an economic production lot size model with price discounting for

Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 11, 531 - 554

An Economic Production Lot Size Model with

Price Discounting for Non-Instantaneous

Deteriorating Items with Ramp-Type Production

and Demand Rates

Garima Garg*, Bindu Vaish** and Shalini Gupta ***

* Raj Kumar Goel Institute of Technology, Ghaziabad.

DD-154, Aavantika, Near Campus School, Chiranjeev Vihar. City-Ghaziabad, State- Uttar Pradesh, Country- India, Pin Code-250002

[email protected]

** D. N. College, Delhi-Meerut Road City- Meerut, State- Uttar Pradesh, Country- India, Pin Code-201002

[email protected] .

*** Raj Kumar Goel Institute of Technology, 5th km. Stone, Delhi-Meerut Road

City-Ghaziabad, State- Uttar Pradesh, Country- India, Pin Code-250003 [email protected]

Abstract

This paper deals with an economic production lot size model with price discounting. Deterioration is non-instantaneous and Weibull in nature. Initially when production begins demand and production rates are variable, after some time deterioration starts and demand and production rates becomes constant. In such a

532 G. Garg, B. Vaish and S. Gupta

situation discount on the selling price is given to boost up the demand and to reduce the loss due to deterioration. The model determines optimal discount to be given on unit selling price during deterioration so as to maximize the total profit. Numerical examples are presented to illustrate the model and sensitivity analysis is also reported.

Keywords: Ramp-type demand rate, non-instantaneous deterioration, price discounting 1. INTRODUCTION

Many researchers considered the constant demand rate in their inventory

models, but the assumption of constant demand rate is usually valid in the mature stage of a product’s life cycle. Generally speaking, the demand rate of any product is always in a dynamic state. It is observed that demand of some new products and many more increases at the beginning and after some time the demand ultimately stabilizes and become constant. This kind of demand pattern seems to be quite realistic and is termed as “ramp type”. Thus, in case of ramp type demand, the demand increases at the beginning and then the market grows into a stable stage such that the demand becomes constant until the end of the inventory cycle.

Hill (1995) first considered the inventory models for increasing demand followed by a constant demand. Wu (2001) developed an inventory model with ramp type demand rate, weibull distributed deterioration and partial backlogging. Giri et al. (2003) extended the ramp type demand inventory model with a more generalized weibull deterioration distribution. Manna & Chaudhuri (2006) presented an EOQ model with ramp type demand rate and time dependent deterioration rate. Panda et al. (2008) developed an inventory models for perishable seasonal products with ramp-type demand. Skouri et al (2009) presented inventory models with ramp type demand rate, partial backlogging & weibull deterioration rate.

In the existing literature, in maximum inventory models for deteriorating items it is assumed that the deterioration occurs as soon as the commodities arrive in inventory. However, in real life, most goods would have a span of maintaining quality or original condition and deterioration starts after that span. This phenomenon is termed as ‘non-instantaneous deterioration’, given by Wu et al. (2006). This type of phenomena can be commonly observed in food stuffs, fruits, green vegetables and fashionable goods, which have a span of maintaining fresh quality & during that period there is almost no spoilage and after some time some of the items will start to

Economic production lot size model 533 decay. For these kinds of items the assumption that the deterioration starts from the instant of arrival in stock may cause retailers to make inappropriate replenishment policy due to overvalue of the total annual relevant inventory cost. Thus it is necessary to consider the inventory problems for non-instantaneous deteriorating items. Lin and Shi (1999) classified inventory models into two categories: decay models & finite lifetime models. Castro & Alfa (2004) proposed a lifetime replacement policy in discrete time for a single unit system. Ouyang et al. (2006b) developed an inventory model for non-instantaneous deteriorating items with permissible delay in payments. Chang et al. (2010) developed optimal replenishment policies for non-instantaneous deteriorating items with stock-dependent demand.

Demand of goods may vary with time or with price or even with the instantaneous level of inventory displayed in a supermarket. In recent years, there is a spate of interest in the problem of finding the economic replenishment policy for an inventory system having a time-dependent demand pattern. Probably Silver and Meal (1969) were first to develop an EOQ model with time-varying demand rate. Resh et al. (1976) and Donaldson (1977) were among the first few the first who studied a model with linearly time varying demand. Many other researchers have studied deteriorating inventory models with time varying demand under a variety of modeling assumptions such as Dave and Patel (1981), Sachan (1984), Goyal (1987), Goswami and Chaudhuri (1991) and Yang et al. (2001).Khanra and Chaudhuri (2003) developed an EOQ model with shortages over a finite-time horizon, assuming a quadratic demand pattern.

