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Yugoslav Journal of Operations Research 14 (2004), Number 2, 231-246 RETAILER’S REPLENISHMENT POLICIES UNDER CONDITIONS OF PERMISSIBLE DELAY IN PAYMENTS Yung-Fu HUANG Department of Business Administration, Chaoyang University of Technology, Taichung, Taiwan, R.O.C. Received: July 2003 / Accepted: February 2004 Abstract: Goyal (1985) is frequently cited when the inventory systems under conditions of permissible delay in payments are discussed. Goyal implicitly assumed that: 1. The unit selling price and the unit purchasing price are equal; 2. At the end of the credit period, the account is settled. The retailer starts paying for higher interest charges on the items in stock and returns money of the remaining balance immediately when the items are sold. But these assumptions are debatable in real-life situations. The main purpose of this paper is to modify Goyal’s model to allow the unit selling price and the unit purchasing price not necessarily be equal to reflect the real-life situations. Furthermore, this paper will adopt different payment rule. We assume that the retailer uses sales revenue during the permissible credit period to make payment to the supplier at the end of the credit period. If it is not enough to pay off the purchasing cost of all items, the retailer will pay off the remaining balance by taking loan from the bank. So, the retailer starts paying for the interest charges on the amount of loan from the bank after the account is settled. Then the retailer will return money to the bank at the end of the inventory cycle. Under these conditions, we model the retailer’s inventory system as a cost minimization problem to determine the retailer’s optimal cycle time and optimal order quantity. Four cases are developed to efficiently determine the optimal cycle time and the optimal order quantity. Numerical examples are given to illustrate these cases. Comparing with Goyal’s model, we also find that the optimal cycle times in this paper are not longer than those of Goyal’s model. Keywords: EOQ, permissible delay in payments, trade credit. 232 Y.-F Huang / Retailer’s Replenishment Policies Under Conditions 1. INTRODUCTION The classical economic order quantity (EOQ) model assumes that the retailer’s capital is unconstrained and the retailer must be paid for the items as soon as the items are received. However, in practice the supplier will offer the retailer a fixed delay period, which is the trade credit period in settling the accounts. Before the end of trade credit period, the retailer can sell the goods, accumulate revenue and earn interest. A higher interest is charged if the payment is not settled by the end of trade credit period. In the real world, the supplier often makes use of this policy to promote his commodities. Several published papers have appeared in the literature which treat inventory problems under varying conditions. Some of these papers are discussed below. Goyal [13] established a single-item inventory model under permissible delay in payments. Chand and Ward [3] analyzed Goyal's problem [13] under assumptions of the classical economic order quantity model, obtaining different results. Chung [9, 10] developed an alternative approach to determine the economic order quantity under condition of permissible delay in payments. Shah [22], Aggarwal and Jaggi [1] considered the inventory model with an exponential deterioration rate under the condition of permissible delay in payments. Chang et al. [5] extended this issue to the varying rate of deterioration. Chu et al. [8] and Chung et al. [12] also investigated the deteriorating items under this condition and developed efficient approach to determine the optimal cycle time. Liao et al. [19] and Sarker et al. [20] investigated this topic with inflation. Jamal et al. [15] and Chang and Dye [4] extended this issue with allowable shortage. Chung [11] developed an alternative approach to modify Shah’s [22] solution. Chang et al. [6] extended this issue with linear trend demand. Chen and Chuang [7] investigated light buyer’s inventory policy under trade credit by the concept of discounted cash flow. Kim et al. [18] developed an optimal credit policy to increase wholesaler’s profits with price-dependent demand functions. Hwang and Shinn [14] modeled an