12. Maths - IJMCAR - Optimal PRICING - RP Tripathi - SS Misra - Paid

International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN 2249-6955 Vol.2, Issue 3 Sep 2012 106-120 Š TJPRC Pvt. Ltd.,

OPTIMAL PRICING AND LOT SIZING POLICY WITH TIME-DEPENDENT DEMAND RATE UNDER TRADE CREDITS 1 1

R.P. TRIPATHI & 2S.S. MISRA

Department of Mathematics, Graphic Era University, Dehradun (Uk), India DRDO, New Delhi, India

ABSTRACT Large number of researcher papers has been published for inventory lot- size models under trade credit financing by assuming that demand rate is constant. But demand rate is often not constant. During the growth stage of the product life cycle, the demand function increases with time. In this paper we extend the constant demand to time- dependent demand. This paper derives the problem of determining the retailer’s optimal price and optimal total profit when the supplier permits delay in payments for an order of a product. In this paper demand rate is considered as a function as a function of price and time.

We also provide the optimal policy for the customer to obtain its maximum annual net profit.

Mathematica software is used for finding optimal price and optimal replenishment time simultaneously.

Finally, numerical examples and sensitivity analysis are given to illustrate the theoretical results.

KEYWORDS:

Inventory, Permissible Delay, Demand Rate, Optimal Pricing, Cycle Time.

INTRODUCTION Large number of research papers/ articles has been presented by researchers in different areas in real life problems, for controlling inventory. One of the important problems faced in inventory management is how to maintain and control the inventories of deteriorating items. The most important concern of the management is to decide when and how much to manufacture so that the total cost associated with the inventory system should be minimum.

In deriving the economic order quantity formula, it is assumed that the buyer must pay for the items as soon as he receives them from a supplier. As a marketing policy, some suppliers offer credit period to the buyer in order to stimulate the demand for the product they produces. Trade credit would play an important role in the conduct of business for many reasons. These credit periods for a supplier, who offers trade credit, it an effective means of price discrimination which circumvent antitrust measures and is also an efficient method to stimulate the demand of the product. To motivate faster payments, stimulate more sales, or reduce credit expanses, the supplier also provides its customers a price reduction.

In classical inventory model the demand rate is considered as constant or depends upon single parameter only, like stock, time etc. But changing market conditions have rendered such a consideration is quite

107

Optimal Pricing and Lot Sizing Policy with TimeDependent Demand Rate Under Trade Credits

unfruitful, since in real life situation, a demand cannot depend on a single parameter. In this paper demand rate is considered as the combination of two factors i.e. price and time.

Goyal [1] established a single inventory model for determining the economic ordering quantity in the case that the supplier offers the retailer the opportunity to delay his payment within a fixed time period. Goyal [1] ignored the difference between the selling price and the purchase cost and concluded that the economic replenishment interval and order quantity increases marginally under trade credits.

Goyal’s model was corrected by Dave [2] by assuming the fact that the selling price is necessarily higher than its purchase cost. Aggarwal and Jaggi [3] extended Goyal’s model for deteriorating items. This model was generalized by Jamal et al. [4] to allow for shortages and deterioration. Teng et al. [5] established optimal pricing and ordering policy under permissible delay in payments. In paper [5] it is considered that selling price is necessarily higher than purchase price. Chang et al. [6] established an EOQ model for deteriorating items under supplier credits linked to order quantity. Huang [7] developed economic order quantity under conditionally permissible delay in payments. The main purpose of paper [7] is to investigate the retailer’s optimal replenishment policy under permissible delay in payments.

Hwang and Shinn [8] developed the model for determining the retailer’s optimal price and lot size simultaneously when the supplier permits delay in payments for an order of a product whose demand rate is a function of constant price. Abad and Jaggi [9] formulated model of seller – buyer relationship, they provided procedure for finding the seller’s and buyers best policies under non – cooperative and cooperative relationship respectively. Lokhandwala et al. [10] established optimal ordering policies under condition of extended payment privileges for deteriorating items. Optimal retailer’s ordering policies in the EOQ model for deteriorating item under trade credit financing in supply chain was established by Mahata and Mahata [11] .In paper [11] authors obtained optimal cycle time to minimize the total variable cost per unit time. Tripathi [12] developed an optimal inventory policy for items having constant demand and constant deterioration rate with trade credit. Many related. Many related articles can be found in Chang and Teng [13] chung [14], Chung and Liao [15], Huang and Hsu [16] , Liao et al.