Discount at the store on commodities helps the supplier to reconnect to customers. The main reasons for discounts include: Firstly, suppliers offer discounts to clear their old stocks and free-up space for the fresh stock. Secondly, a supplier desires to sell more to make more profits. A discount is one of the key factors as it increases the footfall inside the store. Thirdly, discounts lure the customers thus they buy more than usual. Further discount offers also reduces the loss due to deterioration for a supplier. Thus price discount is one of the key factors which influence demand and its effect cannot be ignored. Ardalan (1994) developed an inventory policy where temporary price discounts resulted in increase in demand. Papachristos & Skouri (2003) presented an inventory model for deteriorating items where demand rate is a convex decreasing function of the selling price. Sana & Chaudhuri (2008) presented an EOQ model with delay in payments and price discount offers. Hsu & Yu (2009) proposed an EOQ model for imperfective items under a one-time-only discount. Panda et al. (2009) developed an EOQ model for perishable products with discounted selling price and stock dependent demand. Cardanas-Barron et al. (2010) presented an inventory model for determining the optimal ordering policies with advantage of a one-time discount offer and back orders.

534 G. Garg, B. Vaish and S. Gupta The present paper is an extension of Panda et al. (2009)’s model. They

proposed an EOQ model for an infinite time horizon for perishable products with discounted selling price and stock dependent demand with non-instantaneous constant rate of deterioration. In the present paper an economic production quantity model is considered with ramp type demand and production rate. Deterioration is non-instantaneous and Weibull in nature. In the paper, two cases are considered, first when the deterioration starts during the production period, thus making the demand and production rates constant whereas in the second case the deterioration starts after the production has stopped thus making the demand rate constant. In both the situations when deterioration starts a discount on selling price is given to boost up the demand, to decrease the levels of inventory, so as to reduce loss due to deterioration. The model is developed for finite planning horizon. Profit maximization technique is used to solve the model. Numerical examples are provided to illustrate the optimization procedure. In addition the sensitivity analysis of the optimal solution with respect to parameters of the system is carried out. 2. ASSUMPTIONS AND NOTATIONS The following assumptions and notations are applied in the proposed model:

Assumptions 1. Demand rate follows the pattern ( )D t a bt= + , where a and b are

positive constants, a b> and 0 1b≤ ≤ . 2. The production rate is demand dependent, say ( ) ( )P t D tλ= where 1λ > is a

constant. 3. The system operates for a prescribed period of a planning horizon. 4. Shortages are not allowed. 5. Deterioration is non-instantaneous. 6. Deterioration rate varies with time and follows a two-parameter Weibull

distribution function. 7. There is no repair or replacement of deteriorating items during the period

under consideration. 8. Single item inventory system is considered over a prescribed period of time. 9. Delivery lead time is zero. 10. 2 2(0 1)d d≤ ≤ is the percentage discount offer on unit selling price during

deterioration. ( ) 2

2 21 ndα −= − ( 2n R∈ , the set of real numbers), is the effect of

discounted selling price on demand during deterioration. 2α is determined

Economic production lot size model 535 from prior knowledge of the seller such that the demand rate is influenced with the reduction of selling price. When 2 20, 1d α→ → i.e. the demand of decreased quality items remains same.

In day to day life it is commonly observed that the demand of decreased quality items is increased significantly if a discount is offered on selling price. Since the demand is boosted if the discounts are offered therefore it is concluded that the demand is partially time dependent and partially time and selling price dependent if the discounts are given.

Notations 1. T is the Replenishment cycle. 2. μ is the time at which deterioration starts. 3. ( )Z t is the Weibull deterioration rate function of the stocked item

( ) 1, 0Z t t tγηγ −= > .where η is the scale parameter ( )0η > and γ is the shape

parameter ( )0γ > . 4. ( )I t is the inventory level at any time t.

5. 1t is the decision variable representing the time at which production ceases. 6. Q is the economic production quantity. 7. A is the setup cost. 8. 1C is inventory holding cost per unit per unit time. 9. 4C is the production cost per unit item. 10. 5C is the disposal cost per unit per unit time. 11. s is the selling price per unit where 4s C> .

