research article optimal pricing and ordering policy for

Research ArticleOptimal Pricing and Ordering Policy for Deteriorating Itemswith Stock-and-Price Dependent Demand and Presale Rebate

Lianxia Zhao1 and Jianxin You2

1School of Management Shanghai University Shanghai 200444 China2School of Economic and Management Tongji University Shanghai 200092 China

Correspondence should be addressed to Lianxia Zhao zhaolianxiastaffshueducn

Received 2 August 2016 Accepted 3 November 2016

Academic Editor Xinchang Wang

Copyright copy 2016 L Zhao and J You This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper considers an EOQ inventory model with presale policy for deteriorating items in which the demand rate depends onboth on-hand inventory and selling price Under the assumption that all the presale orders are fully backlogged with waiting-timedependent rebate this study develops several propositions and derives optimal pricing and ordering policy by designing an effectivealgorithm Two numerical examples are also given to illustrate the effectiveness of the algorithm Finally the sensitivity analysis ofthe main parameters is provided

1 Introduction

A large number of products have deteriorating propertywhich manifests the decay or devaluation This kind ofproducts has much shorter life cycle for example someseasonal fresh products that cannot be stored for a long timeeven in modern storage conditions electronic products witha fast updating speed and fashion clothing These productsshould be sold in a much shorter spot-sale period to reducedeteriorating cost and holding cost Presale policy is aneffective marketing strategy to reduce the spot-sale periodwhich is that customers are encouraged to order in advanceto get a rebate related to the preorder time All the orderswill be satisfied when the products launch into the market Inaddition it is crucial to find out what factors mostly impactthe demand of product so we can decide proper orderingquantity

It is obvious that the price of product highly influencesthe demand rate thus the pricing for the product plays animportant role in marketing strategy Meanwhile the stocklevel of product can also affect the demand rate Manyresearchers have focused on such topic Hou and Lin [1]considered an EOQ inventory model for deteriorating itemswith price-and-stock dependent selling rate in which theproposed model allowed shortage and complete backorder

while the shortage cost is constant You and Hsieh [2]developed an inventory model for a seasonal item over afinite planning time by determining the optimal orderingquantity and price settingchanging strategy Chang et al [3]studied the optimal selling price and order quantity underEOQmodel for deteriorating items and the demand dependson the selling price and stock on display Dye and Hsieh[4] developed an inventory model for deteriorating itemswith demand rate depending on selling price and stockand shortages are allowed with the backlogging rate to bea decreasing function of the waiting time while Giri andBardhan [5] studied an integrated single-manufacturer andsingle-retailer supply chain model for deteriorating itemswith stock-and-price dependent demand Relevant referencescan be found in Teng and Chang [6] Panda et al [7] and soforth Furthermore more inventory models for deterioratingitems with a variety of demand rate are summarized inTable 1 The difference between the relevant references andour study can be found in the table

In our study presale rebate depending on waiting time isgiven to encourage customers to order in advance Althoughpresale rebate cost is similar in form to the traditionalshortage cost it is different inmeaning moreover the presalerebate cost in this paper is more complex and realistic thanthe traditional shortage cost which is generally considered to

Hindawi Publishing CorporationScientific ProgrammingVolume 2016 Article ID 1507285 7 pageshttpdxdoiorg10115520161507285

2 Scientific Programming

Table 1 Our paper versus literatures for key assumptions of inventory models with deteriorating items

Authorauthors Demand rate Deterioration rate Shortage or not Shortage cost Decision variablelowast

Skouri et al [8] Time-dependent Weibull distribution Yes Constant 119905 119879Sicilia et al [9] Time-dependent Constant Yes Constant 119905 119879Cheng et al [10] Time-dependent Constant Yes Constant 119905Zhao [11 12] Time-dependent Weibull distribution Yes Constant 119905Begum et al [13] Price-dependent Weibull distribution No No 119901 119879Dye [14] Price-dependent Time-dependent Yes Constant 119901 119905 and 119879Yang et al [15] Price-dependent No Yes Constant 119905 119879B Sarkar and S Sarkar [16] Stock-dependent Time-dependent Yes Constant 119905 119879Padmanabhan and Vrat [17] Stock-dependent Constant Yes Constant 119905 119879Avinadav et al [18] Price-and-time Constant No No 119901 119876 and 119879Maihami and Nakhai Kamalabadi [19] Price-and-time Noninstantaneous Yes Constant 119901 119879Hou and Lin [1] Price-and-stock Constant Yes Constant 119905 119879Giri and Bardhan [5] Price-and-stock No No No 119905 119899This paper Price-and-stock Constant Presale Time-dependent 119901 119905 and 119879lowast119901 = selling price 119905 = replenishmentout of stock time 119876 = order quantity 119879 = inventory cycle and 119899 = the number of replenishment

be constant In addition we assume the demand is price-and-stock dependent according to the actual situation Eventuallywe derive optimal pricing and ordering policy by designingan effective algorithm and provide the sensitivity analysis ofthe parameters to assess their effects on the optimal policy

The rest of the paper is organized as follows we introducesome basic notations and assumptions in Section 2 Section 3establishes an inventory model under presale policy andprovides an effective procedure to find the optimal pricingand presale strategy In Section 4 we use several numericalexamples to illustrate the procedure of the optimal strategiesand analyze the sensitivity of parameters involved Finally weprovide a summary of the paper

2 Notation and Assumptions

The fundamental notation and assumptions used in thispaper are given as below

21 Notation

119888119901 the purchase price of unit item1198600 the fixed cost per order119878 the maximum inventory level119876 the ordering quantity119901 the selling price of unit item120579 the constant deteriorating rate 0 le 120579 ≪ 11199051 presale period1199052 spot-sale period119868(119905) the level of inventory at time 119905 0 le 119905 le 119879 where119879 = 1199051 + 1199052119888119889 the cost of each deteriorated item119888ℎ the inventory holding cost per unit per time119863(119905 119901) the demand rate

