An inventory model for deteriorating items under the condition of permissible delay in payments

In economic order quantity (EOQ) models, it is often assumed that the payment of an order is made on the receipt of items by the inventory system. In this paper, a varying rate of determination and the condition of permissible delay in payments used in conjunction with the economic order quantity model are the focus of discussion. Numerical examples are presented to illustrate the proposed models.

Yugoslav Journal of Operations Research 12 (2002), Number 1, 73-84 AN INVENTORY MODEL FOR ITEMS MS FOR DETERIORATING ITE UNDER THE CONDITION OF PERMISSIBLE DELAY IN PAYMENTS Horng-Jinh CHANG, Chung-Yuan DYE Department of Management Sciences, Tamkang University Tamsui, Taipei, Taiwan, R.O.C Bor-Ren CHUANG Electronics Systems Division, Chung-Shan Institute of Science & Technology Lung-Tan, Tao-Yuan, Taiwan, R.O.C Abstract: In economic order quantity (EOQ) models, it is often assumed that the payment of an order is made on the receipt of items by the inventory system However, such an assumption is not quite practical in the real world Under most market behaviours, it can be easily found that a vendor provides a credit period for buyers to stimulate demand In this paper, a varying rate of determination and the condition of permissible delay in payments used in conjunction with the economic order quantity model are the focus of discussion Numerical examples are presented to illustrate the proposed models Keywords: Inventory, EOQ, deterioration INTRODUCTION In the literature of inventory theory, deteriorating inventory models have been continually modified so as to accommodate the more practical features of real inventory systems Ghare and Schrader [1] were the first to address problems with a constant demand and a deterioration rate Since this introduction, a lot of studies such as Covert and Philip [2], Philip [3], Misra [4], Tadikamalla [5], Dave and Patel [6], Hariga [7], Chen [8], Chakrabarty et al [9], Bhunia and Maiti [10], and Chang and Dye [11] have been made on deteriorating inventory control On the other hand, in the developed mathematical models, it is often assumed that payment will be made to the vendor for the goods immediately after receiving the consignment As pointed out by Aggarwal and Jaggi [12], a permissible delay in 74 H.-J Chang, C.-Y Dye, B.-R Chuang / An Inventory Model for Deteriorating Items payments can be economically worthwhile for buyers In such a case, it is possible for a vendor to allow a certain credit period for buyers to simulate demand so as maximize his own benefits and advantages Recently, several researchers have developed analytical inventory models with the consideration of permissible delay in payments Goyal [13] established a single-item inventory model under the condition of permissible delay in payments Chung [14] presented the discounted cash-flow (DCF) approach for an analysis of the optimal inventory policy in the presence of trade credit Later, Shinn et al [15] extended Goyal's [13] model and considered quantity discounts for freight cost Recently, Chung [16] presented a simple procedure to determine the optimal replenishment cycle to simplify the solution procedure described in Goyal [13] More recently, in order to advance the practical inventory solution, Aggarwal and Jaggi [12] considered an inventory model with a constant deterioration rate under the condition of permissible delay in payments Hwang and Shinn [17] were concerned with a combined price and lot size determination problem for an exponentially deteriorating product when the vendor permits delay in payments Jamal et al [18] extended Aggarwal