APPLICATION ARTICLE

Dynamic pricing model with logarithmic demand

U. K. Khedlekar & D. Shukla

Accepted: 9 June 2012 /Published online: 7 July 2012# Operational Research Society of India 2012

Abstract The demand of an item depends on time and price both. For example,electric fan in winter season has low demand than summer. The competingcompany with lower price gets benefit of better sale during this season. Theoffer of discount in price of product is an effective tool to increase the demandin the declining market. In this paper, we develop a dynamic pricing model forthe product having logarithmic decline price sensitive demand. A useful solu-tion procedure is presented to determine the optimal number of price settingsand respective selling price. It proved there exist an optimal number of pricesettings to provide optimal profit for varying business setup. Results aresupported by a numerical example and a computational procedure is appendedto illustrate the performance of the model. It is shown that the dynamic pricingpolicy has better performance over the static pricing policy.

Keywords Inventory . Dynamic pricing . Price sensitive demand . Optimal pricesettings . Deterioration

AMS subject classification 90 B 05 . 90 B 30 . 90 B 50

1 Introduction

In most of existing inventory models in the literature, it is assumed that item can bestored for long duration to meet the future demand but the quantity received does notmatch with the quantity consumed. The variation may be due to adopting methodology

OPSEARCH (Jan–Mar 2013) 50(1):1–13DOI 10.1007/s12597-012-0093-2

U. K. Khedlekar (*) :D. ShuklaDepartment of Mathematics and Statistics,Doctor Hari Singh Gour Vishwavidyalaya (Central University),Sagar, Madhya Pradesh, Indiae-mail: uvkkcm@yahoo.co.in

D. Shuklae-mail: diwakarshukla@rediffmail.com

of pricing policy. Many products lose their utility and quality after duration, likefashion products have limited sale season; they are out-dated or out of fashionthereafter. Many food products may be rotten after time duration therefore, a pricingpolicy is required to sellout the entire stock before the start of next sale season. Itneeds to implement special sale or price reduction or both for a shorter duration.

It is being advised to managers to control and maintain the inventory according tofuture potential of demand influenced by policies of inventory management andretailers. Expenditure sources like ordering cost, safety features, lead time andnumbers of lots are the integral parts of decision making. Lo [9] presents inventorymodel focusing on ordering cost, safety feature, lead time and number of lots fordeteriorating item.

Roy and Chaudhuri [15] introduce the concept of sale promotion in festival for theclearance of overdue stock and compared it without special sale and found thisoutperforming to earlier. The back order and partial lost sale is investigated due toLo et al. [10] with the impact of leading time on optimal policy and safety. For fewmore related contributions, we refer to Shah and Pandey [18], Urban and Baker [23]and Sabahno [16]. Padmanabhan and Vrat [12] apply the stock dependent demandtheory on deteriorating item whereas Seeletse [17] solve the problem through a casestudy that faces by industry practicenor in real situations and obtain an optimumsolution.

Federgruen and Heching [4] design the replenishment strategy when price of anitem faces uncertainty and follows a general stochastic demand function. Theyanalyzed the simultaneous price and the inventory in an incapacitated system withthe use of stochastic demand for single item. The aspect of inflation and delay inpayment to vendors has been attempted by many authors. Teng and Chang [22]suggest the EOQ model for deteriorating item assuming demand depends on priceand stocks both. Similar approach has been adopted by Hou and Lin [5] on thedeterministic economic order quantity (EOQ) model by taking into account theinflation and time value of money for the deteriorating items with price and stock-dependent selling rate.

Not only price reduction but also the price hike specially on fuel, excise duty,transportation uplifted the rate of inflation. Joglekar [6] used a linear demand functionwith price sensitiveness and allowed retailers to use a continuous increasing pricestrategy in one inventory cycle. He derived the retailer’s optimal profit by ignoring allthe inventory costs. His findings are restricted to the growing market only, neither forstable market nor for declining market.

A research overview presented by Elmaghraby and Keskinocak [3] is based on theabove problem incorporating the future planning as it jointly determines the dynamicpricing and order level both. Bernstein and Federgruen [2] analyze the constantdemand-rate and solved the problem of VMI (Vendor Management Inventory) wherethe replenishment decision-making is transferred to the supplier, but the retailer isable to make own pricing decisions. We refer literature on pricing policy by Ray et al.[14], Natrajan and Rao [11], Pal and Dutta [13], Shukla et al. [20], and Khedlekar [7].