inventory system for retailer’s pricing and lot sizing policy for exponential deteriorating products under the condition of permissible delay in payment. Jamal et al. [16] and Sarker et al. [21] addressed the optimal payment time under permissible delay in payment with deterioration. Khouja and Mehrez [17] investigated the effect of four different supplier credit policies on the optimal order quantity within the EOQ framework. Shawky and Abou-El-Ata [23] investigated the production lot-size model with both restrictions on the average inventory level and trade-credit policy using geometric programming and Lagrange approaches. Teng [25] assumed that the selling price was not equal to the purchasing price to modify the inventory model under permissible delay in payments. Shinn and Hwang [24] determined the retailer’s optimal price and order size simultaneously under the condition of order-size-dependent delay in payments. They assumed that the length of the credit period was a function of the retailer’s order size, and also the demand rate was a function of the selling price. Arcelus et al. [2] modeled the retailer’s profit-maximizing retail promotion strategy, when confronted with a vendor’s trade promotion offer of credit and/or price discount on the purchase of regular or perishable merchandise. From the above literature review, we understand that Goyal [13] is well known in the study of the inventory systems under conditions of permissible delay in payments. Goyal [13] implicitly makes the following assumptions: Y.-F Huang / Retailer’s Replenishment Policies Under Conditions 1. 2. 233 The unit selling price and the unit purchasing price are assumed to be equal. However, in practice, the unit selling price is not lower than the unit purchasing price in general. Consequently, the viewpoint of Goyal [13] is debatable sometimes. At the end of the credit period, the account is settled. The retailer starts paying for higher interest charges on the items in stock and returns money of the remaining balance immediately when the items are sold. What the above statement describes is just one of the ways how the capital of enterprises is arranged. Based on considerations of profits, costs and developments of enterprises, enterprises may invest their capitals to the best advantage. Hence, the arrangement of capital of an enterprise is an important issue to the enterprise itself. This paper tries to consider some alternatives to move capital to match the policy of enterprise. According to the given arguments, this paper will make the following assumptions to modify Goyal’s model. a) The unit selling price and the unit purchasing price are not necessarily equal to match the practical situations. b) The retailer uses sales revenue during the permissible credit period to make payment to the supplier at the end of the credit period. If it is not enough to pay off the purchasing cost of all items, the retailer will pay off the remaining balance by taking loan from the bank. So, the retailer starts paying for the interest charges on amount of loan from the bank after the account is settled. In addition, this paper also assumes that the retailer does not return money to the bank until the end of the inventory cycle. Incorporating the above assumptions (a) and (b), we tried to develop model of the inventory systems under conditions of permissible delay in payments. 2. MODEL FORMULATION AND CONVEXITY The following notation and assumptions will be used throughout: Notation: D = annual demand A = cost of placing one order c = unit purchasing price per item s = unit selling price per item h = unit stock holding cost per item per year excluding interest charges Ie = interest which can be earned per $ in a year Ip = interest charges per $ investment in inventory per year M = the trade credit period in years T = the cycle time in years TVC(T) = the total variable cost per unit time when T > 0 T* = the optimal cycle time of TVC(T) Q*= the optimal order quantity = DT*. Y.