3. FORMULATION AND SOLUTION OF THE MODEL: There may be two scenarios in the discussion of formulation of the present

model according to the position of time μ when demand rate becomes constant under the effect of deterioration.

3.1 Scenario-1: When 10 tμ< <

In this scenario the demand rate becomes constant due to deterioration during production thus making the production rate also constant. Therefore the following

536 G. Garg, B. Vaish and S. Gupta time intervals[ ]0,μ ,[ ]1, tμ and [ ]1,t T are considered separately. During the interval

[ ]0,μ , the inventory level increases due to production as production rate is much greater than demand rate where production and demand rates are variable. At time μ , deterioration starts and thus the inventory level increases due to production rate which is greater than the demand and deterioration until the maximum inventory level is reached at 1t t= . Under the effect of deterioration the production and demand rates becomes constant during the interval[ ]1, tμ . During the interval[ ]1,t T , the production stops and the inventory level decreases due to constant demand and deterioration until the inventory level becomes zero at t T= . The graphical representation of the model is shown below in figure 1.

(Figure 1) Therefore, the inventory level ( )I t at any time t in the interval[ ]0,T can be represented by the following differential equations:

( ) ( )( )1dI t

a btdt

λ= − + 0 t μ≤ ≤ … (1)

( ) ( ) ( ) ( )121

dI ta b t I t

dtγλ α μ ηγ −= − + −

1t tμ ≤ ≤ … (2)

Economic production lot size model 537

( ) ( ) ( )12

dI ta b t I t

dtγα μ ηγ −= − + − 1t t T≤ ≤ … (3)

Using the boundary conditions ( )0 0I = and ( ) 0I T = and solving the above equations the results obtained are:

( ) ( )2

12

btI t atλ⎛ ⎞

= − +⎜ ⎟⎝ ⎠

0 t μ≤ ≤ … (4)

( ) ( ) ( ) ( ){ ( )21 1I t a b t tγλ α μ η μ= − + − −⎡⎣ ( )1 1

1tγ γη μ

γ+ + ⎫

+ − ⎬+ ⎭

( )2

12

bt aγ μη μ⎤⎛ ⎞

+ − + ⎥⎜ ⎟⎝ ⎠⎦

1t tμ ≤ ≤ … (5)

( ) ( ) ( )( )2 1I t a b t T tγα μ η⎡= + − −⎣ ( )1 1

1T tγ γη

γ+ + ⎤

+ − ⎥+ ⎦ 1t t T≤ ≤ … (6)

The economic production quantity can be determined as:

( ) ( )1

0

t

Q P t dt P t dtμ

μ

= +∫ ∫

( )( )2

2 12ba a b tμλ μ λα μ μ

⎛ ⎞= + + + −⎜ ⎟

⎝ ⎠ … (7)

Based on above equations, total profit consists of the following elements: 1. Set up cost is given by rC A= , 2. Holding cost hC is given by

( ) ( ) ( )1

1

10

t T

ht

C C I t dt I t dt I t dtμ

μ

⎡ ⎤= + +⎢ ⎥

⎢ ⎥⎣ ⎦∫ ∫ ∫

( )2 3

1 12 6

a bC μ μλ⎧ ⎫

= − +⎨ ⎬⎩ ⎭

( ) ( ) ( )( )( )

2 21 1

1 212 1 2

t tC a bγμ ηγλ α μ

γ γ

+⎧ −⎪+ − + −⎡ ⎨⎣ + +⎪⎩

( )( ) ( ) ( )1 12

1 1

1 2 1 1t tγ γγ ημ ημηγμ

γ γ γ γ

+ ++ ⎫⎪+ + − ⎬+ + + + ⎪⎭ ( ) ( )12 1

112 1 1

tba tγ γημ ημμ μ

γ γ

+ + ⎤⎧ ⎫⎛ ⎞⎪ ⎪+ + − − − ⎥⎨ ⎬⎜ ⎟ + +⎪ ⎪⎝ ⎠ ⎥⎩ ⎭⎦

( ) ( )( )( )

