International Journal of Engineering Research and Development e-issn: 78-07X, p-issn: 78-800X, www.ijerd.com Volume 5, Issue 1 (February 013), PP. 01-09 An Inventory Model for ime Dependent Deteriorating Items and Holding Cost under Inflations When Vendor Credits o Order Quantity Rajendra Sharma 1, Jasvinder Kaur Department Of Mathe mati cs Graphic Era Uni versit y De hradun, ind ia Abstract:- In this paper we discuss the possible affects of inflations by the supplier on a retailer s replanishment policy for time-dependent deteriorating items with constant demand rate. his optimal order quantity is obtained for all four cases i.e. for case 1, 0<< d, case, d <M, case 3, d M and case 4, M d <. he optimal total relevent cost is obtain for all these four cases which is minimum. Numerical result are used to illustrate the theoretical results. Keywords:- constant demand, Inventory modal, Inflation, Order quantity, ime dependent deteriorating Items, ime dependent holding cost. I. INRODUCION In past few decades; inventory problems for deteriorating items have been studied in large scale. Most of physical goods deteriorate over time. In reality some of the items either decayed are not in a perfect condition to satisfy the demand. Food items, grasses, vegetables, fruits, drugs, pharmaceuticals, fashion goods and electronics substances are a few example of such items in which inefficient deteriorations can take place during the normal storage period of the units and this loss must be taken into account in the classification of the system. he deterioration rate is an important factor of inventory in stock during the storage period, Ghare and Schrader [1] were the first proponents to establish a model for an exponentially deteriorating items. Ghare and Schrader s model was expended by Eovest and Philip [] by taking constant deterioration rate to a two parameter weibull distribution. Misra [3] developed an inventory model for optimal production lot-size model for a system with deteriorating inventory. An order-level lot-size inventory model for deteriorating items was discussed by Shah [4]. Hollier and Mak [5] developed inventory replenishment policies for deteriorating items in a declining market. Heng et al [] extended misra s [3] and Shahi s [4] models to consider a lot-size, orderlevel inventory system with finite replenishment rate, constant demand rate and and exponential decay. Acomplete note on inventory literature for deteriorating inventory models was developed by Goyal and Giri [7] and Raafat [8]. Misra et al [9] developed an inventory model for weibull deteriorating itemswith permissible delay in payments under inflation. In paper [9] misra et al obtained the conditions for concavity of optimality. Later there are several interesting papers related to deteriorations such as Shah and Jaiswal [10], Aggarrwal [11] Hariga [1] and goyal and Giri [13], Dave and Patel [14] and Sachaan [15]. In most of the research papers mentioned above, the effects of inflation are not considered. However from a financial point of view, an inventory referents a capital investment and must complete with other assets for a firm s limited capital funds. hus the effects of inflation can must be ignored is the study of inventory system. Misra [1], Buzacott [17], Bierman and homas [18] developed the inventory model under an inflationary conditions for the EOQ model. Liao et al [19] developed a model with deteriorating items under inflation when a delay in payment in permissible. Other related research papers are Chang [0], Brahmbhatt [1], Chandra and Bahner [], Dattand pal [3], Moon et al [4], Gor and Shah [5], Huo [] and othes. In the present paper, an attempt has been made to develop a deterministic inventory model for timedependent deteriorating items and time-dependent holding cost under inflation when supplier offers delay in payments. Shortages are not allowed. Rest of the paper organized as follows: In the next section proposed assumptions and notations are given following by mathematical formulation in section 3. heoretical results are given in section 4. he numerical examples sensitivities are given in section 5. Finally, conclusions are given in the last section. 1. ASSUMPIONS: II. PROPOSED ASSUMIONS & NOAIONS 1
