 3.16 with C 3 þ qT, we obtain;
# " C 4 C 1 K alðKT cÞbþ1 aððl cÞbþ2 ðcÞbþ2 Þ l b l KT ðaðcÞ 1Þ u ðT; KÞ ¼ a þ 2 ðb þ 1Þðb þ 2Þ T bþ1 2 D
C 4 al l C 2 alð1 KÞ2 T C 3 þ þ q: KT þ 2 T 2 T
ð3:21Þ
Differentiating 3.21 partially w.r.t T and K, we observed that @ uD =@T ¼ @ u=@T ¼ 0 and @ uD =@K ¼ @ u=@K ¼ 0. Hence, the result follows and the proof is complete. h
4. Numerical example Example 4.1. An example is chosen for numerical illustration which represents an inventory system with the following data: C 1 ¼ 2.40 /unit/year, C 2 ¼ 5=unit=year;C 3 ¼ 100=order;C 4 ¼ 8=unit;a ¼ 9000 units/year, l ¼ 2=3 year, a ¼ 0:002; b ¼ 20 and c ¼ 0:6. The optimal solutions are found to be: K ¼ 0:9907; T ¼ 1:893 years;t1 ¼ 1:8754 years;I0 ¼ 9346:8851 units;I ¼ 9452:5145 units and u ¼ 11566:60=year.
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Table 1 Sensitivity analysis of the inventory model parameters. Parameter
C1
C2
C3
C4
a
l
a
b
c
K
% Change
+50 +25 25 50 +50 +25 25 50 +50 +25 25 50 +50 +25 25 50 +50 +25 25 50 +50 +25 25 50 +50 +25 25 50 +50 +25 25 50 +50 +25 25 50 +20 25 50
% Change in T
I0
u
1.61 0.81 +0.82 +1.65 0.00 0.00 0.00 +0.01 +1.05 +0.04 0.04 0.08 +1.05 +0.62 1.02 3.01 0.05 0.03 +0.05 +0.78 6.78 3.43 +3.86 +8.91 3.60 2.00 +2.64 +6.49 18.03 11.12 +20.15 +61.75 +6.55 +3.27 3.26 6.51 16.66 +33.33 +99.97
2.52 1.29 +1.34 +2.73 0.00 0.00 0.00 0.00 +0.10 +0.06 0.06 0.13 +1.71 +1.01 1.62 4.63 +49.87 +24.94 24.94 49.87 +27.58 +15.29 18.08 38.94 3.71 2.06 +2.71 +6.66 20.43 12.52 +22.14 +65.82 +6.25 +3.11 3.08 6.12 +0.003 0.008 +0.02
+27.08 +13.65 13.88 28.00 +.29 +0.14 0.15 0.29 +2.20 +1.44 1.44 2.89 +19.02 +9.56 9.78 20.07 +47.11 +23.57 23.56 45.97 +12.51 +7.59 10.21 23.20 0.91 0.51 +0.71 +1.82 16.29 9.29 +13.87 +35.51 0.77 0.52 +0.84 +2.05 +19.50 22.33 39.79
Example 4.2. Let: C 1 ¼ 0:15 /unit/year, C 2 ¼ 0.02 /unit/year, C 3 ¼ 280=order, C 4 ¼ 1.5/unit, a ¼ 15000 units/year, year, a ¼ 0:001, b ¼ 8, c ¼ 0:4 and K ¼ 0:8. The optimal solutions to the inventory problem are: T ¼ 3:2416 years, t 1 ¼ 2:5933 years, I0 ¼ 18553:5546 units, I ¼ 23416:0230 units and u ¼ 1662.25/year.
l ¼ 1=2
We now proceed to test the responsiveness of the proposed model to changes in the model parameters using Example 4.2. The sensitivity analysis is performed by changing the value of each parameter by 50%, 25%, 25%, 50%, taking one parameter at a time and keeping the remaining parameters unchanged. We assume that insensitive, moderately sensitive, and highly sensitive imply percentage changes are 3 to 3, 20 to +20 and more respectively. This guide for categorizing sensitivity is used by [16]. It can be seen from Table 1 that the solutions T ; I0 and u are highly insensitive to change in C 2 and C 3 while the solutions are highly sensitive to changes in l and b. Furthermore, the ‘initial’ optimal quantity I0 and the optimal inventory cost u change by roughly the same percentage change in a. 5. Conclusion We have presented an inventory model for three-parameter Weibull distribution deteriorating items with ramp-type demand rate. The proposed model is suitable for newly launched product with erratic demand pattern up to a point in time. The Weibull distribution is often used for modeling duration data because it has exponential and Rayleigh distributions as sub-models. In practice, the rate of deterioration of most items increases with time. Cooray [19] suggested that when modeling monotone hazard rates, the weibull distribution may be an initial choice because of its negatively and positively skewed density shape. The location parameter, c, in the model depicts the shelf-life of the item under consideration; an essential feature often ignored by most inventory modelers. Several intrinsic features of the model have demonstrated via
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sensitivity analysis. In particular, the optimal quantity and the total inventory cost change with approximately the same percentage changes in a for the inventory problem in Example 4.2. Also, we give the necessary and sufficient conditions for the optimal solutions to the model. The proposed model can be extended in a number of ways; it would be interesting to relax the assumption of no discount quantity or to consider the parameters as fuzzy/stochastic fuzzy. Addendum Material below came to the attention of the authors at the time of revising this paper. Note 1. Jalan et al [20] presented an EOQ model for items with two-parameter Weibull distribution deterioration and ramp type demand. They allowed shortages in the inventory and obtained the economic order quantity via numerical technique. Note 2. Giri et al [21] reconsidered the model in [20] by taking a ramp