[17], Ouyang et al [18], [19] , Teng et al. [20] and their references.

All the above research papers / articles established their EOQ or EPQ inventory models under trade credit financing by considering that the demand rate is constant. But in case of business, demand rate is not always constant. During the growth stage of the product life cycle, the demand function increases with time. In this paper demand rate is taken as the function of retail price and time.Teng [21] established economic order quantity model with trade credit financing for non- decreasing demand. In paper [21] demand rate is considered as linearly time dependent. Tripathi [22] developed EOQ model with time dependent demand rate and time dependent holding cost function. Tripathi and Kumar [23] presented a model on credit financing in economic ordering policies of time- dependent deteriorating items by considering three different cases. Tripathi and Misra [24] developed an inventory model with shortage, time- dependent demand rate and quantity dependent permissible delay in payment. In paper [24] some important results are obtained. Khanra et al. [25] developed an EOQ for deteriorating items with time – dependent quadratic demand under permissible delay in payment. An inventory model with

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R.P. Tripathi & S.S. Misra

generalized type demand, deterioration and backorder rates was developed by Hung [26]. In paper [26] hung extended their inventory model from ramp type demand rate and weibull deterioration rate to arbitrary deterioration rate in the consideration of partial backorder. Sana [27] presented price- sensitive demand for perishable item- an EOQ model. In paper [27] demand is taken in such a manner that it decreases quadratically with selling price. The objective of paper [27] is to find the optimal ordering quantity and optimal sales prices that maximizes the vander’s total profit. Skouri et al. [28] presented supply chain models for deteriorating products with ramp type demand rate under permissible delay in payments. In paper [28] optimal replenishment policy is obtained for each model.

In this proposed model, the author develops an EOQ model of non- deteriorating items over finite time horizon where demand is function of selling price and time. Mathematical models have been derived under two different circumstances i.e. case I: The credit period m is less than or equal to the cycle time for settling the account and case II: The credit period is greater than the cycle time for settling the account. The numerical examples are given for both the cases. Also sensitivity analysis of the model is given to validate the model for changes in the different parameters.

The rest of the paper is organized as follows: The notations and assumptions used in this paper are given in section 2. In section 3 mathematical formulations is given for each of the two cases i.e. case I and II. Section 4 devoted for determination of optimal pricing and cycle time. Numerical example is cited in section 6. In section 7 optimal solutions for different credit period is provided. In section 8 sensitivity analysis is mentioned. Finally, this closes with concluding remarks and future research directions in section 8.

NOTATIONS AND ASSUMPTIONS The following notations are used throughout this paper: c

:

Unit Purchase Cost

h

:

Inventory Carrying Cost, excluding the capital opportunity cost.

R

:

Capital opportunity cost (as percentage)

m

:

Credit period set by the supplier.

I

:

Earned Interest rate (as a percentage)

S

:

The ordering cost per order

Q

:

Order size

T

:

Replenishment cycle time

q(t)

:

Inventory level at time t

p

:

Unit Retail Price

Îą

:

Scaling Factor ( Îą > 0)

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Optimal Pricing and Lot Sizing Policy with TimeDependent Demand Rate Under Trade Credits

β

:

Index of price elasticity (β > 1)

D

:

The annual demand, as a decreasing function of price and time; we set D(p, t) = α p-βt, where α > 0 and β > 1

Q*

:

optimal order cycle

T = T1

:

optimal cycle time for case I

T = T2

:

optimal cycle time for case II

p = p*

:

optimal retail price

p = p* = p1

:

optimal price for case I

p = p* = p2

:

optimal price for case I

Z(p, T)

:

the total annual profit.