-F Huang / Retailer’s Replenishment Policies Under Conditions 234 Assumptions: 1) 2) 3) 4) 5) 6) Demand rate is known and constant. Shortages are not allowed. Time period is infinite. The lead time is zero. s ≥ c and Ip ≥ Ie. During the time the account is not settled, generated sales revenue is deposited in an interest-bearing account. If the credit period is less than the cycle length, the retailer continues to accumulate revenue and earn interest for the rest of the inventory cycle. At the end of credit period, the retailer pays off the remaining balance by taking loan from the bank if necessary. The retailer does not return money to the bank until the end of inventory cycle. The total annual variable cost consists of the following elements. A (1) Annual ordering cost = . T DTh . 2 (3) Based on the above assumptions (5) and (6), the sales revenue is DsM at the end of the credit period and the remaining balance will be DcT–DsM if the sales revenue is less than the total purchasing cost. Therefore, the remaining balance will be paid off by taking loan from the bank and must be financed at higher interest rate Ip until the end of inventory cycle. But the interest payable will be zero if DsM–DcT ≥ 0. That is, the total sales revenue during the credit period exceeds the total purchasing cost. Hence the interest payable for two cases is obtained as follows: sM (i): T ≥ c (1) Annual interest payable = Ip(DcT–DsM)(T–M) / T. sM (ii): T ≤ c In this case, no interest charges are paid for the items. (2) Annual stock holding cost (excluding interest charges) = (4) According to assumption (6), the interest earned during the credit period and beyond the credit period until the cycle time T. Hence the interest earned for three cases are obtained as follows: sM (i): T ≥ , shown in Figure 1. c In this case, the interest earned must consider two periods. One is the trade credit offered by the supplier within M. At M, the retailer pays the sales revenue DsM to the supplier. Beyond M, the retailer still accumulates the interest earned until the end of inventory cycle.  DsM 2 Ds (T − M ) 2  Annual interest earned =  + (2)  Ie / T . 2  2  Y.-F Huang / Retailer’s Replenishment Policies Under Conditions 235 sM , shown in Figure 2. c In this case, the interest earned is similar to the above case (i). But the total sales revenue during the credit period exceeds the total purchasing cost. Hence the interest earned from the profit DsM – DcT can be accumulated on [ M, T ]. Annual interest earned  DsM 2 Ds (T − M ) 2  = + ( DsM − DcT )(T − M ) + (3)  Ie / T . 2  2  (ii): M ≤ T ≤ $ DsT DsM Time M sM / c T Figure 1: The total accumulation of interest earned when sM / c ≤ T $ DsT DsM DcT Time M T sM / c T sM / c Figure 2: The total accumulation of interest earned when M ≤ T ≤ sM / c (iii): T ≤ M, shown in Figure 3. In this case, the retailer can sell the items and earn interest until the end of the credit period. 236 Y.-F Huang / Retailer’s Replenishment Policies Under Conditions  DsT 2  + DsT ( M − T )  I e / T . Annual interest earned =   2  (4) $ DsT Time T M M Figure 3: The total accumulation of interest earned when T ≤ M From the above arguments, the annual total variable cost for the retailer can be expressed as TVC(T) = ordering cost + stock-holding cost + interest payable – interest earned We show that the annual total variable cost, TVC(T), is given by  TVC1 (T )   TVC (T ) = TVC2 (T )  TVC3 (T )   if if if sM c sM M ≤T ≤ c T ≤M T≥ (5a) (5b) (5c) where A DTh + + [ I p ( DcT − DsM )(T − M )] / T 2 T  DsM 2 Ds (T − M ) 2  − +  Ie / T , 2  2  TVC1 (T ) = TVC2 (T ) = and A DTh  DsM 2 Ds (T − M ) 2  + − + ( DsM − DcT )(T − M ) +  Ie / T T 2 2  2  (6) (7) Y.-F Huang / Retailer’s Replenishment Policies Under Conditions TVC3 (T ) = 237  A DTh  DsT 2 + − + DsT ( M − T )  I e / T . 2 T  2  (8) sM  sM   sM  , we find TVC1   = TVC2   . Similarly, TVC2(M)=TVC3(M). c c    c  Hence TVC(T) is continuous and well-defined. All TVC1 (T ), TVC2 (T ), TVC3 (T ) and TVC(T) are defined on T > 0. Equations (6), (7) and (8) yield At T =  A + DsM 2 ( I p − I e )  h + 2cI p − sI e TVC1′ (T ) = −  ),  + D( 2 2 T   TVC1″ (T ) = 2[ A + DsM 2 ( I p − I e )] >0, T3 h + 2cI e − sI e −A ), TVC2′ (T ) = 2 + D( 2 T 2A TVC2″ (T ) = 3 > 0 , T h + sI e − A ) TVC3′ (T ) = 2 + D( 2 T (9) (10) (11) (12) (13) and TVC3″ (T ) = 2A >0. T3 (14) Equations (10), (12) and (14) imply that all TVC1 (T ), TVC2 (T ) and TVC3 (T ) are convex on T > 0. Then, there are two situations to occur: (1) If s = c, then TVC1′ ( M ) = TVC2′ ( M ) = TVC3′ ( M ) and TVC(T) is convex on T > 0 .  sM  ′  sM  and TVC ′ ( M ) ≠ TVC ′ ( M ) in general. (2) If s > c, then TVC1′  2 3  ≠ TVC2    c   c  Consequently, TVC(T) is piecewise convex but not convex. 