Applied Mathematical Modelling 37 (2013) 9698–9706

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

An Economic order quantity model for Items withThree-parameter Weibull distribution Deterioration,Ramp-type Demand and Shortages

0307-904X/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.apm.2013.05.017

⇑ Corresponding author. Tel.: +234 8035813754.E-mail addresses: sanni_smaila@yahoo.com (S.S. Sanni), chukwuagusi@yahoo.com (W.I.E. Chukwu).

S.S. Sanni ⇑, W.I.E. ChukwuDepartment of Statistics, Faculty of Physical Sciences, University of Nigeria, Nsukka, Enugu, Nigeria

a r t i c l e i n f o a b s t r a c t

Article history:Received 30 May 2012Received in revised form 16 April 2013Accepted 21 May 2013Available online 5 June 2013

Keywords:Inventory modelRamp type demandDifferential equationOptimal policyInstantaneous rate function

In this paper, we develop an economic order quantity inventory model for items withthree-parameter Weibull distribution deterioration and ramp-type demand. Shortagesare allowed in the inventory system and are completely backlogged. The demand rate isdeterministic and varies with time up to a certain point and eventually stabilized andbecomes constant. The instantaneous rate of deterioration is an increasing function of time.We provide simple analytical tractable procedures for deriving the model and give numer-ical examples to illustrate the solution procedure. Our adoption of ramp-type demandreflects a real market demand for newly launched product.

� 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 replen-ish the inventory of an item’’ and ‘‘When to order so as to minimize the total cost’’ [1]. Several inventory models have beendeveloped to answer the above questions. The economic-order-quantity model, hereafter EOQ, was originally developed byFord 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 inventoryand replenishment is instantaneous i.e. the lead time is zero. These assumptions restrict the applicability of the classical EOQmodel. In order to make the basic EOQ model more realistic, many researchers have extended Wilson’s EOQ model by con-sidering time/price varying demand pattern and deterioration rate. It ispertinent to note that the depletion of any inventoryis 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 functionand Weibull function.

Definition 1.1. Deterioration is defined as decay, damage, change or spoilage that prevents items from being used for its originalpurpose. 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 itsdensity is given, for some parameters a > 0; b > 0; and c, by f ðyÞ ¼ abðy� cÞb�1e�aðt�cÞb ; y > 0. The parameters a; b, and c

S.S. Sanni, W.I.E. Chukwu / Applied Mathematical Modelling 37 (2013) 9698–9706 9699

are the the scale, shape and location parameters of the distribution, respectively. For time to deterioration data, the scaleparameter, a represents the characteristic rate of deterioration of items, the shape parameter, b is a measure of the spreadof time-to-deterioration and c is the minimum time such that y P c. A negative c may indicate that deterioration hasoccurred prior to the beginning of the inventory, namely during production, in transit, or prior to actual storage whilec ¼ 0 reduces the density to the case of two-parameter Weibull distribution. The scale and the location parameters havethe same unit as T (i.e. cycle length in time) while the shape parameter is dimensionless. Weibull model is widely used todayfor failure and survival analysis.

Definition 1.3. Let R be the set of real numbers. A function g : R! R defined by gðxÞ ¼ hðxÞ; x < lhðlÞ; x P l

�is called a ramp

function. In particular, gðxÞ ¼ a½x� ðx� lÞHðx� lÞ� where Hðx� lÞ ¼ 0; x < l1; x P l

�and a–0;l > 0 is a ramp-type function.