2 21

1 2 2 1 2T t TC a b

γηγα μγ γ

+⎧ −⎪+ + +⎨ + +⎪⎩ ( )( ) ( ) ( )2 1 1

1 1 1

1 2 1 1t Tt T tγ γ γηγ η η

γ γ γ γ

+ + + ⎫⎪− + − ⎬+ + + + ⎪⎭


3. Disposal cost DC is given by

( ) ( )5 20

T

DC C Q a bt dt a b dtμ

μ

α μ⎡ ⎤⎧ ⎫⎪ ⎪= − + + +⎢ ⎥⎨ ⎬⎢ ⎥⎪ ⎪⎩ ⎭⎣ ⎦

∫ ∫

( ){2

5 12

bC a μλ μ⎛ ⎞

= − +⎜ ⎟⎝ ⎠

( )( )}2 1a b t Tα μ λ λμ μ+ + − − +

4. Production cost pC is given by

( ) ( )1

40

t

pC C P t dt P t dtμ

μ

⎧ ⎫⎪ ⎪= +⎨ ⎬⎪ ⎪⎩ ⎭∫ ∫

( )( )2

4 4 2 12bC a C a b tμλ μ λα μ μ

⎛ ⎞= + + + −⎜ ⎟

⎝ ⎠

5. Sales revenue SR is given by

( ) ( ) ( )2 20

1T

SR s a bt dt d a b dtμ

μ

α μ⎧ ⎫⎪ ⎪= + + − +⎨ ⎬⎪ ⎪⎩ ⎭∫ ∫

( )( )( )2

2 212

bs a d a b Tμμ α μ μ⎧ ⎫

= + + − + −⎨ ⎬⎩ ⎭

Consequently, the total profit of the system can be formulated as:

( )2 1, r h D pTP d t SR C C C C= − − − −

( ) ( )( )22

121

2nbs a d a b Tμμ μ μ− +⎧ ⎫

= + + − + −⎨ ⎬⎩ ⎭

2

4 2bC a μλ μ

⎧− +⎨

⎩

( ) ( )( )}2

2 11 nd a b tμ μ−+ − + − ( )2 3

1 12 6

a bC μ μλ⎛ ⎞

− − +⎜ ⎟⎝ ⎠

( ) ( ) ( )2

1 21 1 nC d a bλ μ−⎡− − − +⎣( )

( )( ) ( )( )

2 2 21 1

2 1 2 1 2t t γ γμ ηγ ηγμ

γ γ γ γ

+ +⎧ −⎪ − +⎨ + + + +⎪⎩1 1

1 1

1 1t tγ γημ ημ

γ γ

+ + ⎫+ − ⎬+ + ⎭

2

2ba μμ

⎛ ⎞+ +⎜ ⎟⎝ ⎠

1 11

1 1 1nt nt

γ γμμγ γ

+ + ⎤⎧ ⎫− − −⎨ ⎬⎥+ +⎩ ⎭⎦


( ) ( ) ( )2

21

1 212

n T tC d a bμ− ⎧ −⎪− − + ⎨

⎪⎩

( )( ) ( )( ) ( ) ( )2 1 12

1 1 1

1 2 1 2 1 1t Tt T tT γ γ γγ ηγ η ηηγ

γ γ γ γ γ γ

+ + ++ ⎫⎪+ − + − ⎬+ + + + + + ⎪⎭

( )2

5 12

bC a μλ μ⎧ ⎛ ⎞⎪− − +⎨ ⎜ ⎟⎪ ⎝ ⎠⎩

( ) ( )( )}2

2 11 nd a b t T Aμ λ λμ μ−+ − + − − + −

… (8)

It is to be noted that here the discount 2d is given on constant unit selling price s of the product. 3.1.1 Solution Procedure

According to equation (8), ( )2 1,TP d t is a function of 2 1d and t . To maximize

the total profit ( )2 1,TP d t , the optimal values of 2d and 1t are obtained by solving the following equations simultaneously.

( )2 1

2

,0

TP d td

∂=

∂ … (9)

And ( )2 1

1

,0

TP d tt

∂=

∂ … (10)

Provided 22 2 2

2 22 1 2 1

0TP TP TPd t d t

⎛ ⎞⎛ ⎞ ⎛ ⎞∂ ∂ ∂− <⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠

… (11)

Equation (9) and (10) are equivalent to:

( )( ) ( )( ){ }2

2 21 1 ns n d a b Tμ μ−− − − + − 4C λ− ( ) ( )( ){ }2 12 2 11 nn d a b tμ μ− −− + −

( ) ( ) ( )2 11 2 21 1 nC n d a bλ μ− −⎡− − − +⎣

( )( )( ) ( )( )

