a stochastic inventory model for deteriorating items with ... fileall these strategies prevent the...

A stochastic inventory model for deteriorating

items with price, promotion effort and quality

level sensitive demand

By

Ashaba D. Chauhana,c and Hardik N. Sonib

(a) Department of Mathematics, D. L. Patel Science College, Himmatnagar, Gujarat, India,

383001.

(b) Chimanbhai Patel Post Graduate Institute of Computer Applications, Ahmedabad,

Gujarat, India, 380015.

(c) Corresponding author: [email protected]

Abstract

A number of models have been proposed to investigate the quantitative deterioration inventories but in practice, product

deterioration can also be qualitative. In this paper, we focussed on qualitative deterioration and to reduce it we consider the

quality investment. Quality investment increases the ability to maintain the freshness of the product. In order to provide general

framework, the quality investment, promotional effort and lot sizing problem where demand function is assumed to be dependent

on selling price, quality level and promotional effort with shortages at the starting period. This paper seeks to maximize the total

profit per unit time by taking decision on the shortage period, inventory period, quality investment cost and promotional effort.

Numerical example and sensitivity analysis on the key parameters are presented to illustrate the model.

Keywords: Inventory, Shortage, Quality investment, Promotion.

JASC: Journal of Applied Science and Computations

Volume VI, Issue III, March/2019

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1. INTRODUCTION

Most of the products undergo deterioration so the items are not suitable for their original purpose. During the

normal storage period deterioration of the product will lead to qualitative and quantitative changes. Quality is one of

the main criteria to evaluate a product’s utility. The seller usually invests in quality investment like functionality,

safety and packaging to improve or maintain the quality level. For example, product’s wholesomeness and aesthetics

is improved by doing food packaging at almost every stage of food chain. Another one, automobile company improves

the car’s paint quality by increasing the longevity or ability to retain the original shade without fading and upgrading

the thickness of paint layer. All these strategies prevent the degradation of quality and satisfy the customer. In this

direction, Xie et al (2011) investigated a quality investment and price decision of a make-to-order supply chain with

uncertain demand in international trade. Qin et al (2014) considered the pricing and lot-sizing problem for products

with quality and physical quantity deteriorating simultaneously. Rabbani et al (2016) formulated an optimal pricing

and replenishment policies for items with simultaneous deterioration of quality and quantity. Recently, Feng (2018)

presented an optimal replenishment model with dynamic pricing and quality investment for perishable products, where

the quality and quantity deterioration simultaneously.

The product’s demand is an important factor for the seller to maximize the profit. Many of the researchers have

worked on price, time dependent demand, but now a days the product’s selling is not based only on the price and time

but also on the quality, promotional effort etc. In different models, researchers use different demand patterns. As Pal

and Maity (2012) explored an inventory model for deteriorating items when demand for the item is dependent on the

selling price with permissible delay in payment and inflation. Mishra (2013) developed an inventory model of

instantaneous deteriorating items which can be controlled by preservation technology and holding cost for time

dependent demand. Shukla (2013) presented an inventory model for deteriorating items considering a parametric

dependent linear function of time and price dependent demand. Cardenas-Barron and Sana (2015) developed a multi-

item EOQ inventory model in a two layer supply chain while demand varies with promotional effort. Liu et al (2015)

formulated an inventory model for perishable foods, in which the demand depends on the price and quality that decays

continuously. Banerjee and Agrawal (2017) presented an inventory model for deteriorating items with freshness and

price dependent demand. Roy and Giri (2018) developed a three-echelon supply chain model with price and two level

quality dependent demand.