type demand pattern for fashionable products which initially increases exponentially with time up to a point after which it becomes steady. Note 3. Jain and Kumar[22] developed an EOQ model for inventory system that starts with shortage. They considered the case where the time to deterioration of items follows three-parameter Weibull distribution, the demand rate is ramp type function of time that increases exponentially up to a point then stays constant and they provided an analytical method for reaching the optimal replenishment policy. The present authors feel that inventory model that starts with shortage is appropriate for faulty inventory system which experience unnecessary delay at the initial stage due to delay in supply, shortage of raw materials, shortage of labour, etc. Such system rarely occurred in practice because of technological advancement and business competition. We are also of the opinion that exponential time-varying demand rate is unrealistic because the demand of any product seldom experiences a rate which is as high as exponential rate. Finally, Two-parameter Weibull distribution instantaneous rate function is appropriate for item with decreasing rate of deterioration only if the initial rate of deterioration is very high or it can be used for system with increasing rate of deterioration only if the initial rate is negligible[see [10,11]]. Acknowledgements We thank the referees for their valuable comments and suggestions which led to a substantial improvement of this paper. The first author would like to thank Yekini Shehu for helpful discussions. Appendix A. Derivation of the total average holding cost We derive the Total average holding cost per cycle directly from the inventory depletion curve as follows:
"Z # Z t1 l C1 HC ¼ IðtÞ dt þ IðtÞ dt T 0 l " Z " # l C1 1 2 atðt cÞbþ1 aððt cÞbþ2 ðcÞbþ2 Þ b b þ I0 expfaðcÞ g dt a t ð1 aðt cÞ Þ þ ¼ 2 ðb þ 1Þðb þ 2Þ T bþ1 0 " # # Z t1 aððt1 cÞbþ1 ðt cÞbþ1 Þ þ alðt 1 tÞð1 að1 cÞb Þ þ dt bþ1 l " " !# C1 1 l3 l2 ðl cÞbþ1 2 bþ1 bþ3 bþ3 ½lðb þ 3Þðl cÞ a a ððl cÞ ðcÞ Þ ¼ 2 3 ðb þ 1Þðb þ 2Þðb þ 3Þ T bþ1 þ
a ðb þ 1Þðb þ 2Þðb þ 3Þ
½lðb þ 3Þðl cÞbþ2 ððl cÞbþ3 ðcÞbþ3 Þ
½ðl cÞbþ3 ðcÞbþ3 ðb þ 3ÞlðcÞbþ2 þ lI0 expfaðcÞb g ðb þ 1Þðb þ 2Þðb þ 3Þ "" # t1 l aðl cÞbþ1 ðt1 lÞ ððt1 cÞbþ2 ðl cÞbþ2 Þ þal lðt 1 Þ þ bþ1 ðb þ 1Þðb þ 2Þ 2 2 h i a : ðb þ 2Þðt 1 ðt1 cÞbþ1 lðl cÞbþ1 Þ ððt 1 cÞbþ2 ðl cÞbþ2 Þ þ ðb þ 1Þðb þ 2Þ
a
Appendix B. Derivation of the inventory level Using the integrating factor(i.e. expfaðt cÞb g) of the first order linear differential equation of (3.1) to multiply both sides of (3.1), we get
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d ½IðtÞ expfaðt cÞb g ¼ at expfaðt cÞb g: dt Solving the above equation using the condition, Ið0Þ ¼ I0 , we obtain
Z t IðtÞeaðtcÞb I0 eaðcÞb ¼ a t½1 þ aðt cÞb dt; 0 " # t2 atðt cÞbþ1 aððt cÞbþ2 ðcÞbþ2 Þ aðtcÞb þ I0 expfaðcÞb g; ¼ a þ IðtÞe ðb þ 1Þðb þ 2Þ 2 bþ1 "
" # # t 2 atðt cÞbþ1 aððt cÞbþ2 ðcÞbþ2 Þ b ) IðtÞ ¼ a þ þ I0 expfaðcÞ g expfaðt cÞb g; ðb þ 1Þðb þ 2Þ 2 bþ1
0 6 t 6 l:
Note that we have used eaðtcÞb 1 þ aðt cÞb on the integrand since 0 < a 1. Similarly, the solutions to the differential Eqs. (3.2) and (3.3), after using the boundary conditions, give (3.5) and (3.6). Hence, the inventory level at any time t 2 ½0; T is:
8h h i i 2 cÞbþ1 cÞbþ2 ðcÞbþ2 Þ > a t2 þ atðt aððtðbþ1Þðbþ2Þ þ I0 expfaðcÞb g expfaðt cÞb g; 0 6 t 6 l; > bþ1 > al ðt 1 tÞ þ ðbþ1Þ > > : t1 6 t 6 T; alðt t 1 Þ; References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
P.K. Gupta, D.S. Hira, Operations Research, S. Chand & Company LTD, New Delhi, 2002. A.C. Hax, D. Candea, Production and Operations Management, Prentice-Hall, Englewood Cliffs, New Jersey, 1984. pp. 135. T.M. Whitin, Theory of Inventory Management, Princeton University Press, New Jersey, 1957. pp. 62–72. H.M. Wagner, T.M. Whitin, Dynamic version of the economic lot size model, Manage. Sci. 5 (1958) 89–96. P.M. Ghare, G.F. Schrader, A model for exponentially decaying inventory, J. Ind. Eng. 14 (1963) 238–243. W.A. Donaldson, Inventory replenishmemt policy for a linear trend in demand: an analytical solution, Oper. Res. Q. 28 (1977) 663–670. E.A. Silver, A simple inventory replenishment decision rule for a linear trend in demand, J. Oper. Res. Q. 30 (1979) 71–75. M. Deb, K.A. Chuadhuri, A note on the heuristic for replenishment of trended inventories considering shortages, J. Oper. Res. Soc. 38 (1987) 459–463. R.P. Covert, G.C. Philip, An EOQ model for items with Weibull distribution deterioration, AIIE Trans. 5 (1973) 323–326. G.C. Philip, A generlized EOQ model for items with Weibull distribution deterioration, AIIE Trans. 6 (1974) 159–162. B.C. Chakrabarty, B.C. Giri, K.S. Chuadhuri, An EOQ model for items with Weibull distribution deterioration, shortages and trended demand: an extension of Philip’s model, Comput. Oper. Res. 25 (1998) 649–657. S.K. Ghosh, K.S. Chaudhuri, An order-level Inventory model for a deteriorating item with Weibull distribution deterioration, time-quadratic demand and shortages, Adv. Model. Optim. 6 (1) (2004) 21–35. S.S. Sanni, An economic order quantity inventory model with time dependent weibull deterioration and trended demand, M.Sc. thesis, University of Nigeria, 2012. B. Mandal, A.K. Pal, Ordel level inventory system with ramp type demand rate for deteriorating items, J. Interdisciplinary Math. 1 (1998) 49–66. K.S. Wu, L.Y. Ouyang, A replenishment policy for deteriorating items with ramp type demand rate, Proc. Natl. Sci. Counc. ROC (A) 24 (2000) 279–286. G.P. Samanta, J. Bhowmick, A deterministic inventory system with Weibull distribution deterioration and ramp-type demand rate, J. App. Stat. Anal. 3 (2) (2010) 92–114 (Electronic). F. Raafat, Survey of literature on continuous deteriorating inventory models, J. Opl. Res. Soc. 42 (1991) 27–37. R. Li, H. Lan, J.R. Mawhinney, A rewiew on deteriorating inventory study, J. Service Sci. Manage. 3 (2010) 117–129. K. Cooray, Generalization of the Weibull distribution: the odd Weibull family, Stat. Model. 6 (2006) 265–277. A.K. Jalan, B.C. Giri, K.S. Chaudhuri, An EOQ model for items with Weibull distribution deterioration, shortages and ramp type demand, Recent Development in Operations Research, Narosa Publishing House, India, 2001. pp. 207–223. B.C. Giri, A.K. Jalan, K.S. Chaudhuri, Economic order quantity with Weibull distribution deterioration, shortage and ramp type demand, Int. J. Syst. Sci. 4 (2003) 237–243. S. Jain, M. Kumar, An EOQ model for items with ramp type demand, three parameter Weibull distribution deterioration and starting with shortage, Yugoslav J. Oper. Res. 20 (2010) 249–259.
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
An Economic order quantity model for Items with Three-parameter Weibull distribution Deterioration, Ramp-type Demand and Shortages S.S. Sanni ⇑, W.I.E. Chukwu Department of Statistics, Faculty of Physical Sciences, University of Nigeria, Nsukka, Enugu, Nigeria
a r t i c l e
i n f o
Article history: Received 30 May 2012 Received in revised form 16 April 2013 Accepted 21 May 2013 Available online 5 June 2013 Keywords: Inventory model Ramp type demand Differential equation Optimal policy Instantaneous rate function
a b s t r a c t In this paper, we develop an economic order quantity inventory model for items with three-parameter Weibull distribution deterioration and ramp-type demand. Shortages are allowed in the inventory system and are completely backlogged. The demand rate is deterministic and varies with time up to a certain point and eventually stabilized and becomes constant. The instantaneous rate of deterioration is an increasing function of time. We provide simple analytical tractable procedures for deriving the model and give numerical examples to illustrate the solution procedure. Our adoption of ramp-type demand reflects a real market demand for newly launched product. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Inventory control policy is concerned basically with two decisions ; ‘‘How much to order (produce or purchase) to replenish the inventory of an item’’ and ‘‘When to order so as to minimize the total cost’’ [1]. Several inventory models have been developed to answer the above questions. The economic-order-quantity model, hereafter EOQ, was originally developed by Ford W. Harris in 1913, but R. H. Wilson a consultant who applied it extensively, is given credit for his in-depth analysis [2]. The basic EOQ model assumes a constant demand rate and an infinite planning horizon, there is no deterioration of inventory and replenishment is instantaneous i.e. the lead time is zero. These assumptions restrict the applicability of the classical EOQ model. In order to make the basic EOQ model more realistic, many researchers have extended Wilson’s EOQ model by considering time/price varying demand pattern and deterioration rate. It ispertinent to note that the depletion of any inventory is due mainly to demand and partly to deterioration of the item. In what follows we give definition of terms central to the paper. In particular, we define deterioration, ramp-type function and Weibull function. Definition 1.1. Deterioration is defined as decay, damage, change or spoilage that prevents items from being used for its original purpose. Some examples of items that deteriorate are fashion goods, foods, mobile phones, chemicals, automobiles, drugs, etc.