Z*(p, T)

:

optimal total annual profit

Z1*(p,T)

:

optimal annual profit for case I

Z2*(p,T)

:

optimal annual profit for case II

In addition, the following assumptions are being made: (i)

We assume that the demand is a constant elasticity function of the price and time.

(ii)

Shortages are not allowed.

(iii)

The inventory system involves only one item.

(iv)

The supplier proposes a certain credit period and sales revenue generated during the credit period is deposited in an interest bearing account with rate I. At the end of the period, the credit is settled and the retailer starts paying the capital opportunity cost for the items in stock with rate R (R ≥ I).

(v)

Replenishments are instantaneous with a known and constant lead time.

(vi)

Time horizon is infinite.

The total annual profit consists of: (a) the sales revenue, (b) the purchasing cost,

(c) ordering

cost, (d) inventory carrying cost, (e) capital opportunity cost.

MATHEMATICAL FORMULATION The level of inventory I(t) decreases gradually mainly to meet demands. Hence the variation of inventory with respect to time can be described by the following differential equation:

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R.P. Tripathi & S.S. Misra

dq(t) where, a = α p −β = − D(p, t) = − at, 0 ≤ t ≤ T , dt With boundary condition q(T) = 0. Therefore the solution of (1) is given by q(t) =

a (T 2

2

− t 2 ),

(1)

0 ≤ t ≤ T

(2)

and the order size is

Q=

aT 2 2

(3)

Inventory level Q

Inventory level Q

q(t)

q(t)

m

m

time

(a)

time

(b)

Fig.1: Credit Period (m) Verses Replenishment Cycle Time (T)

Now, we formulate the annual net profit Z (p,T). The annual net profit consists of the following elements: (a).

Annual sales revenue = p T

T

∫ at dt = 0

apT 2

(b).

Annual purchasing cost = cQ = caT T 2

(c) .

Annual inventory carrying cost = h T

(d).

Annual ordering cost = s T Annual capital opportunity cost

(e).

T

∫ q(t)dt = 0

haT 3

2

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Optimal Pricing and Lot Sizing Policy with TimeDependent Demand Rate Under Trade Credits

Case 1: m ≤ T (Fig. 1(a)): As products are sold, the sales revenue is used to earn interest with annual rate I during the credit period m. The average number of products in stock earning 2 m m interest during time (0,m) is 1 ∫ D(p, t)t dt = 1 ∫ at 2 dt = am , and the interest earned per order m 0 m 0 3



2



becomes  am  mcI . When the credit has to be settled, the product still in stock is assumed to be  

 3 

financed with annual rate R. The average number of product during (m, T) is

1 T−m

T

∫ q(t)dt,

the

m

T

interest payable per order is cR ∫ q(t)dt . Thus the annual capital opportunity cost = m

T

cR ∫ q(t)dt − m

T

3

am cI 3

=

acR 6

 m3  2T 2 − 3Tm +  T 

 am 3 cI −  3T 

Case 2: m > T (Fig. 1(b)): In this case all the sales revenue is used to earn interest with annual rate I during the credit period m. The annual capital opportunity cost T T = − cI  at 2 dt + (m − T ) at dt  = acI  T − m  T ∫ ∫   T 0 2  3  0 

The annual net profit Z (p,T) can be expressed as Z(p,T) = Sales revenue – Purchasing cost – Ordering cost – Inventory carrying cost – Capital opportunity cost.

The Z (p,T) has two different expressions as follows: Case 1: m ≤ T

Z 1 (p, T ) =

apT acT s ahT − − − 2 2 T 3

2

−

acR 6

 m 3  acIm  2T 2 − 3Tm + + T  3T 

3

(4)

Case 2: m > T 2

aIcT  T   − m 2 3  Hence the total annual profit Z(p,T) is written as Z 2 (p, T ) =

apT acT s ahT − − − 2 2 T 3

−

 Z 1 (p, T ) for m ≤ T Z (p, T ) =   Z 2 (p, T ) for m > T

(5)