Promotion refers to the entire set of activities, which communicate the product, brand or service to the user. The

idea is to make people aware, attract and induce to buy the product, in preference over others. Normally, the seller

promotes the product to attract the customer by giving discount, coupon, free gift etc., but in today’s era seller promotes



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the product by press releases, incentive trips, online banner advertisement, social networking, websites and blogs. In

this direction, Palanivel and Uthayakumar (2015) presented an EPQ model for deteriorating items involving

probabilistic deterioration where the demand is dependent on sales team’s initiatives. Gahan and Pattnaik (2017)

analysed the instantaneous EOQ profit optimization model of deteriorating items for the impact of variable ordering

cost and promotional effort cost for leveraging profit margins in finite planning horizons. Palanivel et al (2017)

considered a two-warehouse (owned and rented) inventory problem for a non-instantaneous deteriorating item with

inflation and time value of money over a finite planning horizon. Rajan and Uthayakumar (2017) analysed an EOQ

inventory model with promotional effort dependent demand and back ordering under delay in payments. Soni and

Suthar (2018) investigated an inventory model of pricing and inventory decisions for non-instantaneous

deteriorating items with partially backlogging where demand is price and promotional effort dependent and

stochastic in nature. Recently, Soni and Chauhan (2018) developed a joint pricing, inventory and preservation

decision making problem for time dependent deterioration and partially backlogged subject to price dependent

stochastic demand and promotional effort.

The considered model incorporates the following features: (1) Quality and quantity deterioration of the product

simultaneously. (2) Price, quality level and promotional effort sensitive demand rate. (3) Quality investment for quality

deterioration. (4) Inventory starts with shortages and ends with zero.

The remainder of this paper is organized as follows. Section 2 lists the notation and assumptions used in this

paper. Section 3 develops the mathematical modelling. Section 4 provides a numerical example and sensitivity analysis

to illustrate the proposed model. Finally, a conclusion is made in Section 5.

2. NOTATIONS AND ASSUMPTIONS

2.1 Notations

Decision variables

1t The duration of shortage period

2t The length of time after which inventory reaches to zero

ξ The quality investment cost per unit

ρ The promotional effort, 1ρ ≥

Parameters

θ The decay rate of quantity

O The Ordering cost per order



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C The purchasing cost per unit

hC The holding cost per unit per time unit

bC The backorder cost per unit per time unit

lC

The lost sale cost per unit

p The selling price per unit

Variables

( )1I t The level of negative inventory at time t , 10 t t≤ ≤

( )2I t The level of positive inventory at time t , 1 2t t t≤ ≤

α The decay rate of quality

( )f ξ The proportion of reduced quality rate

( )N t The quality level at time

Q The ordering quantity per cycle

( )( ), ,D p N tρ The demand rate

( )1 2, , ,t t ρ ξΠ The total profit per time unit of the inventory system

2.2 Assumptions

(These assumptions are mainly adopted from Soni and Chauhan 2018):

1. The inventory system involves single deteriorating item.

2. Demand rate ( )( ), ,D p N tρ is a function of the selling price p , promotional effort ρ and quality level ( )N t

as

( )( ) ( ) ( ), , ,D p N t d p N tρ ρ= +

( ) ( )( )( )( )1

0 1

11

1

f t td d p e

α ξµ

ρ

− − − = − + − +

+ (1)

Where 0 1, ,d d µ andα are positive constant.

3. Deterioration involves both quality and physical quantity. Qualitative deterioration is instantaneous.

4. The quality investment cost per unit time for reducing the instantaneous deterioration rate, 0 wξ≤ ≤ , where w

is the maximum cost of investment in quality investment and quality investment cost is

( ) ( )1 af e

ξξ α −= − (2)

Where α and a are positive constant.



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5. The promotional effort cost is an increasing function of the promotional effort and the basic demand

( ) ( )( )1

2

0

1 , ,K D p N t dt

λ

ρ ρ

− , where 0K > and λ is a constant. Both the market demand and the

cost of promotional effort will increase as the promotional effort increases.

6. Shortages are allowed. The inventory model starts with shortages and ends with zero inventory. During stock-out

period some fraction of demand is backordered and rest of the demand is partially backlogged which is decreasing

function of time t , denoted as ( )v t .