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. Wagnerand Whitin[4] developed a dynamic version of the classical EOQ model. Thereafter, Ghare and Schrader[5] developed a mod-el for exponentially deteriorating inventory. They proposed, explicitly for the first time, the differential equation governingthe variation in the inventory system ; dIðtÞ=dt þ hIðtÞ ¼ �DðtÞ. Donaldson[6] provided a somehow complicated analyticalsolution procedure for the basic inventory policy for the case of positive linear trend in demand. Silver[7] formulated a heu-ristic for deteriorating inventory model with time-dependent linear demand. Deb and Chuadhuri[8] extended the deterio-rating inventory model with linear demand by incorporating shortages in the inventory. Covert and Philip [9] presentedan inventory model where the time to deterioration is described with two-parameter Weibull distribution. Philip[10] gen-eralized the model in [9] by considering a three-parameter Weibull distribution deterioration, no shortages and a constantdemand. Chakrabarty et al. [11] proposed an EOQ model with three-parameter Weibull distribution deterioration, shortagesand linear demand rate and obtained infinite series representation for the initial inventory level and the average total costequation. Ghosh and Chaudhuri [12] developed an inventory model for two-parameter Weibull deteriorating items, withshortages and quadratic demand rate and gave infinite series representation for the initial inventory level and the total aver-age variable cost equation. Sanni [13] developed an inventory model with three-parameter Weibull deterioration, shortagesand quadratic demand rate. He derived explicit equations for the initial inventory level and the average total cost by approx-imating exponential functions by the first two terms of the Tailor series. An order-level inventory model for deterioratingitems with ramp type demand rate was discussed by Mandal and Pal [14]. They considered an EOQ model for items withconstant rate of deterioration, ramp-type demand rate and no shortage allowed in the system and obtained an approximatesolution 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 differ-ent cases. This model was further generalized by Samanta and Bhowmick [16] by taking two parameter Weibull distributionto represent the time to deterioration and allowed shortages in the inventory.They studied two cases; where the inventorystarts with shortages and the case where the system starts without shortages and derived the EOQ for the respective sys-tems. 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 itemswith three parameter Weibull Distribution deterioration, ramp type demand rate and shortages are allowed and completelybacklogged. The research focus of this paper is to develop a mathematical model for the system, provide an optimal replen-ishment policy for the model and establish the necessary and sufficient conditions for the optimal policy. The Weibull dis-tribution is suitable for items whose rate of deterioration increases with time and the location parameter c, in the three-parameter 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 whichthe demand increases as they are launched into the market and after some time the demand stabilizes and becomesconstant.

2. Assumptions and notation

The mathematical model in this work is developed on the basis of the following assumptions and notation.Notation

C1 : inventory holding cost per unit per unit time.C2 : shortage cost per unit per unit time.C3 : ordering cost per order.C4 : unit cost.

DðtÞ : demand rate at any time t P 0.T : cycle length.Io : size of the initial inventory.

9700 S.S. Sanni, W.I.E. Chukwu / Applied Mathematical Modelling 37 (2013) 9698–9706

hðtÞ ¼ abðt � cÞb�1 : 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 dete-rioration, t P c

t1 : time during which there is no shortage.T� : optimal value of T.I�o : optimal value of Io.t�1 : optimal value of t1.

Assumptions

(i) The inventory system under consideration deals with single item. This assumption ensures that a single item is iso-lated 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

that l > 0, and Hðt � lÞ is a Heaviside unit function of time defined as follows: Hðt � lÞ ¼ 1; t P l0; t < l

�. Also, a stands

for the initial demand rate and l is a fixed point in time. The implication of the ramp-type demand rate is that demandvaries 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 ispermitted.

(v) Replenishment is instantaneous and lead time is zero.The assumption of zero lead time is made so that the period ofshortage 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

f ðtÞ ¼ abðt � cÞb�1e�aðt�cÞb ; t > 0. The instantaneous rate function is hðtÞ ¼ abðt � cÞb�1. The implication of the three-parameter Weibull instantaneous rate is that the impact of the already deteriorated items that are received intothe inventory as well as those items that may start deteriorating in future are accounted for. The graph of three-pap-rameter 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 threepossible situations may arise and the situations may be presented as follows:

Let lL be a prescribed interval between orders, thensituation 1. the prescribed time lL starts with inventory level say M1 and ends with inventory level say M2, i.e.