1.1 Replenishment is instantaneous 1. he demand is known and is constant. 1.3 he net discount rate of inflation is constant. 1.4 Shortages are not allowed. 1.5 Holding cost is time dependent i.e. h=h(t)=ht 1. If Q<Q d then the payment for the items received must be made. 1.7 If Q Q d then the delay in payments up to M is permitted. During the trade credit period the account is not settled and generated sales revenue is deposited in an interest bearing account. At the end of credit period, the customer pays off all units ordered and starts paying for the interest charges on the items in stock.. NOAIONS:.1 I(t) : Inventory level at any time t, 0 t.. r : constant rate of inflation, 0< r < 1..3 h : holding cost per unit time i.e. h(t)=ht..4 H : length of planning horizon and H = n, where n is an integer for the number of replenishments to be made during period H and is an interval of time between replenishments..5 D : the demand rate per unit time.. P(t) = pe rt : the selling price per unit time, p is the initial selling price at t=0..7 S(t) =se rt : the ordering cost per order at time t, s is the initial ordering cost at t=0..8 C(t) =ce rt : the purchasing cost at time t, c is the initial purchase price at t=0, c < p..9 I c : interest charged / $ / year by the supplier per order..10 I d : the interest earned / $ /year..11 Q : order quantity..1 Q d : minimum order quantity for which the delay in payments is allowed..13 : the replenishment time interval..14 d : the time interval that Q d units are depleted to zero due to demand only..15 Z(t) : the total relevant cost over (o, H)..1 : time dependent deteriorating items. Note that the total relevant cost consists of (i) cost of placing order, (ii) cost of purchasing, (iii) cost of carrying inventory excluding interest charges, (iv) cost of interest charges for unsold items at t = 0 or after credit period M and (v) interest earned from sales revenue during the credit period. III. MAHEMAICAL FORMULAION AND EQUAIONS he rate of change of inventory with respect to time can be described by the following differential equation: di (t) dt + θt. I t = D, 0 t (1) he solution of (1), with boundary condition I(t) = 0 is I t = D + θ3 D t + θt3. e θt, 0 t () And the order quantity is Q = I 0 = D + θd3 (3) From the above equation (3) we can find the time interval in which Q d units are depleted to zero due to demand only Q d D = d + 3 d θ (4) Hence it is easy to see that the inequality Q < Q d iff < d Again the length of time intervals are all the same, hence we have
I K + t = D + θ3 t + θt3. e θt, 0 k n 1, 0 t (5) For total relevant cost in (0,H), we need following elements (i) cost of placing order S 0 + S + S +. +S n 1 = S erh 1 e r 1 () (ii) cost of purchasing Q C 0 + C + C + + C n 1 = QC erh 1 e r 1 =CD + θ 3 (iii) cost of carrying inventory e rh 1 e r 1 (7) n 1 K=0 C K, O h t. I K + t dt = chd e r 1 3 + θ5 0 (8) (iv) Regarding interest charged and earned, we have the following four possible cases based on the values of, M and d Case I, 0 < < d Since < d ( i.e. Q< Q d ). In this case the interest charges for all unsold items start at the initial time, we obtain the interest payable in (0,H) as n+1 I C C K, I K + t dt = I C cd erh 1 e r 1 k=0 Z 1 = total cost in (0,H) 0 + θ4 8 (9) Z 1 = s + cd + θ3 + chd 3 + θ5 0 + I CcD + θ4 8 e r 1 (10) Case II, d < M In this case there is a permissible delay M which is longer than. As a result there is no interest charged, but the interest earned in (0,H) is n 1 I d P K, Dt dt + D M k=0 0 = I d PD erh 1 e r 1 M (11) Z = total relevant cost in (0,H) Z = s + cd + θ3 + chd 3 + θ5 0 I dpd M e r 1 (1) Case III, d M 3
In this case, is longer than or equal to both d and M then delay in payment is permitted and the total relevant cost includes both the interest charged and the interest earned. he interest payable in (0,H) is n 1 I c C K, I K + t dt k=0 M = I C cd + θ3 M ( M) 1 θ( M) + 4 e r 1 (13) he interest earned in (0,H) is n 1 M I d P K, Dtdt = I d pd erh 1 e r 1 k=0 0 Z 3 = total relevant cost in (0,H) M (14) Z 3 = s + cd + θ3 + I c cd + θ3 + chd 3 + θ5 0 M ( M) 1 θ( M) + 4 + I d PD M e r 1 Case IV, M d (15) In this case, the replenishment time interval is also greater than or equal to both d and M. