5-Maths - IJMCAR -Optimal INVENTORY - RP.Tripathi - Paid

International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN 2249-6955 Vol.2, Issue 3 Sep 2012 35-49 © TJPRC Pvt. Ltd.,

OPTIMAL INVENTORY POLICIES WITH EXPONENTIAL DEMAND RATE AND A CASH DISCOUNT UNDER TRADE CREDIT DEPENDING ON ORDER QUANTITY R.P.TRIPATHI Department of Mathematics, Graphic Era University, Dehradun (Uk) India

ABSTRACT This study develops an EOQ (economic order quantity) model for cash discount optimal inventory policies where demand rate is exponential and deterioration rate is zero. In this model the seller proposes a fixed trade credit M1 or M2 and sales revenue generated during the credit period. During the credit period, the retailer can earn more by selling products. In this model, the seller must be paid for the item as soon as the customer receives them during cash discount and delay in payment. Some of the item may exponentially time dependent in the course of time. In this regard, the author develops an EOQ model for exponential time dependent demand rate. The mathematical model is developed by considering seven different cases for finding total relevant cost. The total relevant cost of the model is minimized.

Taylor’s series expansion (for first order approximation as well as second order approximation) is applied for finding closed form solution. Furthermore, five different results have been discussed. Finally numerical examples provide the solution procedure to obtain optimal cycle time and optimal total relevant cost.

KEYWORDS: Cash Discount, Order Quantity, Inventory, Trade Credit INTRODUCTION In classical EOQ (Economic order quantity) models it is assumed that the demand rate to be time – dependent or constant but independent of stock – status. However, the consumption rate may be influenced by the stock levels for certain types of inventory particularly consumer goods. Under the classical EOQ model, it is assumed that the retailer must pay for the items upon receiving them. In real world, suppliers frequently offer retailers a fixed time period for payment of the amount owed. Usually, there is no interest charge if the outstanding amount is paid within this fixed period. However, if the payment is not paid in full by the end of the fixed period, interest is charged on the outstanding amount.

The permissible delay period for payment is beneficial to the supplier to attract new customers and price deduction. However, a common practice in industries is to provide the customer either cash discount or permissible delay in payment. The cash discount encourages the customer to pay cash ahead of time to reduce the trade risk.

36

R.P.Tripathi

Goyal [1] derived an EOQ model under the conditions of a permissible delay in payments. In his paper Goyal ignored the difference between the selling price and purchase cost. Agarwal and Jaggi [2], then extended Goyal’s [1] model to consider the deteriorating items. Dave [3] corrected Goyal’s model by assuming the selling price is necessarily higher than its purchase cost. Jamal et al. [4] further generalized the model to allow shortages. Chang [5] proposed an inventory model under a situation that the supplies provide the purchaser a permissible delay in payments, if the quantity of the purchaser’s order is large. Chung [6] developed an alternative approach to determine the economic order quantity under the condition of trade credit being granted. Teng [7] amended Goyal’s model by considering the difference between unit price and unit cost and found it to be the economic sense for a well- established buyer to order smaller quantities and avail itself of the benefits of the permissible delay mere frequently.

Chung and Huang [8] developed an economic production quantity model (EPQ) for a retailer where the supplies offer a permissible delay in payments. Teng et al. [9] developed economic order quantity model with trade credit financing for non- decreasing demand. In paper [9], the main objective is to minimize the profit per unit of time for the first replenishment cycle time and obtained conclusions on important and relevant managerial phenomena. Sarkar [10] developed an EOQ model with delay in payments and time varying deterioration rate. In paper [10] demand and deterioration rate are both time- dependent and retailers are allowed a trade- credit offered by the suppliers to buy more items with different discount rate on the purchasing costs. Ghosh et al. [11] developed optimal price and lot size determination for a perishable product under conditions of finite production, partial backordering and lost sale. Tripathi and Misra [12] presented an inventory model with shortages, time – dependent demand rate and quantity dependent permissible delay in payment. In paper [12] the total average cost per unit time is a concave function of time is obtained.

It has been observed that the demand is usually influenced by the amount of stock displayed in the shelves. Ray and Chaudhuri [13] presented an EOQ model with stock – dependent demand, shortage, inflation and time- discounting. In paper [13] the effects of inflation and the time- value of money are taken into account, considering two separate inflation rates. Hou [14] discussed and presented an EOQ model for deteriorating with stock- dependent consumption rate and shortages under inflation and time discounting and obtained the total cost function is convex. Ouyang et al. [15] developed an EOQ model for deteriorating items under trade credits. In paper [15] authors provided the optimal policy for the customer to obtain its minimum cost when the supplier offers not only a permissible delay but also a cash discount. Sarkar and Moon [16] developed an EPQ model with inflation in an imperfect production system. Sarker [17] extended the EMQ model with variable reliability and inflation. Jaggi, et al. [18] developed optimal order policy for deteriorating items with inflation induced demand. This paper [18] presented the optimal inventory replenishment policy of deteriorating items under inflationary conditions using a discounted cash flow (DCF) approach over a finite planning horizon. An inventory model for deteriorating items with stock- dependent consumption rate and shortages under inflation and time discounting was discussed by Hou [19]. Jaggi et al. [20] developed retailer’s optimal ordering policy under two stage trade credit financing. In paper [20] an inventory model under two levels of trade period

Optimal Inventory Policies with Exponential Demand Rate and Cash Discount Under Trade Credit Depending on Order Quantity

37

policy by assuming the demand is function of credit period offered by retailer to the customers using discounted cash flow (DCF) approach is obtained. Teng [21] developed discounted cash- flow analysis on inventory control under various supplier’s Trade credits.In this model [21] Teng obtained the discounted cash flow (DCF) approach is not only simple to understand but also easy to identify which alternative is less cost.Chang [22] presented an EOQ model with deteriorating items under inflation when supplier credits linked to order quantity. In this paper [22] Chang established an EOQ model for determining items under inflation when the supplier offers a permissible delay to the purchaser if the order quantity is greater than or equal to a predetermined quantity. Hou and Lin [23] developed a cash flow oriented EOQ model with deteriorating items under permissible delay in payments. In paper [23] by applying discounted cash flows approach for problem analysis, the optimal (minimum) annual total relevant cost is obtained.

In most business transaction, the supplier will offer the trade credit mixing cash discount. In general, a supplier is always willing to offer the purchaser either a cash discount or a permissible delay of payments if the purchaser orders a large quantity. Khanra et al. [24] discussed an EOQ model for a deteriorating item with time- dependent quadratic demand under permissible delay in payment. In paper [24] the deterioration rate is assumed to be constant and the time varying demand rate in taken to be a quadratic function of time. Sana [25] discussed price sensitive demand for perishable items – an EOQ model. This paper [25] developed a finite time- horizon deterministic economic order quantity (EOQ) model where the rate of demand decreases quadratically with selling price. An inventory model with generalized type demand, deterioration and backorder rates was discussed by Hung [26]. The paper [26] is the extension of the paper of Skouri et al. [27]. Hung extended their [27] inventory model from ramp type demand rate Weibull deterioration rate to arbitrary demand rate and arbitrary deterioration rate in the consideration of partial backorder. Hsieh and Dye [28] discussed pricing and lot – sizing policies for deteriorating items with partial backlogging under inflation. In paper [28] Hsieh and Dye developed an inventory lot- size model for deteriorating items under inflation using a discounted cash flow (DCF) approach over a finite planning horizon. Liao and Huang [29] presented an inventory model for optimizing the replenishment cycle time for a single deteriorating item under a permissible delay in payments constraints on warehouse capacity. Chung and Liao [30] discussed the optimal order quantity of the EOQ model that is not only dependent on the inventory policy but also on firm credit policy by using a discounted cash- flow (DCF) approach and trade credit depending on the quantity ordered.

This paper deals with the problem of determining the economic order quantity when the supplier provides a cash discount and permissible delay to the customers. In this paper the author assumes that the demand rate is exponential time dependent. Optimal solutions of the total relevant cost have been obtained. Truncated Taylor’s series is used for finding closed form solution. In addition we discuss results for all cases. Truncated Taylor’s series for second order approximation is also used for more appropriate results.

The rest of the paper is organized as follows: In the next section assumption and notations are given followed by mathematical formulation is in section 3. In section 4 theoretical results are given for

38

R.P.Tripathi

finding optimal solution followed by some results is given in section 5. Numerical examples are given in section 6. In addition the second order truncated Taylor’s series approximation for finding optimal solutions are given in section 7 with numerical examples. Finally, we draw the conclusion and future research direction in section 8.

NOTATIONS AND ASSUMPTIONS The following notations are used through this manuscript: A : ordering cost H

: unit stock holding cost, excluding capital opportunity cost

C

: unit purchase cost

Ic

: annual interest charges for inventory item

Ie

: annual interest rate that can be earned, Ic > Ie

Β

: cash discount rate, 0 < β < 1

λ0

: initial annual demand

Q

: order quantity

s

: unit selling price per item, s > c

T

: replenishment cycle time

Td

: time at which the cash discount and permissible delay is permitted i.e.

Td =

1

α

log e (1 +

α Qd ) λ0

( : quantity at which the cash discount and delay in payments is permitted : the period of cash discount in years : the period of trade credit in years M2 > M1

Qd M1 M2

λ eα t

λ > 0, 0 < α < 1

D≡ D(t) = 0 : the demand rate , 0 I(t) : inventory level at time ‘t’ T1, T2, T3, T4, T5, T6, T7 : the optimal cycle time for case 1, 2, 3, 4, 5, 6, and 7 respectively for first order approximation T1’,T2’,T3’,T4’,T5’,T6’,T7’ : the optimal cycle time for case 1, 2, 3, 4, 5, 6, and 7 respectively for second order approximation T* : the optimal cycle time TRCi(T) :annual total relevant cost, i = 1,2,3,4,5,6,7 for 1st approximation TRCi’(T) :annual total relevant cost, i = 1,2,3,4,5,6,7, for 2nd order approximation In addition the following assumptions are being made to derive the EOQ model:

D(t) =λ0 eα t )

1.

2.

3.

Demand rate is time dependent and increasing exponential function ot time ( Lead time is zero Shortages are not allowed

4.

Time period is infinite

5.

The seller proposes a fixed trade- credit period M1 or M2, and the sales revenue generated during the credit period is deposited in an interest- bearing account with rate Ie . The trade credit is settled and the buyer starts paying for the interest charges for the items in stock with rate Ip at the end of the period.

6.

If Q < Qd , the credit period is not permitted else credit period M1 or M2, is permitted

39

Optimal Inventory Policies with Exponential Demand Rate and Cash Discount Under Trade Credit Depending on Order Quantity

7.

If Q ≥ Qd , the supplier offers a cash discount when payment is paid with M1 else the full payment must be paid within M2.

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

dI ( t ) = − λ0 eα t , 0 < α < 1 , 0 ≤ t ≤ T dt (1) With the boundary conditions I(0) = Q , I(T) = 0. The solution of (1) is given by λ 0≤t ≤T I (t ) = 0 (eα T − eα t ) , α (2) and the order quantity Q = I(0) = I ( t ) = λ 0 ( e α T − 1) α (3) The annual total relevant cost TRC(T) consists of the following elements: (a) ordering cost (b) the stock holding cost excluding interest charges (c) cash discount earned if the payment is made at M1 (d) the interest charges for unsold item after trade credit period M2 (e) the interest earned from sales revenue during the credit period The annual ordering cost = A/T (4) The annual stock holding cost = h

T

T

hλ0

α ∫ I (t ) dt = α T {(α T − 1) e

T

2

+ 1}

0

(5) The following cases based on the marketing policy are provided to determine the cash discount, interest charges and earned. (a).When trade policy is not permitted i.e. T < Td We see that if Q < Qd , then the payment must be made immediately.

Case 1. 0 < T < Td Since the replenishment time interval T is shorter than Td (i.e. T < Td) ⇒ Q < Qd, then the seller must be paid for the items as soon as the customer receives them. Therefore, this is the classical EOQ model.

The cost of purchasing = cQ = c λ 0 ( e α T − 1) T αT (6)

The annual cash discount earned =

cI c λ 0 {(α T − 1) e α T + 1} α 2T

(7) Therefore total relevant cost is given by αT TRC1(T) = A + h λ 0 {(α T − 1) e α T + 1} + c λ 0 ( e − 1) + cI c λ 0 {(α T − 1) e α T + 1} αT α 2T T α 2T (8) (b). When marketing policy is permitted i.e. T > Td and payment is paid at time M1.

In this case following three cases arise:

40

R.P.Tripathi

Case 2: Td ≤ M1≤ T In this case, cycle time T is greater than or equal to both Td and M1 , the customer saves βcQ/T, due to

price discount. Therefore the discount rate per year is given by

c (1 − β ) λ 0 ( e α T − 1) αT (9) During [M1, T] period, the customer sells products and deposit’s the revenue into an account that earns Ic per dollar per year. Interest payable per year is

cI c (1 − β ) cI (1 − β ) λ αT αT αM ∫M I (t ) dt = c α 2T 0 {α (T − M 1 ) e − ( e − e 1 )} T 1 T

(10) The interest earned per year is M sI e 1 sI λ λ 0 e α t .tdt = e2 0 (α M 1 − 1) e α M 1 + 1 ∫ α T T 0 (11) Therefore, the annual total relevant cost is given by TRC2(T)

{

=

}

c (1 − β )λ0 ( eα T − 1) cI c (1 − β )λ0 A h λ0 + 2 {(α T − 1)eα T + 1} + + {(α (T − M 1 ))eα T − (eαT − eα M1 } T α T αT α 2T sI λ − e2 0 {(α M 1 − 1) e α M 1 + 1} α T

(12) Case 3: M1≤ Td ≤ T In this case, the cycle time T is also greater than or equal to both Td amdM1. Therefore case 3 is similar

to case 2. Thus the annual total relevant cost is given by: TRC3(T) c (1 − β ) λ 0 ( e α T − 1) cI c (1 − β ) λ 0 A hλ = + 2 0 {(α T − 1) e α T + 1} + + (α (T − M 1 )) e α T − ( e α T − e α M 1 T α T αT α 2T sI λ − e2 0 {(α M 1 − 1) e α M + 1} α T (13)

{

}

1

Case 4: Td ≤ T < M1 In this case, cash discount period M1 is greater than the cycle time T, the customer sells c λ0 α T units in total by the end of the replenishment cycle time T and has c λ 0

α

(e

− 1)

α

( e α T − 1) to pay

the supplier in full by the end of the credit period M1. Thus there is no interest payable. The interest earned per year is: T  sI e λ 0 sI e  T αt αt (α M 1 − 1) e α T − α ( M 1 − T ) + 1  ∫ λ 0 e .tdt + ( M 1 − T ) ∫ λ 0 e dt  = 2 α T 0 T 0  (14) Therefore, the annual total relevant cost is TRC4(T)

{

=

}

c (1 − β )λ0 ( eα T − 1) sI e λ0 A h λ0 + 2 {(α T − 1) eα T + 1} + − 2 {(α M 1 − 1)eα T − α ( M 1 − T ) + 1} αT α T T α T

(15)

41

Optimal Inventory Policies with Exponential Demand Rate and Cash Discount Under Trade Credit Depending on Order Quantity

(c). When trade policy is permitted and payment is paid at time M2. In this situation, there are three cases to discuss: Case 5. Td ≤ M2 ≤ T In this case, there is no cash discount, because the payment is paid at time M2. During the permissible delay period, the buyer sells products and deposits the revenue into an account that earns Ie per dollar per year. Therefore, the interest earn per year is: T sI e sI λ λ 0 e α t .tdt = e2 0 (α T − 1) e α M 2 + 1 ∫ α T T 0 (16) Hence the annual total relevant cost is given by

{

}

TRC5(T) = A hλ0 c λ ( e α T − 1) cI c λ 0 sI λ + 2 {(α T − 1) e α T + 1} + 0 + 2 {α (T − M 2 ) e α T − ( e α T − e α M 2 )} − e2 0 {(α M 2 − 1) e α M 2 + 1} αT α T α T T α T

(17) Case 6. M2 ≤ Td ≤T In this case, the cycle time period is longer than payment time period M2, thus this case is similar to case

5. Therefore, the annual total relevant cost is given by TRC6(T) = c λ ( e α T − 1) cI c λ 0 sI λ A hλ0 + 2 {(α T − 1) e α T + 1} + 0 + 2 {α (T − M 2 ) e α T − ( e α T − e α M 2 )} − e2 0 {(α M 2 − 1) e α M 2 + 1} T α T αT α T α T (18) Case 7. Td ≤ T ≤ M2 In this case cycle time period T is shorter than payment time period M2, therefore there is no cash

discount. The annual purchasing cost is similar to case 4. Thus, the annual total relevant is given by: TRC7(T) = hλ0 A + α 2T T

{( α T

− 1) e α T + 1} +

c λ 0 ( e α T − 1) sI eλ 0 − αT α 2T

{( α M

2

− 1) e α M 2 − α ( M

THEORETICAL RESULTS For very smaller value of α, we can assume eα T ≈ 1 + α T (First order approximation) (20) Hence, the annual total relevant cost will be given by: A T R C 1 (T ) ≈ + ( h + cI c ) λ 0 T + c λ 0 T (21) sI λ M 2 A T R C 2 (T ) ≈ + h λ 0 T + c (1 − β ) λ 0 + cI c (1 − β ) λ 0 (T − M 1 ) − e 0 1 T T (22)

T RC 3 (T ) ≈

sI λ M 2 A + h λ 0T + c (1 − β ) λ 0 + cI c (1 − β ) λ 0 (T − M 1 ) − e 0 1 T T

(23) T R C 4 (T ) ≈ (24)

A + h λ 0 T + c (1 − β ) λ 0 − sI e λ 0 M 1 T

2

− T ) + 1}

(19)

42

TRC 5 (T ) ≈

R.P.Tripathi

sI λ M 2 A + h λ0T + c λ 0 + cI c λ 0 (T − M 2 ) − e 0 2 T T

(25)

TRC 6 (T ) ≈

sI λ M 2 A + h λ 0T + c λ 0 + cI c λ 0 (T − M 2 ) − e 0 2 T T

(26) T R C 7 (T ) ≈

A + h λ 0 T + c λ 0 − sI e λ 0 M T

2

(27) The purpose of this approximation is to obtain the unique closed for optimal solution for the optimal value of T. By taking the first and second order derivatives of T R C i ( T ) , for i= 1, 2,------7, with respect to T, we obtain d (T R C 1 (T )) A = − 2 + ( h + cI c ) λ 0 dT T (28)

sI λ M 2 d(TRC 2 (T )) A = − 2 + h λ 0 + cI c (1 − β ) λ 0 + e 02 1 dT T T (29)

d(TRC 3 (T )) sI λ M 2 A = − 2 + h λ 0 + cI c (1 − β ) λ 0 + e 02 1 dT T T (30) d(T R C 4 (T )) A = − 2 + hλ0 dT T (31) d (T R C 5 ( T )) sI λ M A = − 2 + hλ0 + cI cλ0 + e 0 2 dT T T (32)

2 2

d(TRC 6 (T )) sI λ M 2 A = − 2 + h λ 0 + cI c λ 0 + e 0 2 2 dT T T (33) d(T R C 7 (T )) A = − 2 + hλ0 dT T (34) and d 2 (T R C 1 (T )) 2 A = 3 >0 dT 2 T (35)

d 2 (TRC 2 (T )) 2( A − sI e λ 0 M 1 2 ) = >0 dT 2 T3 (36) d 2 (T R C 3 (T )) 2( A − sI e λ 0 M 1 2 ) = >0 dT 2 T3 (37) d 2 (T R C 4 (T )) 2 A = 3 >0 dT 2 T (38)

d 2 (TR C 5 (T )) 2( A − sI e λ 0 M 2 2 ) = >0 dT 2 T3 (39)

Optimal Inventory Policies with Exponential Demand Rate and Cash Discount Under Trade Credit Depending on Order Quantity

43

d 2 (T R C 6 (T )) 2( A − sI e λ 0 M 2 2 ) = >0 dT 2 T3 (40) d 2 (T R C 7 (T )) 2 A = 3 >0 dT 2 T (41) The optimal (minimum) solution of T is obtained by solving d (T R C i (T )) = 0, i = 1, 2, − − − − − 7. we obtain dT A T * = T1 = λ 0 ( h + cI c ) (42)

T * = T2 = T3 =

A − sI e λ 0 M 1 2 λ0 {h + cI c (1 − β )}

(43)

T * = T4 =

A hλ0

(44)

T * = T5 = T6 =

A − sI e λ 0 M 2 2 λ 0 ( h + cI c )

(45)

T * = T7 =

A hλ0

(46)

RESULTS Result 1.

If A < λ 0 ( h + cI c )T d 2 , then T* =T1 Result 2.

If A ≥ λ 0  {h + cI c (1 − β )} T d 2 + sI e M 1 2  , or   T3.

A ≥ λ 0 {h + sI e + c I c (1 − β )} M 1 2 , then T* = T2 =

Result 3.

A ≥ h λ o T d 2 , and A < h λ 0 M 1 2 , then T* = T4.

Result 4.

If A ≥ λ 0 {( h + cI c )T d 2 + sI e M 2 2 } , and Result 5.

If A ≥ h λ 0 T d 2 , and A ≤ h λ 0 M

2 2

A ≥ λ 0 M 2 2 ( h + cI c + sI e ) , then T* = T5 = T6.

, then T* = T7.

NUMERICAL EXAMPLES (CASE 1 TO 7), FOR FIRST ORDER APPROXIMATION

44

R.P.Tripathi

Case 1. (0 < Td < T): Let A= $100/order, h = $10/unit/year, c = $ 20/ unit, λ0 = 1000 unit/year, Ic = $0.09/$/year, Td = 0.2 year, M1= 0.3 year, Ie = $0.06/$/year, A < λ 0 ( h + cI c ) T d 2 = 4 7 2 ,

T* = T1 = 0.092057 year, ⇒ 0 < Td < T, and TRC1(T*) = $ 22172.5561.

Case 2. (Td ≤ M1≤ T): Let A = $ 1200/order, s = $ 25/unit, Ic = 0.09/$/year, Ie = $0.06/$/year, λ0 = 1000 unit/ year, M1= 0.3 year, c = $ 20/ unit, β =0.3, h = $10/unit/year, Td =0.2 year, A ≥ λ 0 { h + cI c (1 − β )}T d 2 + sI e M 1 2  = 592.6, o r , A ≥ λ 0 [{ h + c I c (1 − β ) } + s I e ] M

2 1

=1164.6, T* = T2 = 0.345424639 year,

⇒ Td ≤ M1≤ T, and

TRC2(T*) = TRC3(T*) = $ 20594.64308.

Case 3. (M1≤ Td ≤ T): Let A = $ 1200/order, s = $ 25/unit, Ic = 0.09/$/year, Ie = $0.06/$/year, λ0 = 1000 unit/ year, M1= 0.2 year, c = $ 20/ unit, β =0.3, h = $10/unit/year, Td =0.4 year, then A ≥ λ 0 { h + cI c (1 − β )}T d 2 + sI e M 1 2  = 1073 .4, o r , A ≥ λ 0 [{ h + c I c (1 − β )} + s I e ] M 1 2 = 5 1 0 .4

T* = T3 = 0.318187585 year, ⇒ M1≤ Td ≤ T, and TRC3(T*) = $ 20551.87441.