119888119904(119909) the presale rebate costprod(1199051 1199052 119901) the average total profit per unit time

22 Assumptions

(i) The replenishment rate is infinite that is replenish-ment is instantaneous

(ii) Assume the customers are loyal and all the presaleorders will not be canceled until the products arelaunched into the market

(iii) The demand rate119863(119905 119901) we assumed it to be

119863(119905 119901) = 119892 (119901) 0 le 119905 le 1199051119892 (119901) + 120572119868 (119905) 1199051 le 119905 le 119879 (1)

where 119892(119901) is a function of 119901 1198921015840(119901) lt 0 and 11989210158401015840(119901) ge0 where 120572 denotes the stock-dependent consumptionrate

(iv) 119888119904(119909) is the presale rebate determined by the waitingtime 119909 that is 119888119904(119909) = 1198880(119890120582119909 minus 1) where 1198880 is positiveconstant and 120582 isin (0 1)

(v) The marginal gross profit with respect to price isdecreasing that is (119901 minus 119888119901)119892(119901) is a strictly concavefunction of 119901

3 Inventory Model with Presale Policy

In this section we consider an inventory model with presalepolicy for deteriorating items and the behavior of the systemis depicted in Figure 1 From 119905 = 0 to 1199051 the total presalequantity accumulates on account of demand which is fullybacklogged and achieves its maximum at time 119905 = 1199051 Duringthe time interval [1199051 1199051 + 1199052) due to customersrsquo demand anddeterioration on-hand inventory level gradually decreases to

Scientific Programming 3

0

Inventorylevel

Time

I (t)

t1 t2

t1 + t2

Figure 1 Graphical representation of inventory level over the cycle

zero According to the above notations and assumptions suchinventory system can be described by following differentialequations

119889119868 (119905)119889119905 = minus119892 (119901) 0 le 119905 le 1199051119889119868 (119905)119889119905 = minus120579119868 (119905) minus 119892 (119901) minus 120572119868 (119905) 1199051 le 119905 le 1199051 + 1199052

(2)

With boundary condition 119868(0) = 0 119868(1199051 + 1199052) = 0The solution of (2) can be obtained as

119868 (119905) = minus119892 (119901) 119905 0 le 119905 le 1199051119868 (119905) = 119892 (119901)

120572 + 120579 (119890(120572+120579)(1199051+1199052minus119905) minus 1) 1199051 le 119905 le 1199051 + 1199052(3)

The maximum inventory level per cycle is

119868 (119905+1 ) = 119892 (119901)120572 + 120579 (119890(120572+120579)1199052 minus 1) (4)

The presale quantity per cycle is

119868 (119905minus1 ) = minus119892 (119901) 1199051 (5)

The ordering quantity over the inventory cycle is

119876 = 119868 (119905+1 ) minus 119868 (119905minus1 ) = 119892 (119901)120572 + 120579 (119890(120572+120579)1199052 minus 1) + 119892 (119901) 1199051 (6)

Based on (3) the total cost per cycle consists of the followingelements

(i) Ordering cost 1198600(ii) Purchase cost

119888119901 [119892 (119901)120572 + 120579 (119890(120572+120579)1199052 minus 1) + 119892 (119901) 1199051] (7)

(iii) Sales revenue

119901[119892 (119901) 1199051 + int1199051+11990521199051

(119892 (119901) + 120572119868 (119905)) 119889119905] (8)

(iv) Deteriorating cost

119888119889 [119892 (119901)120572 + 120579 (119890(120572+120579)1199052 minus 1) minus int1199051+1199052

1199051

(119892 (119901) + 120572119868 (119905)) 119889119905] (9)

(v) Holding cost

119888ℎ119892 (119901)120572 + 120579 int1199051+11990521199051

(119890(120572+120579)(1199051+1199052minus119905) minus 1) 119889119905 (10)

(vi) Presale rebate cost

1198880119892 (119901)int11990510(119890120582(1199051minus1) minus 1) 119905 119889119905 (11)

Therefore the total profit per unit time of the model can beobtained as follows

prod(1199051 1199052 119901) = 11199051 + 1199052 [sales revenue minus holding cost

minus ordering cost minus presale rebate cost

minus deteriorated items cost minus purchase cost]= minus 119892 (119901)

1199051 + 1199052 Δ

(120572 + 120579)2 (119890(120572+120579)1199052 minus 1)+ 11988801205822 (1198901205821199051 minus 12120582211990521 minus 1205821199051 minus 1) minus (119901 minus 119888119901) 1199051minus 1120572 + 120579 (119901120579 + 119888ℎ + 119888119889120579) 1199052 minus 11986001199051 + 1199052

(12)

where Δ = 119888ℎ + (120572 + 120579)119888119901 + 119888119889120579 minus 119901120572Taking the first-order partial derivative of prod(1199051 1199052 119901)

with respect to 1199051 1199052 and 119901 respectively we have120597prod (1199051 1199052 119901)1205971199051 = minusprod(1199051 1199052 119901)1199051 + 1199052 + 119892 (119901)

1199051 + 1199052 [(119901 minus 119888119901)minus 1198880120582 (1198901205821199051 minus 1205821199051 minus 1)]

120597prod (1199051 1199052 119901)1205971199052 = minusprod(1199051 1199052 119901)1199051 + 1199052+ 119892 (119901)1199051 + 1199052 [

119901120579 + 119888ℎ + 119888119889120579120572 + 120579 minus Δ120572 + 120579119890(120572+120579)1199052] 120597prod (1199051 1199052 119901)120597119901 = 11199051 + 1199052

119892 (119901) 120572 minus 1198921015840 (119901) Δ(120572 + 120579)2 [119890(120572+120579)1199052

minus (120572 + 120579) 1199052 minus 1] minus 11988801198921015840 (119901)1205822 (1198901205821199051 minus 12120582211990521 minus 1205821199051minus 1) + 119892 (119901) + 1198921015840 (119901) (119901 minus 119888119901)