Sinha [21] estimates parameters in regression model and the problem of ranking ofbusiness schools solved by Karush-Kuhn-Tucker optimization method. Results arefound satisfactory in quality. Dividing demand rate into multiple segments, Shuklaand Khedlekar [19] introduce three-component demand rate based on marketing

2 OPSEARCH (Jan–Mar 2013) 50(1):1–13

policies for the newly launched deteriorating product. Widyadana and Wee [24]solved the problem by genetic algorithm method for unreliable production systemwith price dependent demand and suggest pricing policy to minimize the loss in profitdue to machine unavailability. Ziaee et al. [27] showed that if vendor managedproblem analytically solved then solution may outperform to the traditional solution.Kumar et al. [8] present a dynamic job-oriented scheduler using real time geneticalgorithm for complex production system.

Alfares et al. [1] develop product-inventory model for both deteriorating item anddeteriorating process considering some realistic aspects in their model such asvarying demand, production rates, quality, and malignance cost. You [25] discussan inventory policy for the product with price and time dependent demand. Heobtained the order size and optimal price both when decision maker has an opportu-nity to adjust price before the end of sale season.

Motivation for this paper is derived due to You [25], You and Hsieh [26], Hou andLin [5] and Lo [9] for applying the adjustment in the selling price dynamically beforeentering into the next cycle. A large proportion of customers are influenced byadvertisements through electronic media, news papers, internet or campaign. Mostlyin declining market, it happens that with decreasing demand, manager has to put extraefforts to uplift the sale level through media and pricing policy. This motivates fordeveloping a dynamic pricing policy for inventory system and is the main focus ofthis paper.

2 Notations and assumptions

We design the proposed model under assumption that shortages are not allowed,replenishment rate is infinite and lead time is zero. Following notations are used todevelop the model:

& L prescribed time horizon.& n number of changes in selling price.& Nmax maximum number of permissible changes in price.& T time interval between any two price changes where T 0 L/n.& Si total sale quantity of product in interval [0, iT], where i 0 1, 2…n.& q quantity required for one prescribed time horizon L.& h holding cost per unit time which is constant.& A per unit purchase cost of product.& co cost required to change the price once.& b contribution of log (t) (decreasing trend) in demand.& pj selling price of product in interval[(j-1)T, jT].& β parameter associate to contribution of price in demand.& θ rate of deterioration of product.& Di(n, p

n) total deteriorated units in time interval [(i-1)T, iT].& D(n, pn) total deteriorated units in system over time horizon L.& H(n, pn) total inventory carrying cost over time horizon L.& R(n, pn) total earned revenue over L.& F(n, pn) net profit over L.

OPSEARCH (Jan–Mar 2013) 50(1):1–13 3

3 Formulation of proposed model

Assumes a product is purchased (or manufactured) at the rate $c per unit for timehorizon L, which is divided into n equal parts T 0 L/n, like [0, T], [T, 2T]……. [(n-1)T,nT] and decreasing optimal prices are p1, p2,……pn in these intervals respectively. Letc0 is cost to change the selling price once and si is sold amount of stock in intervalc[0, iT], i01,2….n. Demand of product is assumed as

d t; pið Þ ¼ a� b logðtÞ � b pið Þ ð1Þ

where i01, 2….n, a >0, b >0, and β>0 are fix for a given business setup such thata� b logðtÞ � b pi > 0 and t>0

si ¼Xij¼1

Z jT

j�1ð ÞTd t; pj� �

dt ¼ a1iT � bTXij¼1

pj � biT log iTð Þ ð2Þ

The total sold amount over time horizon (L 0 nT) is

q ¼ sn ¼ a1nT � bTXnj¼1

pj � bnT log nTð Þ ð3Þ

The objective is to find out optimal number of change in selling price, respec-tive selling prices alongwith optimum profit. Suppose Ii(t) is on hand inventory attime t. The demand function satisfies following differential equation in time interval[(i-1)T, iT].

d

dtIiðtÞ ¼ �d ði� 1ð ÞT þ t; pjÞ ¼ �ða� b log i� 1ð ÞT þ t � b pjÞ ð4Þ

with boundary condition Iið0Þ ¼ q� si�1 and put a1 0 a + b

IiðtÞ ¼ a1 n� iþ 1ð ÞT � bTXnj¼1

pj þ bTXi�1

j¼1

pj � bnT log nTð Þ � a1t þ b tpj

þ bt þ biT � bT log iT � T þ tð Þð5Þ

The holding H(n, pn) is:

H n; pnð Þ ¼ hXni¼1

Z T

0IiðtÞdt ð6Þ

Total deteriorated units D(n, pn) over time [0, T] are

D n; pnð Þ ¼ θXni¼1

Z T

0IiðtÞdt ð7Þ

Due to deterioration required quantity in systemwill be q1 ¼ qþD n; pnð Þ ¼ snþD n; pnð Þ

4 OPSEARCH (Jan–Mar 2013) 50(1):1–13

Total earned revenue R(n,pn) over time L is

R n; pnð Þ ¼Xnj¼1

pj

Z jT

j�1ð ÞTd t; pj� �

dt ð8Þ

Net profit

F n; pnð Þ ¼ R n; pnð Þ � H n; pnð Þ � Aq1 � n c0 ð9ÞOur objective is to maximize F(n,pn) under the condition that

pi <a�b log iTð Þ

b fordemandfunction a� b logðtÞ � b pi > 0

or pj � a�b log jTð Þb þ Zj2 ¼ 0 where Zj2 ¼ Ai

ð10Þ

The suggested objective function is L(n,pn) under the condition (10) with usingLagrangian multiplier λj could be expressed as

L n; pnð Þ ¼ F n; pnð Þ �Xnj¼1

lj pj � a� b log jTð Þb

þ Zj2

� �; n � Nmax

¼a1TXnj¼1

pj � bTXnj¼1

p2j � bTXni¼1

jpj log jTð Þ þ bTXnj¼1

j� 1ð Þpj log jT � Tð Þ

ð11Þ

� hþ Aθð Þ bT 2

4

Xnj¼1

2j� 1ð Þ � bT 2

2

Xnj¼1

j� 1ð Þ2 log jT � Tð Þ � bn2T 2 log nTð Þ( )

� hþ Aθð Þ nT 2a1 nþ 1ð Þ � bTXni¼1

Xnj¼1

pj þ bT 2Xij¼1

n� jð Þpj( )

� hþ Aθð Þ nT 2

2a1 þ bT 2

2

Xnj¼1

pj þ bT 2

4

Xnj¼1

j2 log jTð Þ( )

� nc0

ð12Þ

Theorem 1 When Ai>0, optimal price for interval [(i-1)T, iT] is

pi1 ¼T

2hþ Aθð Þ i� 1

2

� �þ A

2þ 1

2ba1 þ bi log iTð Þ � b i� 1ð Þ log iT � Tð Þð Þ � l i

2bT

ð13ÞIf Ai≤0, then

pi2 ¼a� b log iTð Þ

bð14Þ

Proof To optimize the L(n,pn), we apply Kuhn Tucker optimization condition

@

@piL n; pnð Þ ¼ a1T � 2bTpj þ bT2 hþ Aθð Þ i� 1

2

� �þ bAT � biT log iTð Þ

þ b i� 1ð ÞT log iT � Tð Þ � li ð15Þ

(12)

(11)

OPSEARCH (Jan–Mar 2013) 50(1):1–13 5

@

@liL n; pnð Þ ¼ � pi � a� bi log iTð Þ

bþ Zi

2

� �ð16Þ

@

@ZiL n;pnð Þ ¼ �2liZi ð17Þ

For fixed n, and Ai>0 then p�i¼ pi 1 obtained by (15)

For Ai ¼ Zi2 > 0 , for fix n, selling price is

pi1 ¼T

2hþ Aθð Þ i� 1

2

� �þ A

2þ 1

2ba1 þ bi log iTð Þ � b i� 1ð Þ log iT � Tð Þð Þ � l i

2bT

ð18ÞIf Ai≤0, for fix n, then Eq. 16 provides,

pi2 ¼a� b log iTð Þ

b; a > blog iTð Þ ð19Þ

Theorem 2 R(n, pn) is concave for given n.

Proof Using Eq. 8,

@2

@pi2R n; pnð Þ ¼ @2

@pj2R n; pnð Þ ¼ �2n bT � 0

and

@2

@pi@pjR n; pnð Þ ¼ 0 where i 6¼ j ð20Þ

As per Hesianmatrix of Kuhn-Tucker necessary conditions are Hi1 ¼ @2

@pi2R n; pnð Þ

��� ���; Hi1j j < 0

and

Hi2 ¼@2

@pi2R n; pnð Þ @2

@pi@pjR n; pnð Þ

@2

@pj@piR n; pnð Þ @2

@pj2R n; pnð Þ

����������; Hi2 > 0 ð21Þ

Thus R(n, pn)attaints global maxima.

Theorem 3 F(n, pn)is concave for given n.

Theorem 4 For each i, and for fix n,

si � si�1 > 0; i ¼ 2; 3; . . . :n: ð22ÞProof Using Eq. 2,

si � si�1 ¼ a1T � bTpi � biT log iTð Þ þ b i� 1ð ÞT log iT � Tð Þ¼ T a� bTpi � bT log iTð Þð Þ þ bT � b i� 1ð ÞT log iTð Þ þ b i� 1ð ÞT log iT � Tð Þ¼ T a� b pi � b log iTð Þð Þ þ bT 1þ i� 1ð Þ log i

i�1

� �� �ð23Þ

Since demand during [0, iT] is a� b pi � b log iTð Þð Þ � 0

Si – Si –1 > 0, i = 2, 3, . . . .n.

6 OPSEARCH (Jan–Mar 2013) 50(1):1–13

Hence si � si�1 > 0

Theorem 5 R(n, pi) is a monotonic increasing function for each i, for fix n, and T 0 L/n.

Proof Since R n;pnð Þ ¼Pnj¼1

piR jT

j�1ð ÞT d t; pj� �

dt

Then

R n; pi� �� R n; pi�1

� � ¼ pi a1T � bpiT � bT i log iTð Þ � i� 1ð Þ log i� 1ð ÞTð Þf g

¼ pi a1T � bpiT � bTi log iTð Þ þ bT i� 1ð Þ log i� 1ð ÞTf g

¼ pi si � si�1 þ bT i� 1ð Þ log i� 1ð ÞTf gð24Þ

By using theorem 4, si � si�1 > 0 , we getR n;pið Þ � R n;pi�1ð Þ > 0;8 a; T ; b .Hence R(n, pi) is monotonic increasing function for fix n, and for each i.

Corollary 1 F(n, pi) is monotonic increasing function for each i, for fix n, and T ¼ L=n .

Corollary 2 R(n, pi) and F(n, pi) are maximal at i 0 n, for fix n.

Proof According to theorem 5, R(n, pi) is monotonic increasing function for each i,for fix n and T 0 L/n.

Clearly

Rðn; p1Þ < Rðn; p2Þ; < Rðn; p3Þ < ::::::::::::::Rðn; pn�1Þ < Rðn; pnÞ ð25Þ

Hence R(n, pi) is maximal at i 0 nFor example, if the time horizon L 0 nT is dividing into five equal parts then n05

and i01, 2, 3, 4, 5.

Revenue for period [0, T] is R(5,p1)Revenue for period [0, 2T] is R(5,p2) which greater than R(5,p1)Revenue for period [0, 3T] is R(5,p3) which greater than R(5,p2)Revenue for period [0, 4T] is R(5,p4) which greater than R(5,p3)Revenue for period [0, 5T] is R(5,p5) which greater than R(5,p4)

Thus Revenue for period [0, 5T] is R(5,p5)>R(5,p4) (Revenue for period [0, 4T])i.e. R(5,p5)>R(5,p4)>R(5,p3)>R(5,p2)>R(5,p1) which followed by theorem 5.i.e. R(5,p5) is maximal at i05 for fix n, and hence R(n,pi) is maximal at i0n.When n is fix, then R(n,pi) is maximal at i0n.Similar could be obtained from Corollary 1, F(n,pi) is maximal at i0n.

Theorem 6 Average selling price p is invariant for Ai>0, and for each i and n.

OPSEARCH (Jan–Mar 2013) 50(1):1–13 7

For Ai>0, then by Eq. 13

pi ¼ T

2hþ Aθð Þ i� 1

2

� �þ A

2þ 1

2ba1 þ bi log iTð Þ � b i� 1ð Þ logð iT � Tð Þð Þ � l i

2bT

p ¼ 1

n

Xni¼1

pi ¼ T

4hþ Aθð Þnþ A

2þ a1

2b� b

2nb

Xi¼n

i¼1

bi log iTð Þ �Xi¼n

i¼2

b i� 1ð Þ log iT � Tð Þ !

p ¼ 1

n

Xni¼1

pi ¼ L

4hþ Aθð Þ þ A

2þ a1

2b� b

2blogðLÞ;which is constant:

ð26Þ

Theorem 6, reveals that variation of price does not affect the average price ofproduct in case of dynamic price settings. Although high to low variation of sellingprice generates the excess demand and revenues, but keeps average price same andequal to the static price policy.