2 2 21 1

2 1 2 1 2t t γ γμ ηγ ηγμ

γ γ γ γ

+ +⎧ −⎪ − +⎨ + + + +⎪⎩1 1

1 1

1 1t tγ γημ ημ

γ γ

+ + ⎤⎫+ − ⎬⎥+ + ⎭⎦

( ) ( ) ( )2

21 1

1 2 212

n T tC n d a bμ− − ⎧ −⎪− − + ⎨

⎪⎩


( ) ( ) ( ) ( ) ( ) ( )2 1 12

1 1 1

1 2 1 2 1 1t Tt T tT γ γ γγ ηγ η ηηγ

γ γ γ γ γ γ

+ + ++ ⎫⎪+ − + − ⎬+ + + + + + ⎪⎭

( ) ( )( )2 15 2 2 11 0nC n d a b t Tμ λ λμ μ− −− − + − − + = … (12)

And ( ) ( )( )2

2 4 51 nd a b C Cλ μ−− − + + ( ) ( ) ( ){2

1 2 11 1 nC d a b tλ μ μ−⎡− − − + −⎣1 1

111 1

t tγ γ

γηγ ημημγ γ

+ + ⎫− + − ⎬+ + ⎭

( )2

112

ba t γμμ η⎤⎛ ⎞

+ + − ⎥⎜ ⎟⎝ ⎠ ⎦

( ) ( )2

1 21 nC d a bμ−− − +

( ) ( )1 1

11 1 0

1 1t Tt T Ttγ γ

γηγ ηηγ γ

+ +⎧ ⎫⎪ ⎪− − + − =⎨ ⎬+ +⎪ ⎪⎩ ⎭ … (13) The optimal production quantity Q and the maximum total profit ( )2 1,TP d t are obtained from equations (7) and (8) respectively by substituting the optimal values of

2d and 1t obtained from equations (12) and (13). 3.1.2 Numerical Example A practical model is considered taking the following values for different parameters:a =200 units, b =0.1, T =12 months, s =Rs. 20, λ =2, 2n =2, μ =2.5, η =0.001, γ =3, A =Rs.300 per order, 1C =1.4, 4C =2.85, 5C =0.90.

Using the solution procedure described in the model the optimal results obtained are, *

2d = 0.334207, *1t =6.13105, ( )* *

2 1,TP d t = Rs. 16700.6 and *Q =

4281.25. Thus the production stops at time *1t =6.13105 and when product start to

deteriorate 33.42% discount on unit selling price is given for the remaining period of replenishment cycle in order to obtain the maximum total profit ( )* *

2 1,TP d t = Rs.

16700.6 on optimal production quantity *Q = 4281.25.

4.3.1.3 Sensitivity Analysis To study the effect of changes of the parameters on the optimal total profit

derived by proposed method, a sensitivity analysis is performed considering the numerical example given above. Sensitivity analysis is performed by changing (increasing or decreasing) the parameters by 20% & 50% and taking one parameter at a time, keeping the remaining parameters at their original values. The results are shown in Table 1 for Numerical Example 3.1.2.


Parameter % change % change in

1t % change in

2d % change in

Q % change in

( )2 1,TP d t

a -50 0.0000 0.0000 -0.4994 -0.5084

-20 0.0000 0.0000 -0.1998 -0.2034

20 0.0000 0.0000 0.1998 0.2034

50 0.0000 0.0000 0.4994 0.5084

2n -50 -1.6377 -33.9490 -0.8162 1.5592

-20 -0.0607 -0.6919 -0.4035 0.2186

20 0.0267 0.2887 0.6060 -0.2295

50 -1.8132 -4.3730 -0.8390 -0.3009 η -50 -0.1504 0.1827 -0.0733 -0.1640

-20 -0.0603 0.0631 -0.0323 -0.0552

20 0.0625 -0.0503 0.0397 0.0500

50 0.1672 -0.1015 0.1234 0.1294

s -50 -0.3019 -2.4797 -0.6917 -0.8944

-20 -0.0389 -0.5128 -0.3134 -0.4251

20 0.0199 0.3352 0.3786 0.4975

50 -2.5079 -8.5235 -0.8552 0.5689

1C -50 -0.3431 0.7981 0.1311 -1.2405

-20 -0.0829 0.2791 0.1250 -0.0946

20 0.0522 -0.2573 -0.1123 -0.0118

50 0.0970 -0.6241 -0.2492 -0.0969

(Table 4.1)