7. Replenishment rate is infinite but its order size is finite.

3. MODEL FORMULATION

We consider two time intervals in this inventory model. In the interval [ ]10, t , shortages occur at the starting of

the period which are partially backlogged. This backlogged demand is satisfied at the replenishment point 1t and rest

of the lot size is adjusted up to the time 2t . In the interval [ ]1 2,t t , the inventory level is decreasing only due to demand

and two types of deterioration qualitative and quantitative. Qualitative deterioration is reduced by quality investment.

This process is repeated as mentioned above. The pattern of inventory level is depicted in figure.

Figure 1: Graphical presentation of inventory level

O 1t

2t Backorder

Lost sales

On

han

d I

nven

tory



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During the time interval [ ]1 2,t t the quality status ( )N t starts to decrease over time instantaneously. The quality

status is represented by the following differential equation.

( )

( )( ) ( )dN t

f N tdt

α ξ= − − 1 2t t t≤ ≤ (3)

With boundary condition ( )1 1N t = solving equation (3) yields

( ) ( )( )( )1f t tN t e

α ξ− − −= (4)

The status of negative inventory at any instant of time [ ]10,t t∈ is governed by differential equation:

( )( )( ) 11 ( ), , v t t

dI tD p N t e

dtρ − −= − , 10 t t≤ ≤ (5)

With boundary condition ( )1 0 0I = solving equation (5) yields

( )( )( ) ( )( )11

1

, ,v t tvt

D p N t e eI t

v

ρ − −− −= (6)

The positive inventory level declines due to demand and physical deterioration during time interval [ ]1 2,t t . Based

on this description, the inventory status is represented by the following differential equation.

( )( ) ( )( )2

2 , ,dI t

I t D p N tdt

θ ρ= − − (7)

With boundary condition ( )2 2 0I t = solving equation (4) yields:

( )( )( ) ( )( )2

2

, , 1t t

D p N t eI t

θρ

θ

− −−

= (8)

The seller’s order quantity is

( ) ( )2 1 10Q I I t= −

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

1, , 0 1 , , 1t vtD p N e D p N t e

v

θρ ρ

θ

− − − = −

(9)

The lost sale quantity is



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( ) ( )( ) ( )( )1, , N 1v t t

L t D p t eρ − −= − , 10 t t≤ ≤

The components of total profit of the inventory system are defined as follows:

1. :SR The sales revenue

( )( ) ( )( )2

1 1

0

, ,

t

SR p D p N t dt I tρ= + −

( )( )( )( ) ( )12

0

, , 1, ,

vtt D p N t ep D p N t dt

v

ρρ

− −= − (10)

2. :OC The ordering cost

OC O= (11)

3. :HC The inventory holding cost

( )2

2

0

t

hHC C I t dt=

( )( ) ( )( )22

0

, , 1t tt

h

D p N t eC dt

θρ

θ

− −−

= (12)

4. :PC The purchasing cost

PC CQ=

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

1, , 0 1 , , 1t vtD p N e D p N t e

Cv

θρ ρ

θ

− − − = −

(13)

5. QIC: The quality investment cost

( )1 2QIC t t ξ= + (14)

6. LC: The lost cost

( )1

0

t

lLC C L t dt=

( )( ) ( )( )1

1

0

, , 1

t

v t t

lC D p N t e dtρ − −= − (15)

7. BC: The back logged cost



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( )( )1

1

0

t

bBC C I t dt= −

( )( ) ( )( )111

0

, ,v t tvtt

b

D p N t e eC dt

v

ρ − −− −= − (16)

8. PEC: Promotional effort cost

( ) ( )( )1 2

2

0

1 , ,

t t

PEC K D p N t dt

λ

ρ ρ+

= − (17)

Therefore, the total profit per time unit ( )1 2, , ,t t ρ ξΠ is given by:

( ) ( )1 2

1 2

1, , ,t t SR OC QIC HC LC BC PC PEC

t tρ ξΠ = − − − − − − −

+

( )( )( )( )( )