3. THE DETERMINATION OF THE OPTIMAL CYCLE TIME T* Let TVCi′ (Ti *) = 0 for all i = 1, 2, 3. We can obtain T1 * = 2[ A + DsM 2 ( I p − I e )] D(h + 2cI p − sI e ) if h + 2cI p − sI e > 0 , (15) Y.-F Huang / Retailer’s Replenishment Policies Under Conditions 238 T2 * = 2A D[h + I e (2c − s )] T3 * = 2A . D(h + sI e ) if h + I e (2c − s) > 0 (16) and (17) Then we have the following results. Case 1: a) Suppose that h + 2cI p < sI e . Then T* = ∞. (When T* = ∞, it means that the retailer prefers to keep money of the remaining balance and does not return money to the bank.) b) Suppose that h + 2cI p = sI e . Then (i) If T3* ≥ M, then T* = ∞. (ii) If T3* < M and TVC (T3 *) ≤ − DM [cI p + s ( I p − I e )] , then T* = T3*. (iii) If T3* < M and TVC (T3 *) > − DM [cI p + s ( I p − I e )] , then T* = ∞. Case 2: Suppose that h + 2cI p > sI e and h + 2cI e ≤ sI e . Then sM , then TVC(T*)=min{TVC(T1*), TVC(T3*)}. Hence T* c is T1* or T3* associated with the least cost. sM sM , then TVC(T*)=min{TVC(T3*), TVC( )}. Hence b) If T3* ≤ M and T1 * < c c sM T* is T3* or associated with the least cost. c sM , then T* = T1*. c) If T3* > M and T1 * ≥ c sM sM , then T* = . d) If T3* > M and T1 * < c c a) If T3* ≤ M and T1 * ≥ Based on Case 1 and 2, from now on, we assume h + 2cI e > sI e . Hence h + 2cI p > sI e . Consequently, both T1* and T2* are well-defined. By the convexity of TVCi(T) (i = 1, 2, 3), we see < 0  ′ TVC1 (T ) = 0 > 0  if T < T1 * (18a) if T = T1 * (18b) if T > T1 * (18c) Y.-F Huang / Retailer’s Replenishment Policies Under Conditions 239 < 0  ′ TVC2 (T ) = 0 > 0  if T < T2 * (19a) if T = T2 * (19b) if T > T2 * (19c) < 0  ′ TVC3 (T ) = 0 > 0  if T < T3 * (20a) if T = T3 * (20b) if T > T3 *. (20c) and Equations 18 (a, b, c), 19 (a, b, c) and 20 (a, b, c) imply that TVCi(T) is decreasing on (0, Ti* ] and increasing on [Ti*, ∞) for all i = 1, 2, 3. Equations (9), (11) and (13) yield that 2   c  2    sM    c −2 A + D   h + 2cI p  1 −  + sI e  2   − 1   c    s   s     sM  TVC1′  , = 2  c   sM  2   c  (21) 2  sM  −2 A + D   [h + I e (2c − s )]  sM   c  , TVC2′  =  2  c   sM  2   c  −2 A + DM 2 [h + I e (2c − s )] TVC2′ ( M ) = 2M 2 (22) (23) and TVC3′ ( M ) = −2 A + DM 2 (h + sI e ) . 2M 2 (24) Furthermore, we let 2   c  2    sM    c h cI sI 2 1 ∆1 = −2 A + D  + − + p  e  2   − 1  ,     c    s   s    (25) 2  sM  ∆ 2 = −2 A + D   [h + I e (2c − s )] ,  c  (26) ∆ 3 = −2 A + DM 2 [h + I e (2c − s )] (27) ∆ 4 = −2 A + DM 2 (h + sI e ) . (28) and Y.-F Huang / Retailer’s Replenishment Policies Under Conditions 240 Then, there are two situations to occur: (I) If s > c, equations (25), (26), (27) and (28) yield that ∆1 ≥ ∆ 2 > ∆ 3 and ∆ 4 > ∆ 3 . (II) If s = c, equations (25), (26), (27) and (28) yield that ∆1 = ∆ 2 = ∆ 3 = ∆ 4 = = −2 A + DM 2 (h + cI e ) . The determination of T* of situation (I) is discussed in this section. However, the determination of T* of situation (II) will be discussed in the next section. Case 3: Suppose that s > c. Then (1) If ∆3 ≥ 0, ∆4 > 0, ∆2 > 0 and ∆1 > 0, then TVC(T*) = TVC(T3*) and T* = T3*. (2) If ∆3 < 0, ∆4 ≥ 0, ∆2 ≥ 0 and ∆1 ≥ 0, then TVC(T*) = min{TVC(T2*), TVC(T3*)}. Hence T* is T2* or T3* associated with the least cost. sM )}. (3) If ∆3 < 0, ∆4 ≥ 0, ∆2 < 0 and ∆1 ≥ 0, then TVC(T*) = min{TVC(T3*), TVC( c sM associated with the least cost. Hence T* is T3* or c (4) If ∆3 < 0, ∆4 ≥ 0, ∆2 < 0 and ∆1 < 0, then TVC(T*) = min{TVC(T3*), TVC(T1*)}. Hence T* is T3* or T1* associated with the least cost. (5) If ∆3 < 0, ∆4 < 0, ∆2 ≥ 0 and ∆1 ≥ 0, then TVC(T*) = TVC(T2*) and T* = T2*. sM sM ) and T* = . (6) If ∆3 < 0, ∆4 < 0, ∆2 < 0 and ∆1 ≥ 0, then TVC(T*) = TVC( c c (7) If ∆3 < 0, ∆4 < 0, ∆2 < 0 and ∆1 < 0, then TVC(T*)= TVC(T1*) and T* = T1*. 4. COMPARISON WITH GOYAL’S MODEL In this section, we assume that s = c. Then h + 2cI e > cI e and h + 2cI p > cI e . Hence, equations (15), (16) and (17) can be rewritten as T1 * = 2[ A + DcM 2 ( I p − I e )] (29) D(h + 2cI p − cI e ) and T2 * = T3 * = 2A . D(h + cI e ) (30) Furthermore, equations 5 (a, b, c), (9), (11) and (13) can be reduced to TVC1 (T ) TVC (T ) =  TVC3 (T ) if T ≥M (31a) if T ≤M (31b) and −2 A + DM 2 (h + cI e ) TVC1′ ( M ) = TVC2′ ( M ) = TVC3′ ( M ) = , (32) 2M 2 respectively. Recall TVC(T) to be convex if s = c. Then, we have the following result.