0 < lL < l.situation 2. the prescribe time lL starts with inventory level say M1 and ends with a shortage say M2, i.e. l < lL < t1.situation 3. the prescribed time lL starts with a shortage says M1 and ends with a shortage say M2, 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 demandand shortages. And also establish the necessary and sufficient conditions for this optimal policy. Our model is a variant ofSamanta and Bhowmick [16] and our results extend some results of [16] and many other recently known results in literature.

0γ <

0γ =

0γ >

Rat

e of

det

erio

ratio

n

Time

Increasing rate ( )2 1β> >

Decreasing rate, 0γ <( )1β <

Fig. 1. Rate of deterioration-time relationship for three-parameter Weibull distribution.

S.S. Sanni, W.I.E. Chukwu / Applied Mathematical Modelling 37 (2013) 9698–9706 9701

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 interval½0; t1�, the inventory depletes due mainly to demand and partly to deterioration. At time t ¼ l < t1, the inventory level de-pletes to S units and at t1, the inventory level is zero and all the demand hereafter (i.e. T � t1Þ is completely backlogged. Thetotal number of backordered items is replaced by the next replenishment. The demand varies with time up to a certain timeand become constant. The deterioration rate is described by an increasing function of time hðtÞ ¼ abðt � cÞb�1. A Graphicalrepresentation 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:

dIðtÞdtþ IðtÞabðt � cÞb�1 ¼ �at; 0 6 t 6 l; ð3:1Þ

dIðtÞdtþ IðtÞabðt � cÞb�1 ¼ �al; l 6 t 6 t1; ð3:2Þ

dIðtÞdt¼ �al; t1 6 t 6 T; ð3:3Þ

with boundary conditions Ið0Þ ¼ I0; IðlÞ ¼ S; Iðt1Þ 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

IðtÞ ¼ �at2

2þ atðt � cÞbþ1

bþ 1� aððt � cÞbþ2 � ð�cÞbþ2Þ

ðbþ 1Þðbþ 2Þ

" #þ I0 expfað�cÞbg

" #expf�aðt � cÞbg; 0 6 t 6 l; ð3:4Þ

IðtÞ ¼ al ðt1 � tÞ þ aððt1 � cÞbþ1 � ðt � cÞbþ1Þðbþ 1Þ

" #" #expf�aðt � cÞbg; l 6 t 6 t1; ð3:5Þ

IðtÞ ¼ �alðt � t1Þ; t1 6 t 6 T: ð3:6Þ

Applying the condition IðlÞ ¼ S in Eqs. 3.4 and 3.5, we get

S ¼ �al2

2þ alðl� cÞbþ1

bþ 1� aððl� cÞbþ2 � ð�cÞbþ2Þ

ðbþ 1Þðbþ 2Þ

" #þ I0 expfað�cÞbg

" #expf�aðl� cÞbg; ð3:7Þ

S ¼ al ðt1 � lÞ þ aððt1 � cÞbþ1 � ðl� cÞbþ1Þðbþ 1Þ

" #" #expf�aðl� cÞbg: ð3:8Þ

Eliminate S by equating Eq. 3.7 and 3.8, we obtain the initial inventory level

I0 ¼ a l t1 �l2

� �þ alðt1 � cÞbþ1

bþ 1� aððl� cÞbþ2 � ð�cÞbþ2Þ

ðbþ 1Þðbþ 2Þ

" #expf�að�cÞbg

¼ a l t1 �l2

� �þ alðt1 � cÞbþ1

bþ 1� aððl� cÞbþ2 � ð�cÞbþ2Þ

ðbþ 1Þðbþ 2Þ � að�cÞbl t1 �l2

� �" #: ð3:9Þ

I0

I

S

t1

t μ=

( )0I I−

0

T

Inve

ntor

y le

vel

Time

Depletion curve with detarioration

Without detarioration

Fig. 2. Inventory system for deteriorating items.