( )12

1 2

1 2 0

, , 11, ,

vtt D p N t ep D p N t dt O t t

t t v

ρρ ξ

− −= − − − +

+

( )( ) ( )( )

( )( ) ( )( )2

2 1

1

0 0

, , 1, , 1

t tt t

v t t

h l

D p N t eC dt C D p N t e dt

θρρ

θ

− −

− −−

− − −

( )( ) ( )( )111

0

, ,v t tvtt

b

D p N t e eC dt

v

ρ − −− −+

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

1, , 0 1 , , 1t vtD p N e D p N t e

Cv

θρ ρ

θ

− − − − −

( ) ( )( )1 2

2

0

1 , ,

t t

K D p N t dt

λ

ρ ρ+

− −

(18)

4. NUMERICAL EXAMPLE AND SENSITIVITY ANALYSIS

In this section, a numerical example is given to illustrate the above solution procedure. We consider

( ) ( )( )( )1f t tN t e

α ξ− − −= where ( ) ( )1 a

f eξξ α −= − . The solution of this example and the algorithm was

implemented in Maple 18.

4.1 Numerical example



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Example 1. Consider the following data ( )( ) ( )( )( )( )1

0 1

1, , 1

1

f t tD p N t d d p e

α ξρ µ

ρ

− − − = − + − +

+ ,

0 120d = , 1 0.5, 20d µ= = 80 / /p unit year= , 40α = , 0.1a = , 0.9θ = , $35 / unit/ yearC = ,

$15 / unit/ yearb

C = , $5 / /h

C unit year= , $17 /l

C unit= , $120 /O unit= , 0.6v = , 1.5λ = ,

0.5K = .

With the given data, the optimal results are:*

1 0.1579t = ,*

2 0.1737t = ,* 20.63ξ = ,

* 1.30ρ = ,

* 3407.75Π = and * 31.51Q = .

4.2 Sensitivity analysis

From the previous numerical example, a sensitivity analysis is performed to study the effect of estimating

the parameters on the values of the total profit per unit time. The result can be found by Maple 18 and the results

are presented in Tables.