542 G. Garg, B. Vaish and S. Gupta A careful study of Table 1 reveals the following:

i. 1t is not sensitive to change in the value of parameter a, slightly sensitive to change in the value of parameter 1C , it is moderately sensitive to changes in η and highly sensitive to changes in 2n & s .

ii. 2d is not sensitive to change in the value of parameter a, it is moderately sensitive to changes in η & 1C and highly sensitive to changes in 2n & s .

iii. Q is moderately sensitive to changes in the values of the parameters η & 1C and highly sensitive to changes in a , 2n & s .

iv. ( )2 1,TP d t is moderately sensitive to change in the value of the parameter η

and highly sensitive to changes in a , 2n , s & 1C .

3.2 Scenario-2: When 1t Tμ< <

In this scenario the demand rate becomes constant due to deterioration after the production has stopped. Therefore the following time intervals[ ]10, t ,[ ]1,t μ and

[ ],Tμ are considered separately. During the interval[ ]10, t , the inventory level increases due to production as production rate is much greater than demand rate until the maximum inventory level is reached at 1t t= . At time 1t , the production stops and the inventory level decreases due to demand rate during the interval[ ]1,t μ . At time μdeterioration starts and the inventory level decreases due to constant demand and deterioration until the inventory level becomes zero at t T= . The graphical representation of the model is shown below in figure 2.


(Figure 2) Therefore, the inventory level ( )I t at any time t in the interval[ ]0,T can be represented by the following differential equations:

( ) ( )( )1dI t

a btdt

λ= − + 10 t t≤ ≤ … (14)

( ) ( )dI ta bt

dt= − +

1t t μ≤ ≤ … (15)

( ) ( ) ( )12

dI ta b t I t

dtγα μ ηγ −= − + − t Tμ ≤ ≤ … (16)

Using the boundary condition ( )0 0I = and solving the above equations one obtains:

( ) ( )2

12

btI t atλ⎛ ⎞

= − +⎜ ⎟⎝ ⎠

10 t t≤ ≤ … (17)


( )2 2

11 2 2

bt btI t at atλ⎛ ⎞ ⎛ ⎞

= + − +⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

1t t μ≤ ≤ … (18)

( ) ( )2 2

11 1

2 2bt bI t at a tγ γμλ μ ημ η

⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪= + − + + −⎨ ⎬⎜ ⎟ ⎜ ⎟⎪ ⎪⎝ ⎠⎝ ⎠⎩ ⎭

( )2 a bα μ− +

( ) ( )1 1

1t t t tγ γ γημ η μ μ

γ+ +⎧ ⎫

− − − + −⎨ ⎬+⎩ ⎭ t Tμ ≤ ≤ … (19)

At t T= , ( ) 0I T = , therefore from equation (4.19)

( )2 2

12 11

2 2bt bT at aγ γ μα ημ η λ μ

⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪= + − + − +⎨ ⎬⎜ ⎟ ⎜ ⎟⎪ ⎪⎝ ⎠⎝ ⎠⎩ ⎭

( ) ( ) ( )1 1/1

a b T T T Tγ γ γημ μ η μ μγ

+ +⎧ ⎫+ − − − + −⎨ ⎬+⎩ ⎭ … (20)

The economic production quantity can be determined as

( )1

0

t

Q P t dt= ∫

21

1 2btatλ

⎛ ⎞= +⎜ ⎟

⎝ ⎠ … (21)

Based on above equations, total profit consists of the following elements: 1. Set up cost is given by rC A= , 2. Holding cost hC is given by

( ) ( ) ( )1

1

10

t T

ht

C C I t dt I t dt I t dtμ

μ

⎡ ⎤= + +⎢ ⎥

⎢ ⎥⎣ ⎦∫ ∫ ∫

( )2 3

1 11 1

2 6at btC λ

⎛ ⎞= − +⎜ ⎟

⎝ ⎠

2 321 1

1 1 12 2b t btC a t atμλ μ

⎛ ⎞+ + − −⎜ ⎟

⎝ ⎠

( ) ( )2 2 3 31 1 12 6

a bC t tμ μ⎧ ⎫− − + −⎨ ⎬⎩ ⎭

( ) ( )1 11 1

C T T Tγ γ γημ ημ μ μγ

+ +⎧ ⎫+ − + − − −⎨ ⎬+⎩ ⎭


2 21

1 2 2bt bat a μλ μ

⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪+ − +⎨ ⎬⎜ ⎟ ⎜ ⎟⎪ ⎪⎝ ⎠⎝ ⎠⎩ ⎭