9702 S.S. Sanni, W.I.E. Chukwu / Applied Mathematical Modelling 37 (2013) 9698–9706

The total inventory cost per unit time consists of the the following components:Deterioration cost (DC) in the cycle ½0; t1�;

DC ¼ C4

TI0 �

Z l

0at dt þ

Z t1

laldt

!" #¼ C4

TI0 � al t1 �

l2

� �h i

¼ C4aT

alðt1 � cÞbþ1

bþ 1� aððl� cÞbþ2 � ð�cÞbþ2Þ

ðbþ 1Þðbþ 2Þ � að�cÞbl t1 �l2

� �" #; ð3:10Þ

Shortage cost (SC) in the interval ½t1; T�;

SC ¼ C2

T

Z T

t1

�IðtÞdt ¼ C2alT

:ðT � t1Þ2

2: ð3:11Þ

Ordering cost(OC);

OC ¼ C3

Tð3:12Þ

and the inventory holding cost (HC) in ½0; t1�;

HC ¼ C1

T

Z l

0IðtÞdt þ

Z t1

lIðtÞdt

!: ð3:13Þ

To make the derivation of the cost function simple, we approximate the inventory depletion curve with a straight line. Sim-ilar treatment of the inventory depletion curve with linear approximation can be found in [9,11,12]. Thus, the inventoryholding cost per unit time is approximately;

HC ¼ 12:C1

TI0t1: ð3:14Þ

See Appendix A for ‘exact’ inventory holding cost from Eq. 3.13.Hence, the total inventory cost per unit time, uðT; t1Þ is given by:

uðT; t1Þ ¼C4

TI0 � al t1 �

l2

� �� �þ 1

2:C1

TI0t1 þ

C2alðT � t1Þ2

2Tþ C3

T: ð3:15Þ

We assume t1 ¼ KT; 0 < K < 1. This assumption seems reasonable since the length of the shortage interval is part of the cy-cle length. In addition, this restriction on values of t1 enhances the convexity of the inventory cost function in Eq. 3.15.

Substituting t1 ¼ KT in (3.15), we get

uðT;KÞ ¼ C4

Tþ C1K

2

� �I0 �

C4alT

KT � l2

� �þ C2alðT � KTÞ2

2Tþ C3

T;

¼ aC4

Tþ C1K

2

� �alðKT � cÞbþ1

bþ 1� aððl� cÞbþ2 � ð�cÞbþ2Þ

ðbþ 1Þðbþ 2Þ � l KT � l2

� �ðað�cÞb � 1Þ

" #; ð3:16Þ

� C4alT

KT � l2

� �þ C2alð1� KÞ2T

2þ C3

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 andthen put equal to zero. This gives the necessary condition for minimizing the total inventory cost per time uðT;KÞ. That is;

@u@T¼ C4a

T2

alðKTðbþ 1ÞðKT � cÞb � ðKT � cÞbþ1Þðbþ 1Þ þ aððl� cÞbþ2 � ð�cÞbþ2Þ

ðbþ 1Þðbþ 2Þ � l2

2ðað�cÞb � 1Þ

" #

þ C1alK2

2½aðKT � cÞb � ðað�cÞb � 1ÞÞ� þ C4al2

2T2 þC2alð1� KÞ2

2� C3

T2 ¼ 0; ð3:17Þ

@u@K¼ C4al½aðKT � cÞb � ðað�cÞb � 1Þ�

þ C1a2

alðKTðbþ 1ÞðKT � cÞb þ ðKT � cÞbþ1Þðbþ 1Þ þ aððl� cÞbþ2 � ð�cÞbþ2Þ

ðbþ 1Þðbþ 2Þ � l 2KT � l2

� �ðað�cÞb � 1Þ

" #

� C4alT � C2alð1� 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 totalinventory cost per unit uðT;KÞ provided they satisfy the sufficient condition below.