(13)


For any given selling price 119901 the optimal solution (119905lowast1 119905lowast2 ) isdetermined by the following equations

120597prod (1199051 1199052 119901)1205971199051 = 0120597prod (1199051 1199052 119901)1205971199052 = 0

(14)

By simplifying (14) we have

prod(1199051 1199052 119901) minus 119892 (119901) [(119901 minus 119888119901) minus 1198880120582 (1198901205821199051 minus 1205821199051 minus 1)]= 0

(15)

prod(1199051 1199052 119901) minus 119892 (119901) [119901120579 + 119888ℎ + 119888119889120579120572 + 120579 minus Δ120572 + 120579119890(120572+120579)1199052]= 0

(16)

From the analysis of (15) and (16) we have the followingpropositions

Proposition 1 For any given price 119901 (i) if Δ le 0 then theredoes not exist any optimal solution (1199051 1199052) so as to maximizethe profit prod(1199051 1199052 119901) (ii) if Δ gt 0 then the optimal solution(119905lowast1 119905lowast2 )which solves (15) and (16) simultaneously not only existsbut also is unique

Proof (i) From (15) and (16) we have

Δ120572 + 120579 (119890(120572+120579)1199052 minus 1) = 1198880120582 (1198901205821199051 minus 1205821199051 minus 1) (17)

If Δ lt 0 it is obvious that there does not exist any nonzerofeasible solution to satisfy (17) which means that the optimalsolution of the maximum profitprod(1199051 1199052 | 119901) does not exist

If Δ = 0 by solving (17) we have 1199051 = 0 Substitutingit into (16) we have prod(0 1199052 119901) = ((119901120579 + 119888ℎ + 119888119889120579)(120572 +120579))119892(119901) Thus from (12) we obtain 1198600 = 0 which meansa contradictory Therefore if Δ lt 0 there does not exist anyoptimal solution (1199051 1199052) to minimize the profitprod(1199051 1199052 | 119901)

(ii) Let

119891 (119909) = Δ120572 + 120579 (119890(120572+120579)119909 minus 1) minus 1198880120582 (1198901205821199051 minus 1205821199051 minus 1) (18)

Taking the first-order derivative of (18) with respect to 119909 ityields

1198911015840 (119909) = Δ119890(120572+120579)119909 ge 0 (19)

which means that 119891(119909) is a nondecreasing function in[0 +infin)Similarly we know that the right hand-side of (17) is a

strictly increasing function of 1199051 and goes to infinite as 1199051 rarrinfinOn the other hand from (17) we have Δ119890(120572+120579)1199052(11988911990521198891199051) = 1198880(1198901205821199051 minus 1) gt 0 which means 11988911990521198891199051 gt 0 and1199052 = 1120572 + 120579 ln [(120572 + 120579) 1198880120582Δ (1198901205821199051 minus 1205821199051 minus 1) + 1] (20)

Therefore for any given 1 there exists a unique 2 such that2 = (1(120572 + 120579))ln[((120572 + 120579)1198880120582Δ)(1198901205821 minus 1205821 minus 1) + 1]To prove the uniqueness of the solution substitute (12)

into (15) and let

119865 (1199051) = minus119892 (119901) [ Δ(120572 + 120579)2 (119890(120572+120579)1199052 minus 1) + (119901 minus 119888119901) 1199052

minus 1199052120572 + 120579 (119901120579 + 119888ℎ + 119888119889120579)] minus 1198600 minus1198880119892 (119901)1205822 [1198901205821199051

minus 12120582211990521 minus 1205821199051 minus 1 minus 120582 (1198901205821199051 minus 1205821199051 minus 1) (1199051 + 1199052)] (21)

Taking the first-order derivative of119865(1199051)with respect to 1199051 wehave

119889119865 (1199051)1198891199051 = minusΔ119892 (119901)(120572 + 120579) (119890(120572+120579)1199052 minus 1) 11988911990521198891199051+ 1198880119892 (119901)120582 [(1198901205821199051 minus 1205821199051 minus 1) 11988911990521198891199051+ 120582 (1198901205821199051 minus 1) (1199051 + 1199052)] = 1198880119892 (119901) (1198901205821199051 minus 1) (1199051+ 1199052) ge 0

(22)

which means that 119865(1199051) is an increasing function in [0 +infin)Using (17) it deduces 1199052 = 0 as 1199051 = 0 then we have 119865(0) =minus1198600 lt 0 and lim119905

1rarr+infin119865(1199051) = +infin With the Intermediate

Value Theorem we can find a unique root 119905lowast1 isin [0 +infin) suchthat 119865(119905lowast1 ) = 0

From the above analysis it is concluded that if Δ gt0 the optimal solution (119905lowast1 119905lowast2 ) which solves (15) and (16)simultaneously not only exists but also is unique

Proposition 2 For any given119901 the solution (119905lowast1 119905lowast2 ) of (15) and(16) simultaneously is the globalmaximumof the profit per unittime

Proof Taking the second-order partial derivative ofprod(1199051 1199052 |119901) with respect to 1199051 and 1199052 respectively we have1205972prod(1199051 1199052 | 119901)12059711990521

100381610038161003816100381610038161003816100381610038161003816(11990511199052)=(119905lowast1119905lowast2)

= minus1198880119892 (119901)119905lowast1 + 119905lowast2 (119890120582119905lowast1 minus 1)

lt 0(23)

1205972prod(1199051 1199052 | 119901)12059711990522100381610038161003816100381610038161003816100381610038161003816(11990511199052)=(119905lowast1119905lowast2)

= minusΔ119892 (119901)119905lowast1 + 119905lowast2 119890(120572+120579)119905lowast

2 lt 0 (24)


= 0 (25)