and Jaggi's [12] model to allow for shortages The purpose of this study is to propose a general deterioration rate including the condition of permissible delay in payments to extend the applications of developing mathematical inventory models and fit into more general inventory features This paper is organized as follows In the next section, the assumptions and notations are presented In Section 3, we present the mathematical model and develop the main result of this paper In Section 4, numerical examples including two special cases are provided: first, when the deterioration rate is linear dependent on time, and second, when the distribution of time to deteriorate follows a two-parameter Weibull distribution The method is illustrated by numerical examples, and a sensitivity analysis of the optimal solution with respect to parameters of the system is also carried out, which is followed by the concluding remarks ASSUMPTIONS AND NOTATIONS The mathematical model in this paper is developed on the basis of the following assumptions and notations Assumptions The inventory system involves only one item Replenishment occurs instantaneously at an infinite rate Let θ (t ) be the deterioration rate of the on-hand inventory at time t , where < θ (t ) < and θ ′(t ) ≥ Shortages are not allowed Before the replenishment account has to be settled, the buyer can use sales revenue to earn interest with an annual rate I e However, beyond the fixed credit period, the product still in stock is assumed to be financed with an annual rate Ir , where Ir ≥ Ie H.-J Chang, C.-Y Dye, B.-R Chuang / An Inventory Model for Deteriorating Items 75 Notations R = annual demand (demand rate being constant) A = ordering cost per order I (t) = the inventory level at time t P = unit purchase cost, $/per unit h = holding cost excluding interest charges, $/unit/year Ie = interest which can be earned, $/year Ir = interest charges which are invested in inventory, $/year, Ir ≥ Ie M = permissible delay in settling the account T = the length of replenishment cycle C (T ) = the total reverent inventory cost C1 (T ) = the total reverent inventory cost for T > M in Case C2 (T ) = the total reverent inventory cost for T ≤ M in Case V (T ) = the average total inventory cost per unit time V1 (T ) = the average total inventory cost per unit time for T > M in Case V2 (T ) = the average total inventory cost per unit time for T ≤ M in Case MODEL FORMULATION With the assumptions and notations, the behavior of the inventory system at any time t can be depicted in Fig Figure 1: Credit period vs replenishment cycle 76 H.-J Chang, C.-Y Dye, B.-R Chuang / An Inventory Model for Deteriorating Items Case 1: T > M In this case, it is assumed that the replenishment cycle is larger than the credit period Considering the inventory level at time t , depletion of the inventory occurs due to the effects of demand and deterioration during the replenishment cycle Hence, the variation of inventory level, I ( t ) , with respect to time can be described by the following differential equation: dI (t ) = − R − θ (t ) I (t ), ≤ t ≤ T , dt (1) with boundary condition I (T ) = The solution of (1) may be represented by t I ( t ) = Re t − ∫θ ( t ) dt T − ∫θ (u) du ∫ e dt , 0≤t≤T (2) t First, let g′( x) = θ x and from (2), the cost of holding I ( t ) in stock for a small period of time dt is simply hI (t ) dt Therefore, the inventory holding cost over the T period [0, T ] is h ∫ I ( t ) dt In addition, the deterioration cost during the same period is T  proportional to R  ∫ e g ( t ) dt − T  However, before the replenishment account has to be   0  settled the buyer can use the sales revenue to earn interest with an