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R.P. Tripathi & S.S. Misra

DETERMINATION OF OPTIMAL PRICING AND CYCLE TIME To find an optimal retail price p* and an optimal replenishment cycle time T* which maximizes Z(p,T). Once p* and T* are found, an optimal lot size Q* and optimal annual profit can be obtained by (3) (4) and (5) respectively. Taking the first and second order partial derivatives of Zi(p,T), for i = 1 and 2, with respect to T, and p , we obtain

∂ Z 1 (p, T) ap ac s 2ahT acR  m 3  acIm 3  4T − 3m − 2  − = − + 2 − − ∂T 2 2 3 6  T T  3T 2

(6)

∂ Z 2 (p, T) ap ac s 2ahT acIT acIm = − + 2 − − + ∂T 2 2 3 3 2 T

(7)

∂ Z 1 (p, T) αβ p − β −1T ( p − c ) aT αβ p − β −1 hT 2 =− + + ∂p 2 2 3 αβ p −β −1cR + 6

 2 m3  2T − 3Tm +  T 

(8)

 αβ p −β −1 Im 3 −  3T 

∂ Z 2 (p, T) a  T βhT 2 βIcT  T  = {− (β − 1)p + βc } + +  − m  ∂p p 2 3 2 3 

(9)

 2s ∂ 2 Z 1 (p, T) 2ah 2acR acRm = −  3 + + + 3 3 ∂T 2 T 3T 3 

(10)

3

+

2acIm 3T 3

3

  < 0 

∂ 2 Z 2 (p, T) 2ah acI   2s = − 3 + + <0 3 3  ∂T 2 T

∂ 2 Z1 (p, T) ∂p 2

=−

(11)

(β + 1) − (β − 1)p + βc + hT 2 p2

 

 

3

+

cR  2 m3  2T − 3Tm + 6  T

 cIm 3  a  −  − (β − 1) < 0  3T  p

(12)

∂ 2 Z 2 (p, T) (β + 1) a  − T (β − 1)p + βcT + βhT 2 + βIcT  T − m   − a (β − 1)T < 0 =−    2 2 3 2 3 2p ∂p p2   2 2 ∂ 2 Z 1 (p, T) , ∂ Z 1 (p, T) < 0 , and < 0 ∂T 2 ∂p 2

 ∂ 2 Z 1 (p, T)  ∂T 2 

∂ 2 Z 2 (p, T) ∂ 2 Z 2 (p, T) and , < 0 , and < 0 ∂T 2 ∂p 2 The

optimal

(maximum)

value

  

 ∂ 2 Z1 (p, T)   ∂ 2 Z 1 (p, T)  _  ∂p 2   ∂T ∂ p   

 ∂ 2 Z 2 (p, T)    ∂T 2  

of

(13)

2

  > 0  2

 ∂ 2 Z 2 (p, T)   ∂ 2 Z 2 (p, T)   > 0   _    ∂p 2    ∂T∂p 

T = T *, and p= p*

is

obtained

by

putting,

∂ Z i ( p ,T ) ∂ Z i ( p ,T ) = 0, = 0 , i = 1, 2 . simultaneously for both cases case I and case II. We obtain ∂T ∂p

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Optimal Pricing and Lot Sizing Policy with TimeDependent Demand Rate Under Trade Credits

 4 a ( h + cR ) T 3 − 3 a(p − c + c R m )T 2 − {6 s + ac ( R − 2 I ) m 3 } = 0  , (case: I)  2 2 3 2 3  3 ( β -1 )p T − β {3 c T + 2 ( h + cR ) T − 3 c R m T + c ( R − 2 I ) m } = 0

(14)

 2a ( 2h + cI ) T 3 − 3a(p − c + cIm )T 2 − 6s = 0   3( β -1)p − β {3 c + (2 h + cI )T − 3 cI m } = 0

(15)

, (case: II)

Optimal (maximum) value of p = p* (= p1 for case I) and T = T*(= T1 for case I) and p = p* (= p2 for case II) and T = T*(= T2 for case II) are obtained by solving equations (14) and (15) simultaneously for case I and II respectively.