Table 1. Sensitivity analysis with respect to parameters

Parameter Value of

parameter 1t 2t ρ ξ Π Q

0d

96 0.1852 0.1954 1.40 21.88 2432.71 26.98

108 0.1700 0.1836 1.34 21.22 2917.92 29.33

120 0.1580 0.1738 1.30 20.64 3407.76 31.51

132 0.1481 0.1654 1.27 20.12 3901.26 33.56

144 0.1398 0.1582 1.24 19.66 4397.74 35.49

1d

0.4 0.1512 0.1681 1.28 20.29 3736.40 32.89

0.45 0.1545 0.1709 1.29 20.46 3571.89 32.21

0.5 0.1580 0.1738 1.30 20.64 3407.76 31.51

0.55 0.1617 0.1769 1.32 20.82 3244.03 30.80

0.6 0.1657 0.1802 1.33 21.02 3080.74 30.07

µ

16 0.1611 0.1765 1.26 20.80 3315.65 31.24

18 0.1595 0.1752 1.28 20.72 3361.54 31.38

20 0.1580 0.1738 1.30 20.64 3407.76 31.51

22 0.1564 0.1724 1.32 20.56 3454.26 31.64

24 0.1549 0.1711 1.34 20.48 3501.04 31.77

K

0.4 0.1574 0.1733 1.36 20.61 3413.31 31.48

0.45 0.1577 0.1736 1.33 20.63 3410.30 31.49

0.5 0.1580 0.1738 1.30 20.64 3407.76 31.51

0.55 0.1582 0.1740 1.28 20.65 3405.59 31.52

0.6 0.1584 0.1742 1.26 20.66 3403.72 31.54

λ

1.2 0.1578 0.1740 1.63 20.67 3439.04 31.89

1.35 0.1577 0.1737 1.44 20.64 3421.57 31.65

1.5 0.1580 0.1738 1.30 20.64 3407.76 31.51

1.65 0.1585 0.1743 1.20 20.66 3397.45 31.47

1.8 0.1592 0.1749 1.13 20.69 3390.18 31.49

θ

0.72 0.1486 0.2017 1.29 22.93 3441.51 33.34

0.81 0.1535 0.1868 1.30 21.73 3423.65 32.36

0.9 0.1580 0.1738 1.30 20.64 3407.76 31.51

0.99 0.1621 0.1623 1.30 19.63 3393.53 30.78

1.08 0.1659 0.1521 1.30 18.69 3380.75 30.13

0.48 0.1784 0.1648 1.29 19.37 3434.09 32.49

0.54 0.1675 0.1697 1.30 20.05 3420.22 31.97

0.6 0.1580 0.1738 1.30 20.64 3407.76 31.51



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v 0.66 0.1496 0.1774 1.30 21.17 3396.47 31.11

0.72 0.1421 0.1806 1.30 21.66 3386.19 30.75

a

0.08 0.1585 0.1719 1.30 24.46 3402.74 31.39

0.09 0.1582 0.1729 1.30 22.36 3405.49 31.46

0.1 0.1580 0.1738 1.30 20.64 3407.76 31.51

0.11 0.1577 0.1745 1.30 19.19 3409.66 31.55

0.12 0.1575 0.1752 1.30 17.96 3411.27 31.59

α

32 0.1580 0.1738 1.30 18.41 3409.99 31.51

36 0.1580 0.1738 1.30 19.59 3408.81 31.51

40 0.1580 0.1738 1.30 20.64 3407.76 31.51

44 0.1580 0.1738 1.30 21.59 3406.81 31.51

48 0.1580 0.1738 1.30 22.46 3405.94 31.51

The following results are obtained from Table 1.

1. A constant parameter 0d of the demand function impacts optimal total profit per unit ( )Π , dramatically. Optimal

total profit per unit ( )Π is highly sensitive to the parameters µ and 1d . As 1d is the scaling parameter of the

price and µ is the scaling parameter of the promotional effort in demand function.

2. As the value of parameters K , λ ,θ , v and α increases, the optimal total profit per unit ( )Π decreases. These

effects of these changes are apparent.

3. The optimal order quantity Q is highly sensitive to the constant parameter of the demand 0d , moderately sensitive

to the 1d ,θ and v whereas here no significant impact of µ , K , λ and a on optimal order quantity ( )Q .

4. Optimal promotional effort ( )ρ is sensitive to the parameter 0d and λ . Hence, the error in estimating this

parameter causes erroneous policy about promotional effort.

5. The optimal quality investment cost ( )ξ increases with the increase in 1d , K , v andα , whereas it decreases

with the increase in µ , θ and a . Decision maker has to be more conscious while estimating the parametersθ , a

andα to determine the proper quality investment. Quality investment has much influence of the quality varying

deterioration α and physical deterioration rateθ . Hence, it should be carefully selected.

• To provide a better picture of the optimal total profit, optimal promotional effort, optimal quality investment cost

and optimal order quantity in relation to the parameters, the sensitivity analysis for the model is also depicted in

Figure 2-4.



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Figure 2: Sensitivity analysis of the model with respect to 0 1,d d and µ .

Figure 3: Sensitivity analysis of the model with respect to K and λ .



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Figure 4: Sensitivity analysis of the model with respect to ,vθ and a .

5. CONCLUSION

This paper is concerned with the inventory model of joint inventory, promotional effort and quality investment

for quality deteriorating products where Inventory model initiate with accumulation of shortages. The demand is

characterized on the pricing, quality investment and promotional effort. When we consider all this three together

in demand function then it would have much more impact on total profit per unit time. To maintain or improve

quality level we have to apply quality investment in it. The results obtained in this paper have the following

contributions and important managerial implications. First, the model is appropriate for the products whose quality

and quantity deteriorate simultaneously over time. Second, the main objective of the inventory model is to

maximize the total profit per unit. Third, quality investment for quality deterioration. Next, the numerical example

and sensitivity analysis of the key parameters of the model are provided. This sensitivity analysis reveals that the

promotional effort in demand function is much more effective on profit maximization compared to promotional

effort cost. As well as, parameter of pricing in demand function is also very effective on profit maximization.

References Banerjee S., Agrawal S. (2017) Inventory model for deteriorating items with freshness and price dependent demand: Optimal discounting and ordering

policies. Applied Mathematical Modelling, 52:53-64.

Barron L. E. C., Sana S. S. (2015) Multi-item EOQ inventory model in a two-layer supply chain while demand varies with promotional effort. Applied

Mathematical Modelling, 39(21):6725-6737.

Feng L. (2018) Dynamic pricing, quality investment and replenishment model for perishable items. International Transactions in Operational Research,

1-18. DOI: 10.1111/itor.12505.



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Gahan P., Pattnaik M. (2017) Impact of variable ordering cost and promotional effort cost in deteriorated economic order quantity EOQ model.

International Journal of Advanced Engineering, Management and Science, 3(3):178-185.

Liu G., Zhang J., Tang W. (2015) Joint dynamic pricing and investment strategy for perishable foods with price-quality dependent demand. Annals of

Operations Research, 226(1):397-416.

Mandal P., Giri B. C. (2015) A single-vendor multi-buyer integrated model with controllable lead time and quality improvement through reduction in

defective items. International Journal of Systems Science: Operations &Logistics, 2(1):1-14.

Mishra V. K. (2013) An inventory model of instantaneous deteriorating items with controllable deterioration rate for time dependent demand and holding

cost. Journal of Industrial Engineering and Management, 6(2):495-506.

Pal M., Maity H. K. (2012) An inventory model for deteriorating items with permissible delay in payment and inflation under price dependent demand.

Pakistan Journal of Statistics and Operation Research, 8(3):583-592.

Palanivel M., Uthayakumar R. (2015) A production inventory model with promotional effort, variable production cost and probabilistic deterioration.

International Journal of System Assurance Engineering and Management. 8(S1):290-300.

Palanivel M., Priyan S., Mala P. P. (2017) Two warehouse system for non-instantaneous deterioration products with promotional effort and inflation over

a finite time horizon. Journal of Industrial Engineering International, 14(3):603-612.

Qin Y., Wang J., Wei C. (2014) Joint pricing and inventory control for fresh produce and foods with quality and physical quantity deteriorating

simultaneously. International Journal of Production Economics, 152:42-48.

Rabbani M., Zia N. P., Rafiej H. (2016) Joint optimal dynamic pricing and replenishment policies for items with simultaneous quality and physical

quantity deterioration. Applied Mathematics and Computation, 287-288:149-160.

Rajan R. S., Uthayakumar R. (2017) Analysis and optimization of an EOQ inventory model with promotional efforts and back ordering under delay in

payments. Journal of Management Analytics, 4(2):159-181.

Roy B., Giri B. C. (2018) A three-echelon supply chain model with price and two-level quality dependent demand. RAIRO-Operations

Research. https://doi.org/10.1051/ro/2018066.

Shukla D., Khedlekar U. K., Chandel R. P. (2013) Time and price dependent demand with varying holding cost inventory model for deteriorating items.

International Journal of Operations Research and Information Systems, 4(4):75-95.

Soni H. N., Chauhan A. D. (2018) Joint pricing, inventory and preservation decisions for deteriorating items with stochastic demand and promotional

efforts. Journal of Industrial Engineering International, https://doi.org/10.1007/s40092-018-0265-7.

Soni H. N., Suthar D. N. (2018) Pricing and inventory decisions for non-instantaneous deteriorating items with price and promotional effort stochastic

demand. Journal of Control and Decision, 1-25. https://doi.org/10.1080/23307706.2018.1478327.

Xie G., Yue W., Wang S., Lai K.K. (2011) Quality investment and price decision in a risk-averse supply chain. European Journal of Operational

Research, 214(2):403-410.



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