Definition 1.2. A random variable Y (e.g. the time to deterioration of an item) is said to have a Weibull distribution if its b density is given, for some parameters a > 0; b > 0; and c, by f ðyÞ ¼ abðy cÞb1 eaðtcÞ ; y > 0. The parameters a; b, and c ⇑ Corresponding author. Tel.: +234 8035813754. E-mail addresses: [email protected] (S.S. Sanni), [email protected] (W.I.E. Chukwu). 0307-904X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.05.017
S.S. Sanni, W.I.E. Chukwu / Applied Mathematical Modelling 37 (2013) 9698–9706
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are the the scale, shape and location parameters of the distribution, respectively. For time to deterioration data, the scale parameter, a represents the characteristic rate of deterioration of items, the shape parameter, b is a measure of the spread of time-to-deterioration and c is the minimum time such that y P c. A negative c may indicate that deterioration has occurred prior to the beginning of the inventory, namely during production, in transit, or prior to actual storage while c ¼ 0 reduces the density to the case of two-parameter Weibull distribution. The scale and the location parameters have the same unit as T (i.e. cycle length in time) while the shape parameter is dimensionless. Weibull model is widely used today for failure and survival analysis. hðxÞ; x < l Definition 1.3. Let R be the set of real numbers. A function g : R ! R defined by gðxÞ ¼ is called a ramp hðlÞ; x P l 0; x < l and a–0; l > 0 is a ramp-type function. function. In particular, gðxÞ ¼ a½x ðx lÞHðx lÞ where Hðx lÞ ¼ 1; x P l A variety of ramp-type functions have evolved from the above definition. Inventory models for deteriorating items have been widely studied by researchers. Inventory problem for deteriorating items was first studied by Whitin [3], he considered fashion items decaying at the end of the planning horizon. Wagner and Whitin[4] developed a dynamic version of the classical EOQ model. Thereafter, Ghare and Schrader[5] developed a model for exponentially deteriorating inventory. They proposed, explicitly for the first time, the differential equation governing the variation in the inventory system ; dIðtÞ=dt þ hIðtÞ ¼ DðtÞ. Donaldson[6] provided a somehow complicated analytical solution procedure for the basic inventory policy for the case of positive linear trend in demand. Silver[7] formulated a heuristic for deteriorating inventory model with time-dependent linear demand. Deb and Chuadhuri[8] extended the deteriorating inventory model with linear demand by incorporating shortages in the inventory. Covert and Philip [9] presented an inventory model where the time to deterioration is described with two-parameter Weibull distribution. Philip[10] generalized the model in [9] by considering a three-parameter Weibull distribution deterioration, no shortages and a constant demand. Chakrabarty et al. [11] proposed an EOQ model with three-parameter Weibull distribution deterioration, shortages and linear demand rate and obtained infinite series representation for the initial inventory level and the average total cost equation. Ghosh and Chaudhuri [12] developed an inventory model for two-parameter Weibull deteriorating items, with shortages and quadratic demand rate and gave infinite series representation for the initial inventory level and the total average variable cost equation. Sanni [13] developed an inventory model with three-parameter Weibull deterioration, shortages and quadratic demand rate. He derived explicit equations for the initial inventory level and the average total cost by approximating exponential functions by the first two terms of the Tailor series. An order-level inventory model for deteriorating items with ramp type demand rate was discussed by Mandal and Pal [14]. They considered an EOQ model for items with constant rate of deterioration, ramp-type demand rate and no shortage allowed in the system and obtained an approximate solution for the EOQ. This work was extended by Wu and Ouyang [15] by considering two types of shortages in the model: model that starts with stock and model that starts with shortages and obtained optimal replenishment policy for the different cases. This model was further generalized by Samanta and Bhowmick [16] by taking two parameter Weibull distribution to represent the time to deterioration and allowed shortages in the inventory.They studied two cases; where the inventory starts with shortages and the case where the system starts without shortages and derived the EOQ for the respective systems. For some literature on deteriorating inventory models, see [12,17,18]. In this paper we consider the problem of finding optimal replenishment policy for an inventory system which holds items with three parameter Weibull Distribution deterioration, ramp type demand rate and shortages are allowed and completely backlogged. The research focus of this paper is to develop a mathematical model for the system, provide an optimal replenishment policy for the model and establish the necessary and sufficient conditions for the optimal policy. The Weibull distribution is suitable for items whose rate of deterioration increases with time and the location parameter c, in the threeparameter Weibull distribution, is used here to depict the item shelf-life; an important feature of most deteriorating items. The ramp type demand rate describes the demand of products, such as fashion goods, electronics, automobiles, etc, for which the demand increases as they are launched into the market and after some time the demand stabilizes and becomes constant. 2. Assumptions and notation The mathematical model in this work is developed on the basis of the following assumptions and notation. Notation C1 C2 C3 C4 DðtÞ T Io
: : : : : : :
inventory holding cost per unit per unit time. shortage cost per unit per unit time. ordering cost per order. unit cost. demand rate at any time t P 0. cycle length. size of the initial inventory.