Case 4. (Td ≤ T < M1): Let A = $ 1200/order, s = $ 25/unit, Ic = 0.09/$/year, Ie = $0.06/$/year, λ0 = 1000 unit/ year, M1= 0.3 year, c = $ 20/ unit, β =0.2, h = $10/unit/year, Td =0.2 year, then A > h λ 0 T d 2 = 4 0 0 or A < h λ 0 M 1 2 = 9 0 0 , T* = T4= 0.212132034 year ⇒ Td ≤ T < M1, and TRC4(T*) = $23328.1746.

Case 5. (Td ≤ M2 ≤T): Let A = $ 4000/order, s = $ 25/unit, Ic = 0.09/$/year, Ie = $0.06/$/year, λ0 = 1000 unit/ year, M2= 0.5 year, c = $ 20/ unit, β =0.3, h = $10/unit/year, Td =0.4 year, and A ≥ λ 0 {( h + c I c ) T d 2 + s I e M 2 2 } = 2 2 6 3 , or A ≥ λ 0 {h + c I c + s I e } M 2 2 = 3 3 2 5 ,

T* = T5 = 0.514616856 year, ⇒ Td ≤ M2 ≤T and TRC5(T*) = $ 26975.39365.

Case 6. (M2 ≤ Td ≤ T): Let A = $ 4000/order, s = $ 25/unit, Ic = 0.09/$/year, Ie = $0.06/$/year, λ0 = 1000 unit/ year, M2= 0.4 year, c = $ 20/ unit, β =0.3, h = $10/unit/year, Td =0.5 year, A ≥ λ 0 {( h + cI c )T d 2 + sI e M 2 2 } = 3190 or A ≥ λ 0 {h + c I c + s I e } M 2 2 = 2 1 2 8 ,

T* = T6 = 0.56448567 year

⇒ M2 ≤ Td ≤ T and TRC’5(T*) = $ 32601.86173.

Case 7. (Td ≤ T ≤ M2): Let A = $ 850/order, s = $ 25/unit, Ic = 0.09/$/year, Ie = $0.06/$/year, λ0 = 1000 unit/ year, M2= 0.4 year, c = $ 20/ unit, β =0.3, h = $10/unit/year, Td =0.2 year, A ≥ h λ 0Td 2 = 4 0 0 & A < h λ 0 M 2 2 = 9 0 0 , T* = T7 = 0.291547594 year, ⇒ Td ≤ T ≤ M2, and TRC7(T*) = $ 25230.9519S. The result obtained with first order approximation of exponential terms, is independent of α, which is not

appropriate. Thus, taking the second order approximation of exponential terms i.e.

eα T ≈ 1 + α T +

α 2T 2 etc.

2 (47) Equations (8), (12), (13), (15), (16), (18), (19) and (20) become cI λ T A h λ 0 T (1 + α T ) αT T R C 1 ' (T ) ≈ + + c λ 0 (1 + ) + c 0 (1 + α T ) T 2 2 2 (48) T R C 2 ' (T ) ≈

− (49)

c I (1 − β ) λ 0 A h λ 0 T (1 + α T ) αT + + c λ 0 (1 − β )(1 + )+ c ( T − M 1 )( T + α T T 2 2 2T

sI e λ 0 M 1 2 (1 + α M 1 ) 2T

2

− M 1)

45

Optimal Inventory Policies with Exponential Demand Rate and Cash Discount Under Trade Credit Depending on Order Quantity

cI (1 − β ) λ 0 A h λ 0 T (1 + α T ) αT + + c λ 0 (1 − β )(1 + )+ c (T − M 1 )(T + α T 2 − M 1 ) T 2 2 2T

T R C 3 ' (T ) ≈

sI e λ 0 M 1 2 (1 + α M 1 ) 2T

− (50)

T R C 4 ' (T ) ≈

sI λ (2 M 1 − T + α M 1T ) A h λ 0 T (1 + α T ) αT + + c λ 0 (1 − β )(1 + )− e 0 T 2 2 T

(51) A h λ 0 T (1 + α T ) αT + + c λ 0 (1 + ) + cI c λ 0 (T − M 2 )(T − M T 2 2 sI λ M 2 (1 + α M 2 ) (52) − e 0 2 2T

T R C 5 ' (T ) ≈

TRC6' (T ) ≈

2

+αT 2)

sI λ M 2 (1 + α M 2 ) αT A hλ0T (1 + αT ) + + cλ0 (1 + ) + cI c λ0 (T − M 2 )(T − M 2 + α T 2 ) − e 0 2 T 2 2 2

(53)

TRC 7 ' (T ) ≈

A h λ 0T (1 + α T ) c λ 0 (1 + α T ) sI e λ 0 (2 M 2 − T + α M 2T ) + + − T 2 2 2