S.S. Sanni, W.I.E. Chukwu / Applied Mathematical Modelling 37 (2013) 9698–9706 9703

l1 ¼ C4aal

ðbþ 1ÞT�4T�2ðK�ðbþ 1Þ½K�T�bðK�T� � cÞb�1 þ ðK�T� � cÞb� � K�ðbþ 1ÞðK�T� � cÞbÞh"

�2TðK�T�ðbþ 1ÞðK�T� � cÞb � ðK�T� � cÞbþ1Þi� aððl� cÞbþ2 � ð�cÞbþ2Þ

ðbþ 1Þðbþ 2ÞT�3þ l2ðað�cÞb � 1Þ

2T�3

#

þ C1aK�3l2

abðK�T� � cÞb�1 � C4al2

2T�3þ C3

T�3;

l2 ¼C4aal

T�2ðbþ 1Þ½T�ðbþ 1ÞðK�T�bðK�T� � cÞb�1 þ ðK�T� � cÞbÞ � ðbþ 1ÞðK�T� � cÞb� þ C1al

2½abK�2T�ðK�T� � cÞb�1

þ 2K�ðK�T� � cÞb � 2K�ðað�cÞb � 1Þ� þ C2alð1� K�Þ;

l3 ¼ C4aK�labðK�T� � cb�1Þ

þ C1a2

al½K�ðbþ 1ÞðK�T�bðK�T� � cÞb�1 þ ðK�T� � cÞbÞ þ K�ðbþ 1ÞðK�T� � cÞb�bþ 1

� 2K�lðað�cÞb � 1Þ" #

� C4al

� C2alð1� K�Þ;

l4 ¼ C4aT�labðK�T� � cÞb�1

þ C1a2

al½T�ðbþ 1ÞðT�K�bðK�T� � cÞb�1 þ ðK�T� � cÞbÞ þ T�ðbþ 1ÞðK�T� � cÞb�bþ 1

� 2T�lðað�cÞb � 1Þ" #

þ C2aT�l;

where l0is; i ¼ 1;2 . . . 4, are: @2u=@T2; @2u=@T@K; @2u=@K@T and @2u=@K2 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; l1:l4 � ðl2Þ2 > 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 matrixevaluated at ðT�;K�Þ is positive definite.

The total back-order quantity for the cycle is alðT� � t�1Þ . Thus, the optimal order quantity , I� is:

I� ¼ I�0 þ alðT� � t�1Þ: ð3:20Þ

Corollary 3.1. A change in the inventory ordering cost C3 by the amount qT, where q is the ordering cost per unit of item pro-duced, will leave the optimal ordering quantity unchanged.

Proof. Replacing C3 in 3.16 with C3 þ qT, we obtain;

uDðT;KÞ ¼ aC4

Tþ C1K

2

� �alðKT � cÞbþ1

bþ 1� aððl� cÞbþ2 � ð�cÞbþ2Þ

ðbþ 1Þðbþ 2Þ � l KT � l2

� �ðað�cÞb � 1Þ

" #

� C4alT

KT � l2

� �þ C2alð1� KÞ2T

2þ C3

Tþ q: 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, theresult 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 followingdata:

C1 ¼ 2.40 /unit/year, C2 ¼ 5=unit=year;C3 ¼ 100=order;C4 ¼ 8=unit;a ¼ 9000 units/year, l ¼ 2=3 year, a ¼ 0:002; b ¼ 20and c ¼ 0:6.

The optimal solutions are found to be:K� ¼ 0:9907; T� ¼ 1:893 years;t�1 ¼ 1:8754 years;I�0 ¼ 9346:8851 units;I� ¼ 9452:5145 units and u� ¼ 11566:60=year.

Table 1Sensitivity analysis of the inventory model parameters.

