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
108
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 + βc + 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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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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