Then the Hessian matrix satisfies

|119867| = 1205972prod(1199051 1199052 | 119901)12059711990521100381610038161003816100381610038161003816100381610038161003816(11990511199052)=(119905lowast1119905lowast2)

times 1205972prod(1199051 1199052 | 119901)12059711990522100381610038161003816100381610038161003816100381610038161003816(11990511199052)=(119905lowast1119905lowast2)

minus [ 1205972prod(1199051 1199052 | 119901)12059711990511205971199052100381610038161003816100381610038161003816100381610038161003816(11990511199052)=(119905lowast1119905lowast2)

]2

gt 0

(26)

Therefore we conclude that the stationary point (119905lowast1 119905lowast2 ) is aglobal optimal solution for the considered problem

In the following we will show that the optimal sellingprice also exists and is unique for the problem

Proposition 3 For any given (1199051 1199052) there exists a uniqueoptimal selling price such thatprod(119901 | 1199051 1199052) is maximized

Proof For any given 1199051 and 1199052 the first-order derivation ofprod(119901 | 1199051 1199052) with respect to 119901 is

119889prod (119901 | 1199051 1199052)119889119901= 11199051 + 1199052

119892 (119901) 120572 minus 1198921015840 (119901) Δ(120572 + 120579)2 [119890(120572+120579)1199052


(27)

Because1198921015840(119901) lt 0 we have119892(119901)120572minus1198921015840(119901)Δ gt 0Thus119889prod(119901 |1199051 1199052)119889119901 = 0 provides a solution only if119892(119901)+(119901minus119888119901)1198921015840(119901) lt0Let

119866 (119901) = 11199051 + 1199052 119892 (119901) 120572 minus 1198921015840 (119901) Δ

(120572 + 120579)2 [119890(120572+120579)1199052


(28)

Since the gross profit 119892(119901)(119901 minus 119888119901) is a strictly concavefunction of 119901 which means 21198921015840(119901) + 11989210158401015840(119901)(119901 minus 119888119901) lt 0 wehave

1198661015840 (119901) = 11199051 + 1199052 1198921015840 (119901) 120572 minus 11989210158401015840 (119901) Δ

(120572 + 120579)2 [119890(120572+120579)1199052

minus (120572 + 120579) 1199052 minus 1] minus 119888011989210158401015840 (119901)1205822 (1198901205821199051 minus 12120582211990521 minus 1205821199051minus 1) + 21198921015840 (119901) + 11989210158401015840 (119901) (119901 minus 119888119901) lt 0

(29)

Therefore there exists a unique optimal selling price 119901lowastwhich maximizesprod(119901 | 119905lowast1 119905lowast2 ) and completes the proof

From the above analysis we know that the solution of119892(119901) + (119901 minus 119888119901)1198921015840(119901) = 0 is the lower bound for the optimalselling price 119901 such that 119889prod(119901 | 1199051 1199052)119889119901 = 0 CombiningPropositions 2 and 3 we establish an algorithm to obtain theoptimal policy of the considered model as follows

Algorithm 4

Step 1 Start with 119894 = 0 and let 119901119894 be a solution of 119892(119901) + (119901 minus119888119901)1198921015840(119901) = 0Step 2 Put 119901119894 into (15) and (16) to obtain the correspondingvalues of (119905lowast1 119905lowast2 ) then substitute them into (27) and deter-mine the optimal 119901119894+1Step 3 If |119901119894+1 minus119901119894| le 120576 where 120576 is any small positive numberthen set 119901lowast = 119901119894+1 and (119905lowast1 119905lowast2 119901lowast) is the optimal solution tominimizeprod(1199051 1199052 119901) otherwise set 119894 = 119894 + 1 and go back toStep 2

4 Numerical Examples andSensitivity Analysis

To demonstrate our theoretical results we study severalnumerical examples to explain the algorithm proposed in theabove section

Example 1 Consider an inventory system with the followingdata 1198600 = 25$order 120572 = 003$ 119888119901 = 100$unit 119888119889 = 12$unit 119888ℎ = 10$unit 1198880 = 05$unit 120579 = 02 120582 = 03 and119892(119901) = 50119890minus004119901 where 119901 isin [10 +infin)

For the inventory model with presale policy by usingAlgorithm 4 we have 119905lowast1 = 2090 119905lowast2 = 0552 119901lowast = 37709$119876lowast = 32104$ andprod(119905lowast1 119905lowast2 119901lowast) = 209707$Example 2 Consider an inventory system with the followingdata 1198600 = 25$order 120572 = 005$ 119888119901 = 100$unit 119888119889 = 12$unit 119888ℎ = 10$unit 1198880 = 05$unit 120579 = 002 120582 = 06 and119892(119901) = 100 minus 5119901 where 119901 isin [10 20]

For the inventory model with presale policy by usingAlgorithm 4 we have 119905lowast1 = 1739 119905lowast2 = 0669 119901lowast = 15130$119876lowast = 59014$ and prod(119905lowast1 119905lowast2 119901lowast) = 108783$ The three-di-mensional total profit per unit time graph for any given 119901is shown in Figure 2 and the numerical results indicate thatprod(119901 | 119905lowast1 119905lowast2 ) is strictly concave in 119901 as shown in Figure 3

In order to illustrate the effect of the parameters (120582 120579 120572)on the optimal policy that is optimal pricing the optimalordering quantity and the optimal total profit for the inven-tory models the sensitivity analysis is performed on the baseof Example 2 by changing the value of only one parameter ata time and keeping the rest of the parameters at their initialvalues The results are shown in Table 2

From Table 2 we can observe the following

(1) As120582 increases 119905lowast1 119876lowast andprod(119905lowast1 119905lowast2 119901lowast)will decreasewhile 119905lowast2 and 119901lowast will increase simultaneously which


Table 2 The effect of parameters on the optimal policy for presale model

Para Value of para 119905lowast1 119905lowast2 119901lowast 119876lowast prod(119905lowast1 119905lowast2 119901lowast)120582

02 2792 0480 15086 80609 11332304 2087 0594 15111 65851 11058306 1739 0669 15130 59014 10878308 1517 0725 15246 54882 107431

120579001 1725 0744 15131 60532 109017002 1739 0669 15130 59014 108783003 1749 0608 15128 57236 108542004 1758 0557 15127 56752 108337

120572004 1749 0608 15128 57685 108542005 1739 0669 15130 59014 108783007 1728 0731 15136 60396 109128010 1709 0851 15152 63142 109486

t2

t1

10

15

20

25

30

00

05

10

15

105

100

95

90

prod(t

1t

2|plowast)

Figure 2 Total profit per unit timeprod(1199051 1199052 | 119901lowast)

minus20

14 16 1812

p

20

40

60

80

100

prod(p

|tlowast 1

tlowast 2)

Figure 3 Total profit per unit timeprod(119901 | 119905lowast1 119905lowast2 )

means that if the presale rebate increases the presaleperiod will get shorter to reduce the rebate costMoreover the optimal order quantity and profit will

decrease while the selling price and spot-sale periodwill increase

(2) As 120579 increases 119905lowast2 119901lowast 119876lowast and prod(119905lowast1 119905lowast2 119901lowast) willdecrease and 119905lowast1 will increase It implies that theincrease in deterioration cost can lengthen the presaleperiod and then shorten the spot-sale period Asa result the selling price will decrease for salespromotion moreover the optimal order quantity andprofit will decrease

(3) As 120572 increases 119905lowast1 will decrease and 119905lowast2 119901lowast 119876lowastand prod(119905lowast1 119905lowast2 119901lowast) will increase It means that higherdemand rate in spot-sale period will lead to short-ening the presale period while the selling price theoptimal order quantity and profit will increase

(4) In general the fluctuation of 120582 has more effecton 119905lowast1 prod(119905lowast1 119905lowast2 119901lowast) than that of 120579 and 120572 and thefluctuation of 120582 120579 and 120572 each has more effect on119905lowast2 119876lowast than that of 119901lowast

5 Conclusion

In this paper we study an EOQ inventory model with presalepolicy for deteriorating items in which the demand ratedepends on both on-hand inventory and the price of itemsBy analyzing the inventory model an optimal pricing andinventory policy is proposed We also use several numericalexamples to illustrate the solution procedure Moreover thesensitivity analysis of the parameters is provided to assesstheir effects on the optimal policy of the studied problemFrom the results of numerical experiments we find that120582 and 120579 have a negative effect on the profit of inventorysystem while 120572 has a positive effect on the profit of inventorysystem In addition this paper provides an interesting topicfor further study of inventory models It also can be extendedin other ways that is considering the nonconstant or nonin-stantaneous deterioration rate and others


Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is supported partly by National Natural ScienceFoundation of China (nos 71502100 and 71671125)

References

[1] K-L Hou and L-C Lin ldquoAn EOQ model for deterioratingitems with price- and stock-dependent selling rates underinflation and time value of moneyrdquo International Journal ofSystems Science vol 37 no 15 pp 1131ndash1139 2006

[2] P-S You and Y-C Hsieh ldquoAn EOQmodel with stock and pricesensitive demandrdquoMathematical and Computer Modelling vol45 no 7-8 pp 933ndash942 2007

[3] C-T Chang Y-J Chen T-R Tsai and S-J Wu ldquoInventorymodels with stock- and price-dependent demand for deterio-rating items based on limited shelf spacerdquo Yugoslav Journal ofOperations Research vol 20 no 1 pp 55ndash69 2010

[4] C-Y Dye and T-P Hsieh ldquoDeterministic ordering policy withprice- and stock-dependent demand under fluctuating cost andlimited capacityrdquo Expert Systems with Applications vol 38 no12 pp 14976ndash14983 2011

[5] B C Giri and S Bardhan ldquoSupply chain coordination for adeteriorating item with stock and price-dependent demandunder revenue sharing contractrdquo International Transactions inOperational Research vol 19 no 5 pp 753ndash768 2012

[6] J-T Teng and C-T Chang ldquoEconomic production quantitymodels for deteriorating items with price- and stock-dependentdemandrdquo Computers amp Operations Research vol 32 no 2 pp297ndash308 2005

[7] D PandaM KMaiti andMMaiti ldquoTwowarehouse inventorymodels for single vendor multiple retailers with price and stockdependent demandrdquo Applied Mathematical Modelling vol 34no 11 pp 3571ndash3585 2010

[8] K Skouri I Konstantaras S Papachristos and I Ganas ldquoInven-tory models with ramp type demand rate partial backloggingandWeibull deterioration raterdquoEuropean Journal ofOperationalResearch vol 192 no 1 pp 79ndash92 2009

[9] J Sicilia M Gonzalez-De-la-Rosa J Febles-Acosta and DAlcaide-Lopez-de-Pablo ldquoAn inventorymodel for deterioratingitems with shortages and time-varying demandrdquo InternationalJournal of Production Economics vol 155 pp 155ndash162 2014

[10] M Cheng B Zhang and G Wang ldquoOptimal policy fordeteriorating items with trapezoidal type demand and partialbackloggingrdquoAppliedMathematical Modelling vol 35 no 7 pp3552ndash3560 2011

[11] L Zhao ldquoAn inventory model under trapezoidal type demandWeibull-distributed deterioration and partial backloggingrdquoJournal of Applied Mathematics vol 2014 Article ID 747419 10pages 2014

[12] L Zhao ldquoOptimal replenishment policy for weibull-distributeddeteriorating items with trapezoidal demand rate and partialbackloggingrdquoMathematical Problems in Engineering vol 2016Article ID 1490712 10 pages 2016

[13] R Begum R R Sahoo and S K Sahu ldquoA replenishment policyfor items with price-dependent demand time-proportional

deterioration and no shortagesrdquo International Journal of SystemsScience Principles and Applications of Systems and Integrationvol 43 no 5 pp 903ndash910 2012