( ) ( )( )( )

2 2

1 2 2 1 2T TC a b

γμ ηγα μγ γ

+⎧ −⎪− + −⎨ + +⎪⎩

( )( ) ( ) ( )1 1 1

1 2 1 1T Tγ γ γηγμ ημ ημ

γ γ γ γ

+ + + ⎫⎪+ + − ⎬+ + + + ⎪⎭

3. Disposal cost DC is given by

( ) ( ) ( )1

1

5 20

t T

Dt

C C Q a bt dt a bt dt a b dtμ

μ

α μ⎡ ⎤⎧ ⎫⎪ ⎪⎢ ⎥= − + + + + +⎨ ⎬⎢ ⎥⎪ ⎪⎩ ⎭⎣ ⎦

∫ ∫ ∫

21

5 1 2btC atλ

⎧ ⎛ ⎞⎪= +⎨ ⎜ ⎟⎪ ⎝ ⎠⎩

( )( )2

22ba a b Tμμ α μ μ

⎫⎧ ⎫⎪− + + + −⎨ ⎬⎬⎪⎩ ⎭⎭

4. Production cost pC is given by

4pC C Q= 2

14 1 2

btC atλ⎛ ⎞

= +⎜ ⎟⎝ ⎠

5. Sales revenue SR is given by

( ) ( ) ( )2 20

1T

SR s a bt dt d a b dtμ

μ

α μ⎧ ⎫⎪ ⎪= + + − +⎨ ⎬⎪ ⎪⎩ ⎭∫ ∫

( )( )( )2

2 212

bs a d a b Tμμ α μ μ⎧ ⎫

= + + − + −⎨ ⎬⎩ ⎭

Consequently, the total profit of the system can be formulated as:

( )1 r h D pTP t SR C C C C= − − − −


( )2

5 2bs C a μμ

⎧ ⎫= + +⎨ ⎬

⎩ ⎭( )( ) ( ) ( ){ }2 21

2 5 21 1n na b T s d C dμ μ − + −+ + − − + −

( )2

14 5 1 2

btC C atλ⎧ ⎫

− + +⎨ ⎬⎩ ⎭

( )2 3

1 111

2 6at btCλ

⎛ ⎞− − +⎜ ⎟

⎝ ⎠2 3

21 11 1 12 2

b t btC a t atμλ μ⎛ ⎞

− + − −⎜ ⎟⎝ ⎠

( ) ( )2 2 3 31 1 12 6

a bC t tμ μ⎧ ⎫+ − + −⎨ ⎬⎩ ⎭

( ) ( )1 11 1

C T T Tγ γ γημ ημ μ μγ

+ +⎧ ⎫− − + − − −⎨ ⎬+⎩ ⎭

2 21

1 2 2bt bat a μλ μ

⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪+ − +⎨ ⎬⎜ ⎟ ⎜ ⎟⎪ ⎪⎝ ⎠⎝ ⎠⎩ ⎭

( ) ( ) ( )( )( )

2

2 2

1 212 1 2

n T TC d a bγμ ηγμ

γ γ

+− ⎧ −⎪+ − + −⎨ + +⎪⎩

( )( ) ( ) ( )1 1 1

1 2 1 1T T A

γ γ γηγμ ημ ημγ γ γ γ

+ + + ⎫⎪+ + − −⎬+ + + + ⎪⎭ … (22)

It is to be noted that here the discount 2d is given on constant unit selling price s of the product. 3.2.1 Solution Procedure

According to equation (22), ( )1TP t is a function of 1t . To maximize the total

profit ( )1TP t , the optimal value of 1t is obtained by solving the following equation.

( )1

1

0TP t

t∂

=∂

… (23)

Provided 2

21

0TPt

⎛ ⎞∂<⎜ ⎟∂⎝ ⎠

… (24)

Equation (23) is equivalent to:

Economic production lot size model 547 ( ) ( )1 4 5a bt C Cλ− + + ( )2

1 1 1 1C a b t at btλ μ μ− + − −

( ) ( ) ( )1 11 1 0

1C a bt T T Tγ γ γηλ μ ημ μ μ

γ+ +⎧ ⎫

− + − + − − − =⎨ ⎬+⎩ ⎭ … (25) The optimal production quantity Q and the maximum total profit ( )1TP t are obtained from equations (21) and (22) respectively by substituting the value of 1t obtained from equation (25). 3.2.2. Numerical Example A practical model is considered taking the following values for different parameters:a =900 units, b =0.5, T =6 months, s =Rs. 12, λ =3, 2n =3, μ =4, η =0.001, γ =5, A=Rs.300 per order, 1C =0.5, 4C =1.2, 5C =0.10.

Using the solution procedure described in the model the optimal results

obtained are, *2d = 0.447432, *

1t =3.55467, ( )*1TP t = Rs. 98350 and *Q = 9607.09.

Thus the production stops at time *1t =3.55467 and when product start to deteriorate

44.74% discount on unit selling price is given for the remaining period of replenishment cycle in order to obtain the maximum total profit ( )*

1TP t = Rs. 98350

on optimal production quantity *Q = 9607.09.

3.2.3 Sensitivity Analysis To study the effect of changes of the parameters on the optimal total profit

derived by proposed method, a sensitivity analysis is performed considering the numerical example given above. Sensitivity analysis is performed by changing (increasing or decreasing) the parameters by 20% & 50% and taking one parameter at a time, keeping the remaining parameters at their original values. The results are shown in Table 2 for Numerical Example 3.2.2.


Parameter % change % change in 1t

% change in 2d

% change in Q

% change in ( )1TP t

a -50 0.0000 -0.0005 -0.4995 -0.5009

-20 0.0000 -0.0001 -0.1998 -0.2003

20 0.0000 0.0001 0.1998 0.2003

50 0.0000 0.0002 0.4995 0.5009

2n -50 0.0000 0.5526 0.0000 -0.3220

-20 0.0000 0.1702 0.0000 -0.0992

20 0.0000 -0.1283 0.0000 0.0748

50 0.0000 -0.2700 0.0000 0.1573 η -50 0.7097 0.3320 0.7109 0.5742

-20 0.2839 0.1581 0.2842 0.1977

20 -0.2839 -0.2876 -0.2841 -0.2102

50 -0.7097 0.0000 0.0000 0.0000

s -50 0.0000 0.0000 0.0000 -0.5793

-20 0.0000 0.0000 0.0000 -0.1931

20 0.0000 0.0000 0.0000 0.1931

50 0.0000 0.0000 0.0000 0.5793

1C -50 0.7314 0.0154 0.7327 0.0159

-20 0.1829 0.1014 0.1831 0.1289

20 -0.1219 -0.0927 -0.1220 -0.0943

50 -0.2438 -0.2216 -0.2440 -0.1982

(Table 2)


A careful study of Table 2 reveals the following:

i. 1t is not sensitive to change in the value of parameter s, a & 2n and highly sensitive to changes in η & 1C .

ii. 2d is not sensitive to change in the value of parameter s, it is slightly sensitive to changes in a & 1C , it is moderately sensitive to change in 2n and highly sensitive to changes in η .

iii. Q is not sensitive to change in the value of parameter s & 2n , it is moderately sensitive to changes in the values of the parameter a and highly sensitive to changes in η & 1C .

iv. ( )1TP t is moderately sensitive to changes in the values of the parameters a ,

2n & 1C and highly sensitive to changes in s & η .

4. CONCLUSION In this paper an economic production lot size model is developed with ramp-type production and demand rates. Deterioration is non-instantaneous & Weibull in nature. Initially when production starts, production and demand rates are variable but after some time when product starts to deteriorate the demand rate and thus the production rate become constant. A mathematical model is developed incorporating post deterioration discounts on unit selling price. Optimal discount on unit selling price and optimal time at which production stops are determined for maximizing the total profit. An analytic formulation of the problem on the frame work described above and optimal solution procedure to find optimal discount is presented. Sensitivity analysis with respect to various parameters has been carried out and the results obtained from sensitivity analysis are realistic.

550 G. Garg, B. Vaish and S. Gupta Thus this model incorporates some realistic features that are likely to be

associated with some kind of production units. It can be suitable for some new products launched which includes automobiles, mobile phones, fashionable goods and other products which have more likely the characteristics above.

The present study can be further extended for some other factors involved in

the inventory system.

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Received: September, 2011

an economic production lot size model with price discounting for

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