annual rate I e M during the credit period The interest earn is P I e ∫ R( M − t ) dt Beyond the fixed credit period, the product still in stock is assumed to be financed with an annual rate T Ir and thus the interest payable is P I r ∫ I (t ) dt From the discussion mentioned M above, the total reverent inventory cost can be formulated as follows: C1 (T ) = order cost + holding cost + deterioration cost + interest payable − interest earned T T T  = A + hR ∫ e − g ( t ) ∫ e g (u) du dt + PR  ∫ e g ( t ) dt − T  +   t 0  T + PRI r ∫e M − g (t ) T ∫e t g ( u) M du dt − PRI e ∫ ( M − t ) dt Let V1 (T ) be the average total inventory cost per unit time, then taking the first and second derivatives of V1 (T ) with respect to T yields H.-J Chang, C.-Y Dye, B.-R Chuang / An Inventory Model for Deteriorating Items 77 T T  T  hR  T ∫ e g (T ) − g ( t ) dt − ∫ e − g ( t ) ∫ e g ( u) du dt    dV1 (T ) A t  + =− + dT T T T   PR  Te g ( T ) − ∫ e g ( t ) dt     + + T T T T   PR  I e M + Ir ∫ e g (T ) − g ( t ) dt − Ir ∫ e− g ( t ) ∫ e g (u) du dt    M M t   + 2T and d V1 (T ) dT = 2A T + PRk1 (T ) T + hRk2 (T ) T + PR( − I e M + Ir k3 (T )) T3 , where T k1 (T ) = ∫ e g ( t ) dt + (−2 + Tg′(T ))Te g (T ) , T T T T t T T M t M k2 (T ) = T + ∫ e− g ( t ) ∫ e g (u) du dt + T ( −2 + Tg′(T )) ∫ e g (T ) − g ( t ) dt, k3 (T ) = T + ∫ e− g ( t ) ∫ e g (u) du dt + T (−2 + Tg′(T )) ∫ e g (T ) − g ( t ) dt d V1 (T ) > , we just need to show that k1 (T ), k2 (T ) and k3 (T ) are dT positive for T > M From the above, we have To verify that dk1 (T ) = T (( g′(T ))2 + g′′(T )) e g (T ) dT Since < g′(T ) = θ (T ) < and g′′(T ) = θ ′(T ) ≥ , it is clear that dk1 (T ) = T (( g′(T ))2 + dT + g′′(T ))e g (T ) > Hence, k1 (T ) is a strictly increasing function of T Furthermore, due to k1 (0) = , it is obvious that k1 (T ) > k1 ( M ) > k1 (0) = for T > M > Next, differentiating k2 (T ) with respect to T , we obtain T T   dk2 (T ) = T  g′(T ) + ( g′(T ))2 ∫ e g (T ) − g ( t ) dt + g′′(T ) ∫ e g (T ) − g ( t ) dt  e g (T ) >   dT 0   dk2 (T ) > and k2 (0) = , we also have k2 (T ) > k2 ( M ) > k2 (0) = dT Finally, analogous to the discussion above, for T > M > Since 78 H.-J Chang, C.-Y Dye, B.-R Chuang / An Inventory Model for Deteriorating Items T T   dk3 (T ) = T  g′(T ) + ( g′(T ))2 ∫ e g (T ) − g ( t ) dt + g′′(T ) ∫ e g (T ) − g ( t ) dt  e g (T ) >   dT M M   and k3 ( M ) = M Hence, it is easy to see that k3 (T ) > M for T > M > Thus we have − I e M + Ir k3 (T ) > ( Ir − I e ) M ≥ for T > M > From the analysis carried so far, we can conclude that V1 (T ) is a convex function of T and there exits a unique value of T that minimizes V1 (T ) Besides, by using L'Hospital's rule, it is not difficult to show that T   dV1 (T ) 1 = lim  PRe g (T ) g′(T ) + hR  + ∫ e g (T ) − g ( t ) dt  g′(T ) +   T →∞ dT T →∞     lim T    + PRI r  + ∫ e g (T ) − g ( t ) dt  g′(T )  =    M    =∞ Thus, the optimal value of T should be selected to satisfy dV1 (T ) dV1 (T ) = , otherwise T * = M if dT dT T=M >0 (4) Case 2: T ≤ M In this case, it is assumed that the length of the replenishment cycle is not larger than the credit period The holding cost and deterioration are the same as in case Since T ≤ M , the buyer pays no interest and earns interest during the period [0, M ] T  Note that the interest earned in this case is PI e  ∫ R(T − t ) dt + RT ( M − T )  From this,   0  the total reverent inventory cost can be formulated as C2 (T ) = order cost + holding cost + deterioration cost − interest earned T T T  = A + hR ∫ e − g ( t ) ∫ e g (u) du dt + PR  ∫ e g ( t ) dt − T  −   t 0  T  − PRI e  ∫ (T − t ) dt + T ( M − T )    0  The first and second derivatives of average total cost, V2 (T ) , with respect to T , result in H.-J Chang, C.-Y Dye, B.-R Chuang / An Inventory Model for Deteriorating Items 79 T T  T  hR  T ∫ e g (T ) − g ( t ) dt − ∫ e− g ( t ) ∫ e g ( u) du dt    dV2 (T ) A t  + =− + dT T T T   PR  Te g (T ) − ∫ e g ( t ) dt    PRI  + e + 2 T and d V2 (T ) dT = 2A T + hRk2 (T ) T + PRk1 (T ) T3 Using the fact that k1 ( x) > and k2 ( x) > for < x ≤ M , it is easily shown that V2 (T ) is also a convex function of T and there exists a unique value of T that dV2 (T ) = −∞ , the optimal value of T should be selected T → dT minimizes V2 (T ) Since lim to satisfy dV2 (T ) dV2 (T ) = , otherwise T * = M if dT dT T=M 0, (7) f ( M ) = NUMERICAL EXAMPLES In this section, the optimal solution procedure developed in the previous section is now illustrated with two special cases In the first case, we assume that the deterioration rate is linear dependent on time and is in the following form: 80 H.-J Chang, C.-Y Dye, B.-R Chuang / An Inventory Model for Deteriorating Items θ (t ) = a + bt , < a, b And second, the distribution of time to deteriorate follows a two-parameter Weibull distribution: θ (t ) = αβ t β −1 , < α , where α is the scale parameter and β is the shape parameter 4.1 Linear deterioration rate The exact solution procedure for the case of a linear deterioration rate can be b deduced from the previous analysis by substituting g ( x) = ax + x into the derived mathematical expressions Using Taylor's series expansion, V1 (T ), V2 (T ) and f ( M ) can be rewritten as follows:   A hR T  ∞ (− g (t ))n T ∞ ( g (u)) n PR  T ∞ ( g (t )) n + du  dt + dt − T  ∫ ∑ ∑ ∑ ∫ ∫   T T  n =0 n! n! T  n = n! t n =0   (8)  PRIr T  ∞ (− g (t ))n T ∞ ( g (u)) n PRI e M + ∫  ∑ n! ∫ ∑ n! du dt − T ∫ ( M − t ) dt , T M t n =0  n =0  V1 (T ) =   A hR T  ∞ (− g (t ))n T ∞ ( g (u))n PR  T ∞ ( g (t ))n + du dt + dt − T  ∫ ∑ ∑ ∑   ∫ ∫   T T n =0 n! n! T  n = n!  t n =0  (9)  PRI e  T −  ∫ (T − t ) dt + T ( M − T )   T   V2 (T ) = and ∞ ( g ( M )) n  f ( M ) = − A + hR  M ∑  n =0 n!  M ∞  −∫ ∑  n =0  M ∞ (− g (t )) n dt − n! n =0 ∫ ∑   ( g (u))n du dt  +  n! n =0   (− g (t )) n M ∞ n! ∫ ∑ t (10) M ∞ ∞ ( g ( M )) n  ( g (t ))n  PRI e M dt  + + PR  M ∑ − ∫ ∑  n =0  n! n! n =0   As a and b are very small, the approximation solution can be found by neglecting the second and higher terms of a, b and ab , so we have V1 (T ) ≈  T aT bT   aT bT  PRI e M A + hR  + + + +  + PR  −    T 12   2T 2  + PRIe  M aM bM aM 2T bM 3T T aMT − − − MT + + + − +   T  12 2 + aT bMT bT  − + , 6 12  (11) H.-J Chang, C.-Y Dye, B.-R Chuang / An Inventory Model for Deteriorating Items V2 (T ) ≈  T aT bT   aT bT  A T  + hR  + + +  + PR   − PRI e  M −      T 12 2      81 (12) and f (M) ≈ −A + hRM (6 + a + bM ) PRM (3a + 2bM ) PRM I e + + 12 (13) The procedure for determining the approximate optimal value of T first computes f ( M ) from (13) Then applying the above solution produced by dVi (T ) = 0, i = or , is taken to be the approximate optimal value of T dT Example In order to illustrate the above solution procedure, we consider an inventory system with the following data: R = 1000 units/year, A = $250 per order, P = $100/unit/year, h = $20/unit/year, I e = 0.13/$/year, Ir = 0.15/$/year, M = 30/365 year For the linear deterioration rate case, we let θ (t ) = 0.08 + 0.1t For this case, since f ( M ) = −109.343 < , from (7), we have the optimal value of V (T ) = V1 (T * ) Solving dV1 (T ) = and then putting the obtained value into (11), we have the optimal values dT of T and V (T ) , which are T * = 0.1082 and V (T* ) = 3489.28 4.2 Weibull deterioration rate In this case, it is assumed that the deterioration rate is a two-parameter Weibull distribution: θ (t ) = αβ t β −1 , where < α