Parameter % Change % Change in

T� I�0 u�

C1 +50 �1.61 �2.52 +27.08+25 �0.81 �1.29 +13.65�25 +0.82 +1.34 �13.88�50 +1.65 +2.73 �28.00

C2 +50 0.00 0.00 +.29+25 0.00 0.00 +0.14�25 0.00 0.00 �0.15�50 +0.01 0.00 �0.29

C3 +50 +1.05 +0.10 +2.20+25 +0.04 +0.06 +1.44�25 �0.04 �0.06 �1.44�50 �0.08 �0.13 �2.89

C4 +50 +1.05 +1.71 +19.02+25 +0.62 +1.01 +9.56�25 �1.02 �1.62 �9.78�50 �3.01 �4.63 �20.07

a +50 �0.05 +49.87 +47.11+25 �0.03 +24.94 +23.57�25 +0.05 �24.94 �23.56�50 +0.78 �49.87 45.97

l +50 �6.78 +27.58 +12.51+25 �3.43 +15.29 +7.59�25 +3.86 �18.08 �10.21�50 +8.91 �38.94 �23.20

a +50 �3.60 �3.71 �0.91+25 �2.00 �2.06 �0.51�25 +2.64 +2.71 +0.71�50 +6.49 +6.66 +1.82

b +50 �18.03 �20.43 �16.29+25 �11.12 �12.52 �9.29�25 +20.15 +22.14 +13.87�50 +61.75 +65.82 +35.51

c +50 +6.55 +6.25 �0.77+25 +3.27 +3.11 �0.52�25 �3.26 �3.08 +0.84�50 �6.51 �6.12 +2.05

K +20 �16.66 +0.003 +19.50�25 +33.33 �0.008 �22.33�50 +99.97 +0.02 �39.79

9704 S.S. Sanni, W.I.E. Chukwu / Applied Mathematical Modelling 37 (2013) 9698–9706

Example 4.2. Let: C1 ¼ 0:15 /unit/year, C2 ¼ 0.02 /unit/year, C3 ¼ 280=order, C4 ¼ 1.5/unit, a ¼ 15000 units/year, l ¼ 1=2year, 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, I�0 ¼ 18553:5546 units, I� ¼ 23416:0230 units and u� ¼ 1662.25/year.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 param-eter at a time and keeping the remaining parameters unchanged. We assume that insensitive, moderately sensitive, andhighly sensitive imply percentage changes are �3 to 3, �20 to +20 and more respectively. This guide for categorizing sen-sitivity is used by [16].

It can be seen from Table 1 that the solutions T�; I�0 and u� are highly insensitive to change in C2 and C3 while the solutionsare highly sensitive to changes in l and b. Furthermore, the ‘initial’ optimal quantity I�0 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 de-mand 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 assub-models. In practice, the rate of deterioration of most items increases with time. Cooray [19] suggested that when mod-eling monotone hazard rates, the weibull distribution may be an initial choice because of its negatively and positivelyskewed density shape. The location parameter, c, in the model depicts the shelf-life of the item under consideration; anessential feature often ignored by most inventory modelers. Several intrinsic features of the model have demonstrated via

S.S. Sanni, W.I.E. Chukwu / Applied Mathematical Modelling 37 (2013) 9698–9706 9705

sensitivity analysis. In particular, the optimal quantity and the total inventory cost change with approximately the same per-centage changes in a for the inventory problem in Example 4.2. Also, we give the necessary and sufficient conditions for theoptimal solutions to the model. The proposed model can be extended in a number of ways; it would be interesting to relaxthe 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 ramptype 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 whichinitially 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 thecase where the time to deterioration of items follows three-parameter Weibull distribution, the demand rate is ramp typefunction of time that increases exponentially up to a point then stays constant and they provided an analytical method forreaching the optimal replenishment policy.