4 Solution procedure

Step 1: take n01 and i01Step 2: calculate Ai,

if Ai>0, find pi1 , from (18) (Theorem 1),else Ai≤0, find pi2 , from (19) (Theorem 1),

Step 3: calculate respective R(n,pi), F(n,pi)Step 4: n0n+1, i01 to n+1Step 5: go to step 2, until n0Nmaxi, i01, to Nmaxi

Step 6: find F� ¼ Maxn

i¼1;2::::n�F n; pið Þð Þ , where n � is value on which F(n,pn)is maximum

Step 7: display value n � for conclusion.

n � is optimal number of price setting.

5 Example and effect of parameter variation on optimal policy

Suppose demand function is d t; pj� � ¼ 50� 2 logðtÞ � 0:4ð Þpj

� �and the input data

is as under: Holding cost h00.001 per unit time, c0 0150, β00.1, θ00, L0200,Nmaxi010. We apply the solution procedure 4, to find optimal solution and outputs aregiven in Table 1.

In order to get optimal solution we start with n01 and find R(1,p1), K(1,p1) we getF(1,p1)078644.86 (profit for static pricing policy). Put n02 and find R(2,p2), andF(2,p2)079177.11 and so on. This is to observe that profit function F(n,pn) isincreases till n03, and thereafter decreases (see Table 1). It reveals that optimalprofit is F(3,p3)079193.44 (profit for dynamic pricing policy) for optimal pricesetting at n03 and respective selling prices are p1074.57, p2071.15, p3069.88 ininterval [0, 200/3], [200/3, 400/3] and [400/3, 600/3] respectively. Noticeable result

8 OPSEARCH (Jan–Mar 2013) 50(1):1–13

from Table 1 is that total incurred cost K(n,pn) exists for n02, but optimal profit existsat n03. This reveals that minimum cost does not guarantee to get maximum profit buthigh volume and market capitalization leads to earn more profit.

Dynamic pricing policy provides optimal profit ($79193.44) at n03 which ishigher than static pricing policy optimal profit $78644.86 (see Table 1 and Fig. 4).Holding cost decereases upto n03 and thereafter increases (Fig. 1), total inventorycost is continuiou increases except at n03 (Fig. 2). Optimal profit increases upto n03and thereafter decereases. Revenues increases as optimal price settings increases(Figs. 3 and 4).

Fig. 1 Holding costs for differ-ent price settings

Table 1 Optimal policy

n H(n,pn) R(n,pn) K(n,pn) F(n,pn) Selling prices for different price settings

1 2208.73 182669.36 104024.49 78644.86 71.87

2 1764.26 182907.14 103730.02 79177.11 73.57, 70.17

3a 1681.40a 182990.60a 103797.20a 79193.44a 74.57, 71.15, 69.88

4 1689.24 183033.18 103955.00 79078.18 75.28, 71.85, 70.58, 69.76

5 1733.41 183058.99 104149.17 78909.82 75.83, 72.40, 71.12, 70.29, 69.69

6 1795.77 183076.31 104361.53 78714.78 76.28, 72.85, 71.56, 70.74, 70.12, 69.64

7 1868.53 183088.75 104584.29 78504.45 76.99, 73.23, 71.94, 71.11, 70.50, 70.01, 69.61

8 1947.79 183098.09 104813.55 78284.54 76.99, 73.56, 72.27, 71.44, 70.82, 70.33, 69.93,69.58

9 2031.39 183105.39 105047.15 78058.23 77.29, 73.85, 72.56, 71.72, 71.11, 70.62, 70.21,69.87, 69.57

10 2118.03 183111.24 105283.79 77827.44 77.55, 74.11, 72.82, 71.98, 71.36, 70.87, 70.47,70.12, 69.82, 69.55

a Denotes optimal solution

OPSEARCH (Jan–Mar 2013) 50(1):1–13 9

Fig. 3 Revenues for differentprice settings

Fig. 4 Profit for different pricesettings

Fig. 2 Total costs for differentprice settings

10 OPSEARCH (Jan–Mar 2013) 50(1):1–13

6 Simulation

Optimal number of price settings is adverse sensitive for parameter β, holding cost h,and price setting cost c0 (see Table 2). It reveals the optimal number of price settingsis of decreasing nature when these parameters are increasing. However the set-up costA does not affect the optimal number of price settings. Optimal price setting increaseswhen b increases and provides lesser optimal profit. Moreover, if parameter a is highthen the number of price settings and profit both increases.