NUMERICAL EXAMPLES Case I Example 1. Let h = 15, c = 5, α = 106, s = 50, R = 0.25 (=25%), I = 0.1 (=10%), β = 3, m = 5/365, then p = 20.5124 unit, T = 0.711525 year, Q* = 29.3292 units, and Z(p, T) = $ 252.126.

Case II Example 2. Let h = 15, c = 5, α = 106, s = 50, R = 0.25 (=25%), I = 0.1 (=10%), β = 3, m = 260/365, then p = 17.7399 unit, T = 0.705232 year Q* = 44.5431 units, and Z(p, T) = $ 318.478.

OPTIMAL SOLUTION FOR DIFFERENT CREDIT PERIOD ‘m’ Optimal solution of order quantity Q = Q* and annual profit Z(p, T) = Z*(p, T) for different values of credit period ‘m’

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R.P. Tripathi & S.S. Misra

Table 1. case I m≤T

case II m≥T

m (days)

p = p*

T = T* (in years)

Order quantity Q= Q* units

Annual Profit Z(p, T) (dollars)

05

p1=20.5124

T1= 0.711525

Q1 = 29.3292

252.126

10

p1=18.9728

T1= 0.707076

Q1 = 36.6023

257.846

15

p1=18.9026

T1= 0.706619

Q1 = 36.9637

258.744

20

p1=18.8399

T1= 0.70415

Q1 = 37.0736

259.645

25

p1=18.7733

T1= 0.701665

Q1 = 37.2056

260.554

30

p1=18.7068

T1= 0.699161

Q1 = 37.3358

261.471

35

p1=18.6398

T1= 0.696633

Q1 = 37.4675

262.298

40

p1=18.5722

T1= 0.694075

Q1 =37.6004

263.334

50

p1=18.4350

T1= 0.688856

Q1 =37.8702

265.239

60

p1=18.2945

T1= 0.683465

Q1 =38.1453

267.191

70

p1=18.1499

T1= 0.677859

Q1 =38.4261

269.197

80

p1=18.0002

T1= 0.671993

Q1 =38.714

271.263

100

p1=17.6818

T1= 0.659275

Q1 =39.3119

275.609

120

p1=17.3300

T1= 0.644829

Q1 =39.9451

280.307

140

p1=16.9324

T1= 0.628012

Q1 =40.6210

285.458

160

p1=16.4709

T1= 0.607893

Q1 =41.3497

291.208

180

p1=15.9158

T1= 0.582943

Q1 =42.1441

297.774

200

p1=15.2092

T1= 0.550176

Q1 =43.0184

305.526

260

p2=17.7399

T1= 0.705232

Q2 = 44.5431

318.478

280

p2=17.7685

T1= 0.708955

Q2 = 44.7976

321.857

300

p2=17.7973

T1= 0.712694

Q2 = 45.052

325.227

320

p2=17.8263

T1= 0.716448

Q2 = 45.306

328.586

340

p2=17.8555

T1= 0.720216

Q2 = 45.5596

331.935

360

p2=17.8849

T1= 0.723998

Q2 = 45.8126

335.274

380

p2=17.9146

T1= 0.727795

Q2 = 46.0645

338.601

400

p2= 17.9444

T1= 0.731606

Q2 = 46.3167

341.918

420

p2= 17.9745

T1= 0.73543

Q2 = 46.5674

345.223

440

p2=18.0048

T1= 0.739269

Q2 = 46.8177

348.516

460

p2=18.0353

T1= 0.743121

Q2 = 47.0673

351.797

480

p2=18.0659

T1= 0.746986

Q2 = 47.3169

355.067

500 540 580

p2=18.0968 p2=18.1591 p2= 18.222

T1= 0.750864

Q2 = 47.5650

358.325

T2 = 0.75866

Q2 = 48.0597

364.803

T2 = 0.766506

Q2 = 48.5511

371.23

620

p2=18.2861

T2 = 0.774401

Q2 = 49.0387

377.604

660

p2=18.3506

T2 = 0.782344

Q2 = 49.5239

383.925

700

p2=18.4159

T2 = 0.790334

Q2 = 50.0049

390.191

750

p2=18.4984

T2 = 0.800385

Q2 = 50.6017

397.946

800

p2=18.5819

T2 = 0.810504

Q2 = 51.1929

405.611

850

p2=18.6665

T2 = 0.820689

Q2 = 51.7772

413.185

900

p2=18.7520

T2 = 0.830938

Q2 = 52.3558

420.666

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Optimal Pricing and Lot Sizing Policy with TimeDependent Demand Rate Under Trade Credits