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hðtÞ ¼ abðt cÞb1 : instantaneous rate function for a three-parameter Weibull distribution; where a is the scale parameter, 0 < a 1; b is the shape parameter, b > 0 and c is the location parameter, c > 0. Also, t is time to deterioration, t P c t1 : time during which there is no shortage. T : optimal value of T. Io : optimal value of Io . t1 : optimal value of t 1 . Assumptions (i) The inventory system under consideration deals with single item. This assumption ensures that a single item is isolated from other items and thus preventing item interdependencies. (ii) The time horizon is infinite and a typical planning schedule of cycle of length T is considered. (iii) The demand rate is a ramp type function of time, i.e. DðtÞ ¼ a½t ðt lÞHðt lÞ where a and l are constants such 1; t P l that l > 0, and Hðt lÞ is a Heaviside unit function of time defined as follows: Hðt lÞ ¼ . Also, a stands 0; t < l for the initial demand rate and l is a fixed point in time. The implication of the ramp-type demand rate is that demand varies linearly with time up to some time l and then stays constant. (iv) Shortages in the inventory are allowed and completely backlogged so that at inventory level zero all arrived demand is permitted. (v) Replenishment is instantaneous and lead time is zero.The assumption of zero lead time is made so that the period of shortage is not affected. (vi) Deteriorated unit is not repaired or replaced during a given cycle. (vii) The holding cost, ordering cost, shortage cost and unit cost remain constant over time. (viii) There are no quantity discounts. (ix) The distribution of the time to deterioration of the items follows three-parameter Weibull distribution, i.e b f ðtÞ ¼ abðt cÞb1 eaðtcÞ ; t > 0. The instantaneous rate function is hðtÞ ¼ abðt cÞb1 . The implication of the threeparameter Weibull instantaneous rate is that the impact of the already deteriorated items that are received into the inventory as well as those items that may start deteriorating in future are accounted for. The graph of three-paprameter Weibull distribution- time relationship is given in Fig. 1 below. The essence of the assumptions is to make the complexity of the inventory system malleable to mathematical modelling. The assumptions are selected to give accurate approximation of real life inventory system for newly introduced product. Remark 1. The assumption of zero time is crucial to the mathematics of the model. If a positive lead time, L is assumed three possible situations may arise and the situations may be presented as follows: Let lL be a prescribed interval between orders, then situation 1. the prescribed time lL starts with inventory level say M 1 and ends with inventory level say M 2 , i.e. 0 < lL < l. situation 2. the prescribe time lL starts with inventory level say M 1 and ends with a shortage say M 2 , i.e. l < lL < t 1 . situation 3. the prescribed time lL starts with a shortage says M 1 and ends with a shortage say M 2 , i.e. t1 < lL < T. These three situations affect the shortage period and thus complicate the mathematics of the model if a positive lead time L > 0 is assumed. Motivated by the on-going research on inventory models for deteriorating items, it is our purpose in this paper to provide optimal inventory policy for the EOQ model with three-parameter Weibull distribution deterioration, ramp-type demand and shortages. And also establish the necessary and sufficient conditions for this optimal policy. Our model is a variant of Samanta and Bhowmick [16] and our results extend some results of [16] and many other recently known results in literature.
Rate of deterioration
Increasing rate ( 2 > β > 1)
γ <0 γ =0 γ >0
Decreasing rate, γ < 0 ( β < 1)
Time Fig. 1. Rate of deterioration-time relationship for three-parameter Weibull distribution.
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3. Mathematical model and analysis At the start of the cycle, the inventory level reaches its maximum I0 units of item at time t ¼ 0. During the time interval ½0; t1 , the inventory depletes due mainly to demand and partly to deterioration. At time t ¼ l < t1 , the inventory level depletes to S units and at t 1 , the inventory level is zero and all the demand hereafter (i.e. T t1 Þ is completely backlogged. The total number of backordered items is replaced by the next replenishment. The demand varies with time up to a certain time and become constant. The deterioration rate is described by an increasing function of time hðtÞ ¼ abðt cÞb1 . A Graphical representation of the considered inventory system is given below (Fig. 2). The changes in the inventory at any time t are governed by the differential equations:
dIðtÞ þ IðtÞabðt cÞb1 ¼ at; dt
0 6 t 6 l;
ð3:1Þ
dIðtÞ þ IðtÞabðt cÞb1 ¼ al; dt
l 6 t 6 t1 ;
ð3:2Þ
dIðtÞ ¼ al; dt
t1 6 t 6 T;
ð3:3Þ
with boundary conditions Ið0Þ ¼ I0 ; IðlÞ ¼ S; Iðt1 Þ ¼ 0 and IðTÞ ¼ 0. Using the assumptions and the boundary conditions in Eq. 3.1, Eq. 3.2 and Eq. 3.3, we have
"
" # # t 2 atðt cÞbþ1 aððt cÞbþ2 ðcÞbþ2 Þ þ I0 expfaðcÞb g expfaðt cÞb g; þ ðb þ 1Þðb þ 2Þ 2 bþ1 " " ## bþ1 aððt1 cÞ ðt cÞbþ1 Þ expfaðt cÞb g; l 6 t 6 t 1 ; IðtÞ ¼ al ðt1 tÞ þ ðb þ 1Þ
IðtÞ ¼ a
IðtÞ ¼ alðt t 1 Þ;
0 6 t 6 l;
t 1 6 t 6 T:
ð3:4Þ ð3:5Þ ð3:6Þ
Applying the condition IðlÞ ¼ S in Eqs. 3.4 and 3.5, we get
"
"
S ¼ a "
l2 2
þ
alðl cÞbþ1 bþ1
"
aððl cÞbþ2 ðcÞbþ2 Þ
#
# b
þ I0 expfaðcÞ g expfaðl cÞb g;
ðb þ 1Þðb þ 2Þ
ð3:7Þ
##
S ¼ al ðt1 lÞ þ
aððt1 cÞbþ1 ðl cÞbþ1 Þ ðb þ 1Þ
expfaðl cÞb g:
ð3:8Þ
Eliminate S by equating Eq. 3.7 and 3.8, we obtain the initial inventory level
"
# l alðt1 cÞbþ1 aððl cÞbþ2 ðcÞbþ2 Þ I0 ¼ a l t 1 þ expfaðcÞb g ðb þ 1Þðb þ 2Þ 2 bþ1 " # l alðt1 cÞbþ1 aððl cÞbþ2 ðcÞbþ2 Þ l : aðcÞb l t 1 þ ¼ a l t1 ðb þ 1Þðb þ 2Þ 2 bþ1 2
ð3:9Þ
Inventory level
Depletion curve with detarioration
I0 Without detarioration
S I
t1 t=μ
0
( I − I0 ) T Fig. 2. Inventory system for deteriorating items.