[14] C-Y Dye ldquoJoint pricing and ordering policy for a deterioratinginventory with partial backloggingrdquo Omega vol 35 no 2 pp184ndash189 2006

[15] C-T Yang L-YOuyangH-F Yen andK-L Lee ldquoJoint pricingand ordering policies for deteriorating item with retail price-dependent demand in response to announced supply priceincreaserdquo Journal of Industrial and Management Optimizationvol 9 no 2 pp 437ndash454 2013

[16] B Sarkar and S Sarkar ldquoAn improved inventory model withpartial backlogging time varying deterioration and stock-dependent demandrdquo Economic Modelling vol 30 pp 924ndash9322013

[17] G Padmanabhan and P Vrat ldquoEOQ models for perishableitems under stock dependent selling raterdquo European Journal ofOperational Research vol 86 no 2 pp 281ndash292 1995

[18] T Avinadav A Herbon and U Spiegel ldquoOptimal inventorypolicy for a perishable item with demand function sensitive toprice and timerdquo International Journal of Production Economicsvol 144 no 2 pp 497ndash506 2013

[19] R Maihami and I Nakhai Kamalabadi ldquoJoint pricing andinventory control for non-instantaneous deteriorating itemswith partial backlogging and time and price dependentdemandrdquo International Journal of Production Economics vol136 no 1 pp 116ndash122 2012

Submit your manuscripts athttpwwwhindawicom

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Table 1 Our paper versus literatures for key assumptions of inventory models with deteriorating items

Authorauthors Demand rate Deterioration rate Shortage or not Shortage cost Decision variablelowast

Skouri et al [8] Time-dependent Weibull distribution Yes Constant 119905 119879Sicilia et al [9] Time-dependent Constant Yes Constant 119905 119879Cheng et al [10] Time-dependent Constant Yes Constant 119905Zhao [11 12] Time-dependent Weibull distribution Yes Constant 119905Begum et al [13] Price-dependent Weibull distribution No No 119901 119879Dye [14] Price-dependent Time-dependent Yes Constant 119901 119905 and 119879Yang et al [15] Price-dependent No Yes Constant 119905 119879B Sarkar and S Sarkar [16] Stock-dependent Time-dependent Yes Constant 119905 119879Padmanabhan and Vrat [17] Stock-dependent Constant Yes Constant 119905 119879Avinadav et al [18] Price-and-time Constant No No 119901 119876 and 119879Maihami and Nakhai Kamalabadi [19] Price-and-time Noninstantaneous Yes Constant 119901 119879Hou and Lin [1] Price-and-stock Constant Yes Constant 119905 119879Giri and Bardhan [5] Price-and-stock No No No 119905 119899This paper Price-and-stock Constant Presale Time-dependent 119901 119905 and 119879lowast119901 = selling price 119905 = replenishmentout of stock time 119876 = order quantity 119879 = inventory cycle and 119899 = the number of replenishment

be constant In addition we assume the demand is price-and-stock dependent according to the actual situation Eventuallywe derive optimal pricing and ordering policy by designingan effective algorithm and provide the sensitivity analysis ofthe parameters to assess their effects on the optimal policy

The rest of the paper is organized as follows we introducesome basic notations and assumptions in Section 2 Section 3establishes an inventory model under presale policy andprovides an effective procedure to find the optimal pricingand presale strategy In Section 4 we use several numericalexamples to illustrate the procedure of the optimal strategiesand analyze the sensitivity of parameters involved Finally weprovide a summary of the paper

2 Notation and Assumptions

The fundamental notation and assumptions used in thispaper are given as below

21 Notation

119888119901 the purchase price of unit item1198600 the fixed cost per order119878 the maximum inventory level119876 the ordering quantity119901 the selling price of unit item120579 the constant deteriorating rate 0 le 120579 ≪ 11199051 presale period1199052 spot-sale period119868(119905) the level of inventory at time 119905 0 le 119905 le 119879 where119879 = 1199051 + 1199052119888119889 the cost of each deteriorated item119888ℎ the inventory holding cost per unit per time119863(119905 119901) the demand rate

119888119904(119909) the presale rebate costprod(1199051 1199052 119901) the average total profit per unit time

22 Assumptions

(i) The replenishment rate is infinite that is replenish-ment is instantaneous

(ii) Assume the customers are loyal and all the presaleorders will not be canceled until the products arelaunched into the market

(iii) The demand rate119863(119905 119901) we assumed it to be

119863(119905 119901) = 119892 (119901) 0 le 119905 le 1199051119892 (119901) + 120572119868 (119905) 1199051 le 119905 le 119879 (1)

where 119892(119901) is a function of 119901 1198921015840(119901) lt 0 and 11989210158401015840(119901) ge0 where 120572 denotes the stock-dependent consumptionrate

(iv) 119888119904(119909) is the presale rebate determined by the waitingtime 119909 that is 119888119904(119909) = 1198880(119890120582119909 minus 1) where 1198880 is positiveconstant and 120582 isin (0 1)

(v) The marginal gross profit with respect to price isdecreasing that is (119901 minus 119888119901)119892(119901) is a strictly concavefunction of 119901

3 Inventory Model with Presale Policy

In this section we consider an inventory model with presalepolicy for deteriorating items and the behavior of the systemis depicted in Figure 1 From 119905 = 0 to 1199051 the total presalequantity accumulates on account of demand which is fullybacklogged and achieves its maximum at time 119905 = 1199051 Duringthe time interval [1199051 1199051 + 1199052) due to customersrsquo demand anddeterioration on-hand inventory level gradually decreases to


0

Inventorylevel

Time

I (t)

t1 t2

t1 + t2




(2)


119868 (119905) = minus119892 (119901) 119905 0 le 119905 le 1199051119868 (119905) = 119892 (119901)

120572 + 120579 (119890(120572+120579)(1199051+1199052minus119905) minus 1) 1199051 le 119905 le 1199051 + 1199052(3)