The present authors feel that inventory model that starts with shortage is appropriate for faulty inventory system whichexperience 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 theopinion that exponential time-varying demand rate is unrealistic because the demand of any product seldom experiences arate which is as high as exponential rate. Finally, Two-parameter Weibull distribution instantaneous rate function is appro-priate for item with decreasing rate of deterioration only if the initial rate of deterioration is very high or it can be used forsystem 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:

HC ¼ C1

T

Z l

0IðtÞdt þ

Z t1

lIðtÞdt

" #

¼ C1

T�aZ l

0

12

t2ð1� aðt � cÞbÞ þ atðt � cÞbþ1

bþ 1� aððt � cÞbþ2 � ð�cÞbþ2Þ

ðbþ 1Þðbþ 2Þ þ I0 expfað�cÞbg" #

d

"t

þZ t1

lalðt1 � tÞð1� að1� cÞbÞ þ aððt1 � cÞbþ1 � ðt � cÞbþ1Þ

bþ 1

" #dt

#

¼ C1

T�a

12

l3

3� a

l2ðl� cÞbþ1

bþ 1� 2ðbþ 1Þðbþ 2Þðbþ 3Þ ½lðbþ 3Þðl� cÞbþ1 � ððl� cÞbþ3 � ð�cÞbþ3Þ�

!" #"

þ aðbþ 1Þðbþ 2Þðbþ 3Þ ½lðbþ 3Þðl� cÞbþ2 � ððl� cÞbþ3 � ð�cÞbþ3Þ�

� aðbþ 1Þðbþ 2Þðbþ 3Þ ½ðl� cÞbþ3 � ð�cÞbþ3 � ðbþ 3Þlð�cÞbþ2� þ lI0 expfað�cÞbg

�

þal t1

2� lðt1 �

l2Þ þ aðl� cÞbþ1ðt1 � lÞ

bþ 1� ððt1 � cÞbþ2 � ðl� cÞbþ2Þ

ðbþ 1Þðbþ 2Þ

" #"

þ aðbþ 1Þðbþ 2Þ ðbþ 2Þðt1ðt1 � cÞbþ1 � lðl� cÞbþ1Þ � ððt1 � cÞbþ2 � ðl� cÞbþ2Þ

h i��:

Appendix B. Derivation of the inventory level

Using the integrating factor(i.e. expfaðt � cÞbg) of the first order linear differential equation of (3.1) to multiply both sidesof (3.1), we get

9706 S.S. Sanni, W.I.E. Chukwu / Applied Mathematical Modelling 37 (2013) 9698–9706

ddt½IðtÞ expfaðt � cÞbg� ¼ �at expfaðt � cÞbg:

Solving the above equation using the condition, Ið0Þ ¼ I0, we obtain

IðtÞeaðt�cÞb � I0eað�cÞb ¼ �aZ t

0t½1þ aðt � cÞb�dt;

IðtÞeaðt�cÞb ¼ �at2

2þ atðt � cÞbþ1

bþ 1� aððt � cÞbþ2 � ð�cÞbþ2Þ

ðbþ 1Þðbþ 2Þ

" #þ I0 expfað�cÞbg;

) IðtÞ ¼ �at2

2þ atðt � cÞbþ1

bþ 1� aððt � cÞbþ2 � ð�cÞbþ2Þ

ðbþ 1Þðbþ 2Þ

" #þ I0 expfað�cÞbg

" #expf�aðt � cÞbg; 0 6 t 6 l:

Note that we have used eaðt�cÞb � 1þ 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:

IðtÞ ¼�a t2

2 þatðt�cÞbþ1

bþ1 � aððt�cÞbþ2�ð�cÞbþ2Þðbþ1Þðbþ2Þ

h iþ I0 expfað�cÞbg

h iexpf�aðt � cÞbg; 0 6 t 6 l;

al ðt1 � tÞ þ aððt1�cÞbþ1�ðt�cÞbþ1Þðbþ1Þ

h ih iexpf�aðt � cÞbg; l 6 t 6 t1;

�alðt � t1Þ; t1 6 t 6 T;

8>>><>>>:

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