Time
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The total inventory cost per unit time consists of the the following components: Deterioration cost (DC) in the cycle ½0; t1 ;
" !# Z l Z t1 C4 C4 h li ¼ DC ¼ at dt þ al dt I0 I0 al t1 T T 2 0 l " # bþ1 bþ2 bþ2 C 4 a alðt1 cÞ aððl cÞ ðcÞ Þ l ; aðcÞb l t 1 ¼ T ðb þ 1Þðb þ 2Þ bþ1 2
ð3:10Þ
Shortage cost (SC) in the interval ½t1 ; T;
SC ¼
C2 T
Z
T
IðtÞ dt ¼
t1
C 2 al ðT t1 Þ2 : : T 2
ð3:11Þ
Ordering cost(OC);
OC ¼
C3 T
ð3:12Þ
and the inventory holding cost (HC) in ½0; t1 ;
C1 HC ¼ T
Z l
IðtÞ dt þ
0
Z
!
t1
IðtÞ dt :
ð3:13Þ
l
To make the derivation of the cost function simple, we approximate the inventory depletion curve with a straight line. Similar treatment of the inventory depletion curve with linear approximation can be found in [9,11,12]. Thus, the inventory holding cost per unit time is approximately;
HC ¼
1 C1 : I0 t 1 : 2 T
ð3:14Þ
See Appendix A for ‘exact’ inventory holding cost from Eq. 3.13. Hence, the total inventory cost per unit time, uðT; t1 Þ is given by:
uðT; t1 Þ ¼
C4 l 1 C 1 C 2 alðT t 1 Þ2 C 3 þ : þ : I0 t 1 þ I0 al t 1 2 T T 2 2T T
ð3:15Þ
We assume t 1 ¼ KT; 0 < K < 1. This assumption seems reasonable since the length of the shortage interval is part of the cycle length. In addition, this restriction on values of t1 enhances the convexity of the inventory cost function in Eq. 3.15. Substituting t 1 ¼ KT in (3.15), we get
C4 C1K C 4 al l C 2 alðT KTÞ2 C 3 I0 þ þ ; þ KT 2 T T 2 2T T " # C 4 C 1 K alðKT cÞbþ1 aððl cÞbþ2 ðcÞbþ2 Þ l b l KT ðaðcÞ 1Þ ; þ ¼a 2 ðb þ 1Þðb þ 2Þ T bþ1 2
uðT; KÞ ¼
ð3:16Þ
C 4 al l C 2 alð1 KÞ2 T C 3 þ : KT þ 2 T 2 T
We now proceed to determine T and K optimally by treating them as decision variables. The Inventory cost per unit time,
uðT; KÞ being a function of two variables T and K has to be partially differentiated with respect to T and K separately and then put equal to zero. This gives the necessary condition for minimizing the total inventory cost per time uðT; KÞ. That is; " # @ u C 4 a alðKTðb þ 1ÞðKT cÞb ðKT cÞbþ1 Þ aððl cÞbþ2 ðcÞbþ2 Þ l2 b þ ðaðcÞ 1Þ ¼ 2 ðb þ 1Þ ðb þ 1Þðb þ 2Þ @T 2 T þ
C 1 alK 2 C 4 al2 C 2 alð1 KÞ2 C 3 ½aðKT cÞb ðaðcÞb 1ÞÞ þ þ 2 ¼ 0; 2 2 2T 2 T
ð3:17Þ
@u ¼ C 4 al½aðKT cÞb ðaðcÞb 1Þ @K " # C 1 a alðKTðb þ 1ÞðKT cÞb þ ðKT cÞbþ1 Þ aððl cÞbþ2 ðcÞbþ2 Þ l b þ l 2KT ðaðcÞ 1Þ þ 2 ðb þ 1Þ ðb þ 1Þðb þ 2Þ 2 C 4 alT C 2 alð1 KÞT ¼ 0:
ð3:18Þ
The solutions of Eq. 3.17 and Eq. 3.18, solving simultaneously, give the optimal values T and K which minimize the total inventory cost per unit uðT; KÞ provided they satisfy the sufficient condition below.
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" l1 ¼ C 4 a
al
ðb þ 1ÞT
4
h b1 b b T 2 ðK ðb þ 1Þ½K T bðK T cÞ þ ðK T cÞ K ðb þ 1ÞðK T cÞ Þ
b
2TðK T ðb þ 1ÞðK T cÞ ðK T cÞ þ
bþ1
# i aððl cÞbþ2 ðcÞbþ2 Þ l2 ðaðcÞb 1Þ Þ þ ðb þ 1Þðb þ 2ÞT 3 2T 3
C 1 aK 3 l C al2 C abðK T cÞb1 4 3 þ 33 ; 2 2T T
l2 ¼
C 4 aal T 2 ðb þ 1Þ
½T ðb þ 1ÞðK T bðK T cÞ
b1
b
b
þ ðK T cÞ Þ ðb þ 1ÞðK T cÞ þ
C 1 al b1 ½abK 2 T ðK T cÞ 2
b