SENSITIVITY ANALYSIS Case I Table 2(a) Variation of ‘h’ keeping parameters same as in Example 1. h

p = p1

T = T1

Order quantity

Annual Profit Z(p, T) (in

(in years)

Q

dollars)

20

21.4042

T2 = 0.673564

Q1 = 23.1329

162.027

21

24.0797

T2 = 0.672688

Q1 = 16.2048

145.336

22

22.296

T2 = 0.673629

Q1 = 20.4706

134.601

23

23.1879

T2 = 0.676295

Q1 = 18.3425

123.288

24

24.9716

T2 = 0.680629

Q1 = 14.8748

112.99

25

25.8634

T2 = 0.686601

Q1 = 13.6245

103.09

30

28.5389

T2 = 0.741777

Q1 = 11.836

61.8775

35

35.6737

T2 = 0.848628

Q1 = 7.93159

36.2490

Table 2(b) p = p1

c

Variation of ‘c’ keeping parameters same as in Example 1.

T = T1 Order quantity Annual Profit Z(p, T) (in years)

Q units

(in dollars)

6

26.323

0.849565

Q1 = 19.7860

197.292

7

27.143

0.986479

Q1 = 24.3317

175.033

8

24.2809

1.12239

Q1 = 44.0011

96.1106

9

79.5446

1.25739

Q1 = 1.57064

10.4392

10

19.0268

1.39159

Q1 = 140.571

-760.626

11

53.558

1.5251

Q1 = 7.56996

89.0645

12

not valid

---------

-------------

13

97.1236

Q1 = 1.74941

33.0302

Table 2(c)

not valid 1.79039

Variation of ‘s’ keeping parameters same as in Example 1.

s

p = p1

T = T1 (in years)

Order quantity Q units

Annual Profit Z(p, T) (in dollars)

55

19.4584

0.735182

Q1 = 36.6808

250.047

60

20.0664

0.758842

Q1 = 35.6340

343.195

65

21.953

0.7782543

Q1 = 28.6241

231.014

70

22.296

0.806318

Q1 = 29.3293

225.205

75

22.5933

0.830197

Q1 = 29.8809

219.794

80

22.296

0.854207

Q1 = 32.9166

217.906

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R.P. Tripathi & S.S. Misra

Case II Table 3(a) Variation of ‘h’ keeping parameters same as in Example 2. h

p = p2

T = T2

Order quantity

Annual Profit Z(p, T)

(in years)

Q units

(in dollars)

20

20.2806

0.656021

Q2 = 25.7966

200.117

21

20.870222

0.652857

Q2 = 23.4437

182.856

22

21.4922

0.651458

Q2 = 21.3747

167.119

23

22.1495

0.651711

Q2 = 19.5429

152.73

24

22.8451

0.653538

Q2 = 17.9115

139.542

25

23.5824

0.656885

Q2 = 16.4507

127.429

26

24.3651

0.661722

Q2 = 15.1362

116.283

27

25.1977

0.668042

Q2 = 13.9475

106.013

28

26.0848

0.675858

Q2 = 12.8682

96.5373

29

27.0319

0.685205

Q2 = 11.8845

87.7868

30

28.0454

0.696138

Q2 = 10.9844

79.700

31

29.1323

0.708734

Q2 = 10.1581

72.2231

Table 3(b)

Variation of ‘c’ keeping parameters same as in Example 2.

c

p = p2

T = T2 (in years)

Order quantity Q units

1

3.375845

0.155053

Q2 = 312.452

1606.9

2

7.40305

0.30253

Q2 = 112.791

812.406

3

10.9429

0.443034

Q2 = 74.894

538.205

4

14.3861

0.577105

Q2 = 55.931

400.947

Table 3(c) s

p = p2

Annual Profit Z(p, T) (in dollars)

Variation of‘s’ keeping parameters same as in Example 2.