119868 (119905+1 ) = 119892 (119901)120572 + 120579 (119890(120572+120579)1199052 minus 1) (4)


119868 (119905minus1 ) = minus119892 (119901) 1199051 (5)





119888119901 [119892 (119901)120572 + 120579 (119890(120572+120579)1199052 minus 1) + 119892 (119901) 1199051] (7)

(iii) Sales revenue

119901[119892 (119901) 1199051 + int1199051+11990521199051

(119892 (119901) + 120572119868 (119905)) 119889119905] (8)


119888119889 [119892 (119901)120572 + 120579 (119890(120572+120579)1199052 minus 1) minus int1199051+1199052

1199051

(119892 (119901) + 120572119868 (119905)) 119889119905] (9)

(v) Holding cost

119888ℎ119892 (119901)120572 + 120579 int1199051+11990521199051

(119890(120572+120579)(1199051+1199052minus119905) minus 1) 119889119905 (10)


1198880119892 (119901)int11990510(119890120582(1199051minus1) minus 1) 119905 119889119905 (11)





1199051 + 1199052 Δ


(12)




120597prod (1199051 1199052 119901)1205971199052 = minusprod(1199051 1199052 119901)1199051 + 1199052+ 119892 (119901)1199051 + 1199052 [

119901120579 + 119888ℎ + 119888119889120579120572 + 120579 minus Δ120572 + 120579119890(120572+120579)1199052] 120597prod (1199051 1199052 119901)120597119901 = 11199051 + 1199052

119892 (119901) 120572 minus 1198921015840 (119901) Δ(120572 + 120579)2 [119890(120572+120579)1199052


(13)



120597prod (1199051 1199052 119901)1205971199051 = 0120597prod (1199051 1199052 119901)1205971199052 = 0

(14)



(15)


(16)







(ii) Let



1198911015840 (119909) = Δ119890(120572+120579)119909 ge 0 (19)




into (15) and let






(22)







100381610038161003816100381610038161003816100381610038161003816(11990511199052)=(119905lowast1119905lowast2)


lt 0(23)



2 lt 0 (24)


= 0 (25)






]2

gt 0

(26)





119889prod (119901 | 1199051 1199052)119889119901= 11199051 + 1199052

119892 (119901) 120572 minus 1198921015840 (119901) Δ(120572 + 120579)2 [119890(120572+120579)1199052


(27)


119866 (119901) = 11199051 + 1199052 119892 (119901) 120572 minus 1198921015840 (119901) Δ

(120572 + 120579)2 [119890(120572+120579)1199052


(28)


1198661015840 (119901) = 11199051 + 1199052 1198921015840 (119901) 120572 minus 11989210158401015840 (119901) Δ

(120572 + 120579)2 [119890(120572+120579)1199052


(29)



Algorithm 4













02 2792 0480 15086 80609 11332304 2087 0594 15111 65851 11058306 1739 0669 15130 59014 10878308 1517 0725 15246 54882 107431

120579001 1725 0744 15131 60532 109017002 1739 0669 15130 59014 108783003 1749 0608 15128 57236 108542004 1758 0557 15127 56752 108337

120572004 1749 0608 15128 57685 108542005 1739 0669 15130 59014 108783007 1728 0731 15136 60396 109128010 1709 0851 15152 63142 109486

t2

t1

10

15

20

25

30

00

05

10

15

105

100

95

90

prod(t

1t

2|plowast)


minus20

14 16 1812

p

20

40

60

80

100

prod(p

|tlowast 1

tlowast 2)







5 Conclusion



Competing Interests


Acknowledgments


References




























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




0

Inventorylevel

Time

I (t)

t1 t2

t1 + t2




(2)


119868 (119905) = minus119892 (119901) 119905 0 le 119905 le 1199051119868 (119905) = 119892 (119901)

120572 + 120579 (119890(120572+120579)(1199051+1199052minus119905) minus 1) 1199051 le 119905 le 1199051 + 1199052(3)


119868 (119905+1 ) = 119892 (119901)120572 + 120579 (119890(120572+120579)1199052 minus 1) (4)


119868 (119905minus1 ) = minus119892 (119901) 1199051 (5)





119888119901 [119892 (119901)120572 + 120579 (119890(120572+120579)1199052 minus 1) + 119892 (119901) 1199051] (7)

(iii) Sales revenue

119901[119892 (119901) 1199051 + int1199051+11990521199051

(119892 (119901) + 120572119868 (119905)) 119889119905] (8)


119888119889 [119892 (119901)120572 + 120579 (119890(120572+120579)1199052 minus 1) minus int1199051+1199052

1199051

(119892 (119901) + 120572119868 (119905)) 119889119905] (9)

(v) Holding cost

119888ℎ119892 (119901)120572 + 120579 int1199051+11990521199051

(119890(120572+120579)(1199051+1199052minus119905) minus 1) 119889119905 (10)


1198880119892 (119901)int11990510(119890120582(1199051minus1) minus 1) 119905 119889119905 (11)





1199051 + 1199052 Δ


(12)




120597prod (1199051 1199052 119901)1205971199052 = minusprod(1199051 1199052 119901)1199051 + 1199052+ 119892 (119901)1199051 + 1199052 [

119901120579 + 119888ℎ + 119888119889120579120572 + 120579 minus Δ120572 + 120579119890(120572+120579)1199052] 120597prod (1199051 1199052 119901)120597119901 = 11199051 + 1199052

119892 (119901) 120572 minus 1198921015840 (119901) Δ(120572 + 120579)2 [119890(120572+120579)1199052


(13)



120597prod (1199051 1199052 119901)1205971199051 = 0120597prod (1199051 1199052 119901)1205971199052 = 0

(14)



(15)


(16)







(ii) Let



1198911015840 (119909) = Δ119890(120572+120579)119909 ge 0 (19)




into (15) and let






(22)







100381610038161003816100381610038161003816100381610038161003816(11990511199052)=(119905lowast1119905lowast2)


lt 0(23)



2 lt 0 (24)


= 0 (25)






]2

gt 0

(26)





119889prod (119901 | 1199051 1199052)119889119901= 11199051 + 1199052

119892 (119901) 120572 minus 1198921015840 (119901) Δ(120572 + 120579)2 [119890(120572+120579)1199052


(27)


119866 (119901) = 11199051 + 1199052 119892 (119901) 120572 minus 1198921015840 (119901) Δ

(120572 + 120579)2 [119890(120572+120579)1199052


(28)


1198661015840 (119901) = 11199051 + 1199052 1198921015840 (119901) 120572 minus 11989210158401015840 (119901) Δ

(120572 + 120579)2 [119890(120572+120579)1199052


(29)



Algorithm 4













02 2792 0480 15086 80609 11332304 2087 0594 15111 65851 11058306 1739 0669 15130 59014 10878308 1517 0725 15246 54882 107431

120579001 1725 0744 15131 60532 109017002 1739 0669 15130 59014 108783003 1749 0608 15128 57236 108542004 1758 0557 15127 56752 108337

120572004 1749 0608 15128 57685 108542005 1739 0669 15130 59014 108783007 1728 0731 15136 60396 109128010 1709 0851 15152 63142 109486

t2

t1

10

15

20

25

30

00

05

10

15

105

100

95

90

prod(t

1t

2|plowast)


minus20

14 16 1812

p

20

40

60

80

100

prod(p

|tlowast 1

tlowast 2)







5 Conclusion



Competing Interests


Acknowledgments


References




























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





120597prod (1199051 1199052 119901)1205971199051 = 0120597prod (1199051 1199052 119901)1205971199052 = 0

(14)



(15)


(16)







(ii) Let



1198911015840 (119909) = Δ119890(120572+120579)119909 ge 0 (19)




into (15) and let






(22)







100381610038161003816100381610038161003816100381610038161003816(11990511199052)=(119905lowast1119905lowast2)


lt 0(23)



2 lt 0 (24)


= 0 (25)






]2

gt 0

(26)





119889prod (119901 | 1199051 1199052)119889119901= 11199051 + 1199052

119892 (119901) 120572 minus 1198921015840 (119901) Δ(120572 + 120579)2 [119890(120572+120579)1199052


(27)


119866 (119901) = 11199051 + 1199052 119892 (119901) 120572 minus 1198921015840 (119901) Δ

(120572 + 120579)2 [119890(120572+120579)1199052


(28)


1198661015840 (119901) = 11199051 + 1199052 1198921015840 (119901) 120572 minus 11989210158401015840 (119901) Δ

(120572 + 120579)2 [119890(120572+120579)1199052


(29)



Algorithm 4













02 2792 0480 15086 80609 11332304 2087 0594 15111 65851 11058306 1739 0669 15130 59014 10878308 1517 0725 15246 54882 107431

120579001 1725 0744 15131 60532 109017002 1739 0669 15130 59014 108783003 1749 0608 15128 57236 108542004 1758 0557 15127 56752 108337

120572004 1749 0608 15128 57685 108542005 1739 0669 15130 59014 108783007 1728 0731 15136 60396 109128010 1709 0851 15152 63142 109486

t2

t1

10

15

20

25

30

00

05

10

15

105

100

95

90

prod(t

1t

2|plowast)


minus20

14 16 1812

p

20

40

60

80

100

prod(p

|tlowast 1

tlowast 2)







5 Conclusion



Competing Interests


Acknowledgments


References




























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in








]2

gt 0

(26)





119889prod (119901 | 1199051 1199052)119889119901= 11199051 + 1199052

119892 (119901) 120572 minus 1198921015840 (119901) Δ(120572 + 120579)2 [119890(120572+120579)1199052


(27)


119866 (119901) = 11199051 + 1199052 119892 (119901) 120572 minus 1198921015840 (119901) Δ

(120572 + 120579)2 [119890(120572+120579)1199052


(28)


1198661015840 (119901) = 11199051 + 1199052 1198921015840 (119901) 120572 minus 11989210158401015840 (119901) Δ

(120572 + 120579)2 [119890(120572+120579)1199052


(29)



Algorithm 4













02 2792 0480 15086 80609 11332304 2087 0594 15111 65851 11058306 1739 0669 15130 59014 10878308 1517 0725 15246 54882 107431

120579001 1725 0744 15131 60532 109017002 1739 0669 15130 59014 108783003 1749 0608 15128 57236 108542004 1758 0557 15127 56752 108337

120572004 1749 0608 15128 57685 108542005 1739 0669 15130 59014 108783007 1728 0731 15136 60396 109128010 1709 0851 15152 63142 109486

t2

t1

10

15

20

25

30

00

05

10

15

105

100

95

90

prod(t

1t

2|plowast)


minus20

14 16 1812

p

20

40

60

80

100

prod(p

|tlowast 1

tlowast 2)







5 Conclusion



Competing Interests


Acknowledgments


References




























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






02 2792 0480 15086 80609 11332304 2087 0594 15111 65851 11058306 1739 0669 15130 59014 10878308 1517 0725 15246 54882 107431

120579001 1725 0744 15131 60532 109017002 1739 0669 15130 59014 108783003 1749 0608 15128 57236 108542004 1758 0557 15127 56752 108337

120572004 1749 0608 15128 57685 108542005 1739 0669 15130 59014 108783007 1728 0731 15136 60396 109128010 1709 0851 15152 63142 109486

t2

t1

10

15

20

25

30

00

05

10

15

105

100

95

90

prod(t

1t

2|plowast)


minus20

14 16 1812

p

20

40

60

80

100

prod(p

|tlowast 1

tlowast 2)







5 Conclusion



Competing Interests


Acknowledgments


References




























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Competing Interests


Acknowledgments


References




























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



research article optimal pricing and ordering policy for

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