þ 2K ðK T cÞ 2K ðaðcÞb 1Þ þ C 2 alð1 K Þ; l3 ¼ C 4 aK labðK T cb1 Þ " # b1 b b C 1 a al½K ðb þ 1ÞðK T bðK T cÞ þ ðK T cÞ Þ þ K ðb þ 1ÞðK T cÞ b þ 2K lðaðcÞ 1Þ C 4 al 2 bþ1 C 2 alð1 K Þ; b1
l4 ¼ C 4 aT labðK T cÞ " # b1 b b C 1 a al½T ðb þ 1ÞðT K bðK T cÞ þ ðK T cÞ Þ þ T ðb þ 1ÞðK T cÞ b þ 2T lðaðcÞ 1Þ þ C 2 aT l; 2 bþ1 0
where li s; i ¼ 1; 2 . . . 4, are: @ 2 u=@T 2 ; @ 2 u=@T@K; @ 2 u=@K@T and @ 2 u=@K 2 respectively. The sufficient condition is:
l1 > 0;
l4 > 0;
l1 :l4 l2 :l3 > 0:
ð3:19Þ
If l2 ¼ l3 , the condition reduces to:
l1 > 0;
l4 > 0;
2
l1 :l4 ðl2 Þ > 0:
Remark 2. We can put (3.19) in equivalent form as; the total cost per unit time uðT; KÞ is minimized if its Hessian matrix evaluated at ðT ; K Þ is positive definite. The total back-order quantity for the cycle is alðT t1 Þ . Thus, the optimal order quantity , I is:
I ¼ I0 þ alðT t 1 Þ:
ð3:20Þ
Corollary 3.1. A change in the inventory ordering cost C 3 by the amount qT, where q is the ordering cost per unit of item produced, will leave the optimal ordering quantity unchanged. Proof. Replacing C 3 in 3.16 with C 3 þ qT, we obtain;
# " C 4 C 1 K alðKT cÞbþ1 aððl cÞbþ2 ðcÞbþ2 Þ l b l KT ðaðcÞ 1Þ u ðT; KÞ ¼ a þ 2 ðb þ 1Þðb þ 2Þ T bþ1 2 D
C 4 al l C 2 alð1 KÞ2 T C 3 þ þ q: KT þ 2 T 2 T
ð3:21Þ
Differentiating 3.21 partially w.r.t T and K, we observed that @ uD =@T ¼ @ u=@T ¼ 0 and @ uD =@K ¼ @ u=@K ¼ 0. Hence, the result follows and the proof is complete. h
4. Numerical example Example 4.1. An example is chosen for numerical illustration which represents an inventory system with the following data: C 1 ¼ 2.40 /unit/year, C 2 ¼ 5=unit=year;C 3 ¼ 100=order;C 4 ¼ 8=unit;a ¼ 9000 units/year, l ¼ 2=3 year, a ¼ 0:002; b ¼ 20 and c ¼ 0:6. The optimal solutions are found to be: K ¼ 0:9907; T ¼ 1:893 years;t1 ¼ 1:8754 years;I0 ¼ 9346:8851 units;I ¼ 9452:5145 units and u ¼ 11566:60=year.
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Table 1 Sensitivity analysis of the inventory model parameters. Parameter
C1
C2
C3
C4
a
l
a
b
c
K
% Change
+50 +25 25 50 +50 +25 25 50 +50 +25 25 50 +50 +25 25 50 +50 +25 25 50 +50 +25 25 50 +50 +25 25 50 +50 +25 25 50 +50 +25 25 50 +20 25 50
% Change in T
I0
u
1.61 0.81 +0.82 +1.65 0.00 0.00 0.00 +0.01 +1.05 +0.04 0.04 0.08 +1.05 +0.62 1.02 3.01 0.05 0.03 +0.05 +0.78 6.78 3.43 +3.86 +8.91 3.60 2.00 +2.64 +6.49 18.03 11.12 +20.15 +61.75 +6.55 +3.27 3.26 6.51 16.66 +33.33 +99.97
2.52 1.29 +1.34 +2.73 0.00 0.00 0.00 0.00 +0.10 +0.06 0.06 0.13 +1.71 +1.01 1.62 4.63 +49.87 +24.94 24.94 49.87 +27.58 +15.29 18.08 38.94 3.71 2.06 +2.71 +6.66 20.43 12.52 +22.14 +65.82 +6.25 +3.11 3.08 6.12 +0.003 0.008 +0.02
+27.08 +13.65 13.88 28.00 +.29 +0.14 0.15 0.29 +2.20 +1.44 1.44 2.89 +19.02 +9.56 9.78 20.07 +47.11 +23.57 23.56 45.97 +12.51 +7.59 10.21 23.20 0.91 0.51 +0.71 +1.82 16.29 9.29 +13.87 +35.51 0.77 0.52 +0.84 +2.05 +19.50 22.33 39.79
Example 4.2. Let: C 1 ¼ 0:15 /unit/year, C 2 ¼ 0.02 /unit/year, C 3 ¼ 280=order, C 4 ¼ 1.5/unit, a ¼ 15000 units/year, year, a ¼ 0:001, b ¼ 8, c ¼ 0:4 and K ¼ 0:8. The optimal solutions to the inventory problem are: T ¼ 3:2416 years, t 1 ¼ 2:5933 years, I0 ¼ 18553:5546 units, I ¼ 23416:0230 units and u ¼ 1662.25/year.