Table 2 Sensitiveness with respect to different parameters

Sensitiveparameters

n q H(n, pn) R(n, pn) K(n, pn) F(n, pn) Selling prices for differentprice settings

β00.3 3 2943 1721.53 262747.63 119877.29 142870.33 92.75, 88.18, 86.48

β00.4 3 2542 1681.40 182990.61 103797.16 79193.44 74.57, 71.15, 69.88

β00.5 2 2141 1724.13 131853.59 87649.89 44203.70 62.86, 60.15

β00.6 2 1740 1684.01 95137.88 71569.77 23568.11 55.72, 53.47

β00.7 2 1339 1643.88 66615.21 55489.64 11125.56 50.62, 48.70

β00.8 2 938 1603.76 43213.19 39409.52 3803.67 46.79, 45.12

h0 .002 3 2538 1680.88 182829.83 103636.64 79193.18 74.58, 71.20, 69.97

h0 .005 3 2526 1679.33 182344.39 103155.09 79189.29 74.63, 71.35, 70.22

h00.01 3 2506 1676.73 181524.94 102352.49 79172.44 74.72, 71.60, 70.63

h00.05 2 2346 1739.76 174442.38 95865.52 78576.86 74.79, 73.84

C00200 2 2542 1764.26 182907.13 103830.02 79077.11 73.57, 70.17

C00400 2 2542 1764.26 182907.13 104230.02 78677.11 73.57, 70.17

C00600 2 2542 1764.26 182907.13 104630.02 78277.11 73.57, 70.17

C00900 1 5041 2208.73 182669.36 104774.49 77894.86 71.87

A050 3 2142 1641.40 164950.61 109173.60 55777.01 79.57, 76.15, 74.88

A060 3 1742 1601.40 142910.61 106550.04 36360.56 84.57, 81.15, 79.88

A070 3 1342 1561.40 116870.61 95926.48 20944.12 89.57,, 86.15, 84.88

A080 3 942 1521.40 86830.61 77302.92 9527.68 94.57, 91.15, 89.80

A090 3 542 1481.40 52790.61 50679.37 2111.24 99.57, 96.15, 94.88

a050 3 2142 1681.40 182990.61 103797.16 79193.44 74.57, 71.15, 69.88

a075 3 5042 2598.07 520218.36 204713.83 315504.53 105.82, 102.40, 101.13

a0100 3 7542 3514.73 1013696.10 305630.49 708065.62 137.07, 133.65, 132.38

a0125 4 10042 4314.24 1663466.40 406580.00 1256886.44 169.03, 165.60, 164.33,163.51

a0150 4 12423 5189.24 2469444.20 507455.00 1961989.20 200.28, 196.85, 195.58,194.76

a0175 4 15042 6064.24 3431671.90 608330.00 2823341.96 231.53, 228.10, 226.83,226.01

b02 3 2142 1681.40 182990.60 103797.20 79193.44 74.57, 71.15, 69.88

b04 3 1687 1689.99 104532.20 69631.51 34900.68 66.62, 59.75, 57.19

b06 4 833 1888.76 45347.67 35806.05 9541.62 60.82, 50.49, 46.61

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7 Conclusion, recommendations and future research

Dynamic pricing policy is discussed with logarithmic decreasing and price sensitivedemand. Optimal profit is better for n >1 (dynamic policy) than n01. The proposedmodel helps to stay the product in the declining market as well as to compete withother products. It is found that dynamic pricing policy with logarithmic demandpattern produces higher return at the smaller values of β parameter. A characteristicproperty has been established justifying that average price remains unchanged even ifthe number of price setting changes.

The suggested model encourages inventory managers to consider the demandfunction depending on time and price both. The time alone or price alone may notbe better strategy for all items. Inventory managers are suggested to decide their mostfrequent cost and deterioration level to get maximum profit. It is advised to inventorymanagers to keep β parameter low, rather than lowering cost for earning more profit.Also, the high initial demand and low holding cost together provide higher profit tothe inventory holder. Inventory managers may apply the proposed model for productshaving logarithmic demand with price sensitiveness, for example cosmetic products.

For further research one can introduce more model parameters such as timedependent holding cost, two parameter logarithmic demand function, and thendevelop the dynamic replenishment/pricing policy for deteriorating/amelioratingitems. It further needs to study that how model performs with shortage in the system.

Acknowledgement Authors are thankful for rigorous review of the manuscript by reviewer and provid-ing useful suggestions. I am thankful to Dr. Hari Sing Gour Central University Sagar India, for providingme University platform for doing research. The motivation for this article is derived from a nationalworkshop on “Inventory Modeling” held at Banasthali Vidyapeeth, Rajasthan, India.

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