T = T2

Order quantity

Annual Profit Z(p, T)

(in years)

Q units

(in dollars)

45

17.4569

0.686394

Q2 = 44.2808

326.448

40

17.1723

0.667449

Q2 = 43.9866

334.673

35

16.8855

0.648363

Q2 = 43.6581

343.17

30

16.596

0.629097

Q2 = 43.2907

351.961

25

16.3032

0.609605

Q2 = 42.8794

361.071

20

16.0061

0.589832

Q2 = 42.4990

370.527

15

15.7037

0.56971

Q2 = 41.9056

380.364

10

15.3949

0.549153

Q2 = 41.3262

390.622

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Optimal Pricing and Lot Sizing Policy with TimeDependent Demand Rate Under Trade Credits

All the above observations can be sum up as follows: From Table 1. It can be easily seen that: a)

Increase of credit period ‘m’ results decrease of p = p* = p1, T =T* = T1, slight increase of order quantity Q = Q* = Q1 and increase of annual profit Z(p, T) = Z1*(p, T) ,( for case I).

b) Increase of credit period ‘m’ results slight increase of p = p* = p2, T =T* = T2, and order quantity Q = Q* = Q2 and increase of annual profit Z(p, T) = Z2*(p, T) , (for case I). From Table 2 (a), we observe that: c)

Increase of holding cost ‘h’ results increase of p = p* = p1, slight increase of T = T* = T1, decrease of order quantity Q = Q* = Q1 and decrease of Z(p, T) = Z1*(p, T) .

From Table 2 (b), we observe that:

d) Increase of ‘c’ results increase of p = p* = p1, T =T* = T1, and order quantity Q = Q* = Q1 increases then decreases while Z(p, T) = Z1*(p, T) is not uniform. From Table 2(c), we observe that: e)

Increase of‘s’ results slight increase of p = p* = p1, T =T* = T1, decrease of order quantity Q = Q* = Q1, while annual profit increases then decreases.

From Table 3(a), we observe that:

f)

Increase of ‘h’ results, increase of p = p* = p2, slight increase of T =T* = T2, decrease of order quantity Q = Q* = Q2, and decrease of annual profit Z(p, T) = Z2*(p, T) .

g) From Table 3(b), we observe that: h) Increase of ‘c’ results increase of p = p* = p2, T =T* = T2, but decrease of order quantity Q = Q* = Q2, and decrease of annual profit Z(p, T) = Z2*(p, T). i)

Decrease of ‘s’ results, slight decrease of p = p* = p2, decrease of T =T* = T2 slight decrease of order quantity Q = Q* = Q2, an increase of annual profit Z(p, T) = Z2*(p, T) .

Note: Solution is not valid for every value of ‘h’ ‘c’ and‘s’. Solution exists only for limited values of ‘h’ and ‘c’; beyond the limit solution does not exist. The managerial and marketing point of view it application is limited.

CONCLUSION ANDFUTURE RESEARCH In this paper author developed an optimal pricing and lot size model for a retailer when supplier provides a permissible delay in payments. We derive the first and second order conditions for finding the optimal price and optimal cycle time. In this method shortages are not allowed. Numerical examples are studied to illustrate the proposed model with varying credit period ‘m’. The sensitivity of the solution to changes in the values of different parameters has also been discussed for both the cases i.e. case I and case II. The managerial point of view this model is applicable for limited range, parameters etc.

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118

This paper can be extended for several ways. For instance we may extend the model for continuously variable holding costs such as stock dependent holding cost model. Also we could consider the demand as a function of quantity as well as stock- dependent. Finally, we could generalize the model to allow for shortages, cash discount and inflation rates etc.

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