l ¼ 1=2
We now proceed to test the responsiveness of the proposed model to changes in the model parameters using Example 4.2. The sensitivity analysis is performed by changing the value of each parameter by 50%, 25%, 25%, 50%, taking one parameter at a time and keeping the remaining parameters unchanged. We assume that insensitive, moderately sensitive, and highly sensitive imply percentage changes are 3 to 3, 20 to +20 and more respectively. This guide for categorizing sensitivity is used by [16]. It can be seen from Table 1 that the solutions T ; I0 and u are highly insensitive to change in C 2 and C 3 while the solutions are highly sensitive to changes in l and b. Furthermore, the ‘initial’ optimal quantity I0 and the optimal inventory cost u change by roughly the same percentage change in a. 5. Conclusion We have presented an inventory model for three-parameter Weibull distribution deteriorating items with ramp-type demand rate. The proposed model is suitable for newly launched product with erratic demand pattern up to a point in time. The Weibull distribution is often used for modeling duration data because it has exponential and Rayleigh distributions as sub-models. In practice, the rate of deterioration of most items increases with time. Cooray [19] suggested that when modeling monotone hazard rates, the weibull distribution may be an initial choice because of its negatively and positively skewed density shape. The location parameter, c, in the model depicts the shelf-life of the item under consideration; an essential feature often ignored by most inventory modelers. Several intrinsic features of the model have demonstrated via
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sensitivity analysis. In particular, the optimal quantity and the total inventory cost change with approximately the same percentage changes in a for the inventory problem in Example 4.2. Also, we give the necessary and sufficient conditions for the optimal solutions to the model. The proposed model can be extended in a number of ways; it would be interesting to relax the assumption of no discount quantity or to consider the parameters as fuzzy/stochastic fuzzy. Addendum Material below came to the attention of the authors at the time of revising this paper. Note 1. Jalan et al [20] presented an EOQ model for items with two-parameter Weibull distribution deterioration and ramp type demand. They allowed shortages in the inventory and obtained the economic order quantity via numerical technique. Note 2. Giri et al [21] reconsidered the model in [20] by taking a ramp type demand pattern for fashionable products which initially increases exponentially with time up to a point after which it becomes steady. Note 3. Jain and Kumar[22] developed an EOQ model for inventory system that starts with shortage. They considered the case where the time to deterioration of items follows three-parameter Weibull distribution, the demand rate is ramp type function of time that increases exponentially up to a point then stays constant and they provided an analytical method for reaching the optimal replenishment policy. The present authors feel that inventory model that starts with shortage is appropriate for faulty inventory system which experience unnecessary delay at the initial stage due to delay in supply, shortage of raw materials, shortage of labour, etc. Such system rarely occurred in practice because of technological advancement and business competition. We are also of the opinion that exponential time-varying demand rate is unrealistic because the demand of any product seldom experiences a rate which is as high as exponential rate. Finally, Two-parameter Weibull distribution instantaneous rate function is appropriate for item with decreasing rate of deterioration only if the initial rate of deterioration is very high or it can be used for system with increasing rate of deterioration only if the initial rate is negligible[see [10,11]]. Acknowledgements We thank the referees for their valuable comments and suggestions which led to a substantial improvement of this paper. The first author would like to thank Yekini Shehu for helpful discussions. Appendix A. Derivation of the total average holding cost We derive the Total average holding cost per cycle directly from the inventory depletion curve as follows:
"Z # Z t1 l C1 HC ¼ IðtÞ dt þ IðtÞ dt T 0 l " Z " # l C1 1 2 atðt cÞbþ1 aððt cÞbþ2 ðcÞbþ2 Þ b b þ I0 expfaðcÞ g dt a t ð1 aðt cÞ Þ þ ¼ 2 ðb þ 1Þðb þ 2Þ T bþ1 0 " # # Z t1 aððt1 cÞbþ1 ðt cÞbþ1 Þ þ alðt 1 tÞð1 að1 cÞb Þ þ dt bþ1 l " " !# C1 1 l3 l2 ðl cÞbþ1 2 bþ1 bþ3 bþ3 ½lðb þ 3Þðl cÞ a a ððl cÞ ðcÞ Þ ¼ 2 3 ðb þ 1Þðb þ 2Þðb þ 3Þ T bþ1 þ
a ðb þ 1Þðb þ 2Þðb þ 3Þ
½lðb þ 3Þðl cÞbþ2 ððl cÞbþ3 ðcÞbþ3 Þ
½ðl cÞbþ3 ðcÞbþ3 ðb þ 3ÞlðcÞbþ2 þ lI0 expfaðcÞb g ðb þ 1Þðb þ 2Þðb þ 3Þ "" # t1 l aðl cÞbþ1 ðt1 lÞ ððt1 cÞbþ2 ðl cÞbþ2 Þ þal lðt 1 Þ þ bþ1 ðb þ 1Þðb þ 2Þ 2 2 h i a : ðb þ 2Þðt 1 ðt1 cÞbþ1 lðl cÞbþ1 Þ ððt 1 cÞbþ2 ðl cÞbþ2 Þ þ ðb þ 1Þðb þ 2Þ
a
Appendix B. Derivation of the inventory level Using the integrating factor(i.e. expfaðt cÞb g) of the first order linear differential equation of (3.1) to multiply both sides of (3.1), we get
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d ½IðtÞ expfaðt cÞb g ¼ at expfaðt cÞb g: dt Solving the above equation using the condition, Ið0Þ ¼ I0 , we obtain
Z t IðtÞeaðtcÞb I0 eaðcÞb ¼ a t½1 þ aðt cÞb dt; 0 " # t2 atðt cÞbþ1 aððt cÞbþ2 ðcÞbþ2 Þ aðtcÞb þ I0 expfaðcÞb g; ¼ a þ IðtÞe ðb þ 1Þðb þ 2Þ 2 bþ1 "
" # # t 2 atðt cÞbþ1 aððt cÞbþ2 ðcÞbþ2 Þ b ) IðtÞ ¼ a þ þ I0 expfaðcÞ g expfaðt cÞb g; ðb þ 1Þðb þ 2Þ 2 bþ1
0 6 t 6 l:
Note that we have used eaðtcÞb 1 þ aðt cÞb on the integrand since 0 < a 1. Similarly, the solutions to the differential Eqs. (3.2) and (3.3), after using the boundary conditions, give (3.5) and (3.6). Hence, the inventory level at any time t 2 ½0; T is:
8h h i i 2 cÞbþ1 cÞbþ2 ðcÞbþ2 Þ > a t2 þ atðt aððtðbþ1Þðbþ2Þ þ I0 expfaðcÞb g expfaðt cÞb g; 0 6 t 6 l; > bþ1 >
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