(54) Note that the purpose of this second order approximation is to obtain the unique closed form and appropriate optimal solution for the optimal T. By taking the first and second order derivatives '

of TRCi (T ) , for I = 1, 2, ------7, with respect to T, we obtain (h + cI c )λ0 c λ 0α d ( T R C '1 ( T )) A = − 2 + (1 + 2 α T ) + dT T 2 2 (55)

c λ (1 − β ) cI c (1 − β ) λ 0 d (TRC '2 (T )) A hλ0 M 2 =− 3 + (1 + 2α T ) + 0 + (1 + 2α T − α M 1 − 12 ) dT T 2 2 2 T sI e λ 0 M 1 2 (1 + α M 1 ) + 2T 2 (56) d ( T R C ' 3 ( T )) hλ0 c λ 0 (1 − β ) c I c (1 − β ) λ 0 A M 12 = − 3 + (1 + 2 α T ) + + (1 + 2 α T − α M 1 − ) 2 dT

T

2

2

2

sI λ M 2 (1 + α M 1 ) + e 0 1 2 2T (57) cα (1 − β ) λ 0 sI e λ 0 (α M 1 − 1) d (TRC '4 (T )) A h λ 0 (1 + 2α T ) =− 2 + ++ − dT T 2 2 2 (58) d ( T R C '5 ( T )) hλ0 c λ 0α A = − 2 + (1 + 2 α T ) + + cI c λ 0 ( 2 T − 2 M 2 + 3α T 2 − 2 α M 2 T ) dT T 2 2 sI λ M 2 (1 + α M 2 ) + e 0 2 2 2T (59) d (T R C '6 (T )) A hλ0 c λ 0α =− 2 + (1 + 2α T ) + + cI c λ 0 (2T − 2 M 2 + 3α T 2 − 2α M 2 T ) dT T 2 2 s I e λ 0 M 2 2 (1 + α M 2 ) +

(60)

2T

2

T

46

R.P.Tripathi

d (T R C ' 7 (T )) h λ 0 (1 + 2α T ) c λ 0α sI e λ 0 (α M 2 − 1) A =− 2 + + − dT T 2 2 2 (61) d 2 (TRC '1 (T )) 2 A = 3 + α λ 0 ( h + cI c ) > 0 dT 2 T (62) c I (1 − β ) λ 0 s I λ M 2 (1 + α M 1 ) d 2 ( T R C ' 2 ( T )) 2A M 12 = 3 + h λ 0α + + c (α + )− e 0 1 3 > 0 2 3 dT

T

2

T

T

(63) d 2 ( T R C '3 ( T )) c I c (1 − β ) λ 0 M 2A = + h λ 0α + + (α + dT 2 T3 2 T

2 1 3

)−

sI eλ0 M

(1 + α M 1 )

2 1

T

3

> 0

(64) d 2 ( T R C ' 4 ( T )) 2A = 3 + h λ 0α > 0 dT 2 T

(65) d 2 (T R C '6 (T )) 2 A − sI e λ 0 M 2 2 (1 + α M 2 ) = + α h λ 0 + 2 cI c λ 0 (1 + 3α T − α M 2 ) > 0 dT 2 T3 (66) d 2 (TRC '6 (T )) 2 A − sI e λ 0 M 2 2 (1 + α M 2 ) = + α h λ 0 + 2 cI c λ 0 (1 + 3α T − α M 2 ) > 0 dT 2 T3 2 ' (67) d (T R C 7 (T )) = − 2 A + α h λ 0 > 0 dT 2 T3 (68) The optimal (minimum) solution of T** is obtained by solving d (T R C ' i (T )) = 0, i = 1, 2, − − − − − 7. we obtain dT 2 α ( h + c I c ) λ 0 T 3 + ( h + c I c + cα ) λ 0 T 2 − 2 A = 0 (69) 2 α {h + c I c (1 − β ) } λ 0 T 3 + {h + c I c (1 − β )α + c I c (1 − β )(1 − α M 1 ) } λ 0 T 2 − {2 A + c I c (1 − β ) λ 0 M 1 2 − s I e λ 0 M 1 2 (1 − α M 1 ) } = 0

(70) 2 α {h + cI c (1 − β )} λ 0 T 3 + {h + cI c (1 − β )α + cI c (1 − β )(1 − α M 1 )} λ 0 T

2

− {2 A + cI c (1 − β ) λ 0 M 1 2 − sI e λ 0 M 1 2 (1 − α M 1 )} = 0

(71) 2 α h λ 0 T 3 + λ 0 {h + c (1 − β ) α + s I e (1 − α M 1 ) } T 2 − 2 A = 0 (72) 2α ( h + cI c ) λ 0 T 3 + λ 0 {h + cI c + cα (1 + 3 I c M 2 )} T 2 − {( 2 A + cI c λ 0 M 2 2 − sI e λ 0 M 2 2 (1 + α M 2 )} = 0 (73) 2 α ( h + c I c ) λ 0 T 3 + λ 0 {h + c I c + cα (1 + 3 I c M 2 )} T (74) 2α h λ 0T

3

+ λ 0 {h + c α + s I e (1 − α M 2 )} T

2

2

− {( 2 A + c I c λ 0 M

2 2

− sI eλ0 M

2 2

(1 + α M 2 )} = 0

− 2A = 0

(75)

NUMERICAL EXAMPLES (CASE 1 TO 7): FOR SECOND ORDER APPROXIMATION Case 1.(0 < Td < T): Let A= $100/order, h = $10/unit/year, c = $ 20/ unit, λ0 = 1000 unit/year, Ic = $0.09/$/year, Td = 0.1 year, M1= 0.3 year, Ie = $0.06/$/year, then, then T* = T1’= 0.12429 year, ⇒ 0 < Td < T, and TRC’1(T*) = $ 21604.58312.

Optimal Inventory Policies with Exponential Demand Rate and Cash Discount Under Trade Credit Depending on Order Quantity

47

Case 2. ( Td ≤ M1≤ T): Let A = $ 1200/order, s = $ 25/unit, Ic = 0.09/$/year, Ie = $0.06/$/year, λ0 = 1000 unit/ year, M1= 0.3 year, c = $ 20/ unit, β =0.3, h = $10/unit/year, Td =0.2 year, ⇒ T* = T2’= 0.448961 year, which verify, Td ≤ M1≤ T and TRC’2(T*) = $19005.81213.

Case 3. ( M1 ≤ Td ≤ T): Let A = $ 1200/order, s = $ 25/unit, Ic = 0.09/$/year, Ie = $0.06/$/year, λ0 = 1000 unit/ year, M1= 0.2 year, c = $ 20/ unit, β =0.3, h = $10/unit/year, Td = 0.3 year, ⇒ T* = T3’= 0.449819 year, ⇒ M1 ≤ Td ≤ T and TRC’2(T*) = $19148.14346.

Case 4. ( Td ≤ T ≤ M1): Let A = $ 100/order, s = $ 25/unit, Ic = 0.09/$/year, Ie = $0.06/$/year, λ0 = 1000 unit/ year, M1= 0.6 year, c = $ 20/ unit, β = 0.2, h = $10/unit/year, Td =0.4 year, T* = T4’= 0.549917 ⇒ Td ≤ T ≤ M1,and TRC’4(T*) = $18727.06422.

Case 5 (Td ≤ M2 ≤ T) : Let A = $ 850/order, s = $ 25/unit, Ic = 0.09/$/year, Ie = $0.06/$/year, λ0 = 1000 unit/ year, year, c = $ 20/ unit, β =0.3, h = $10/unit/year, Td =0.7 year, M2 = 0.9 year, T* = T5’ = 1.13006 year ⇒ Td ≤ M2 ≤ T and TRC’5(T*) = $26846.6993.

Case 6.(M2 ≤ Td ≤ T) : Let A = $ 850/order, s = $ 25/unit, Ic = 0.09/$/year, Ie = $0.06/$/year, λ0 = 1000 unit/ year, c = $ 20/ unit, β =0.3, h = $10/unit/year, Td =0.9 year, M2=0.7 year, T* = T6’ = 1.13445 year ⇒ M2 ≤ Td ≤ T and TRC’5(T*) = $27310.48096.

Case 7. (Td ≤ T ≤ M2): Let A = $ 850/order, s = $ 25/unit, Ic = 0.09/$/year, Ie = $0.06/$/year, λ0 = 1000 unit/ year, M2=0.9 year, c = $ 20/ unit, β =0.3, h = $10/unit/year, Td =0.2 year, ⇒ T*= T7 = 0.328655 year ⇒ Td ≤ T ≤ M2 and TRC’7(T*) = $ 23306.30387

CONCLUSIONS This model incorporates some realistic features that are likely to be associated with major kind of inventory. Since the inventory systems always need to invest large capital to buy inventories, which is highly correlated to the return of investment. This model is very useful in business in industries where the effect of inflation and time value of money is applied. In this study the economic order quantity under the condition of cash discount and permissible delay in payments is discussed. The demand rate is exponential time- dependent. The cash discount and delay in payments is not permitted if Q < Qd. Otherwise, the cash- discount period M1 and the fixed credit period M2 is permitted. Theoretical results have been developed by applying Truncated Taylor’s series expansion of the first order approximation and second order approximation. Seven cases are explored in this study in which results 1 to 5 provide the solution procedure to obtain optimal (minimum) T*. Numerical examples are given to illustrate the proposed model.

The model proposed in this paper can be extending in several ways. For instance, we may extend the model for deteriorating items. This paper can also be extending by considering the demand as a function of price and stock dependent. Finally. We could generalize the model for with stochastic demand when the supplier provides credit period and cash discount.

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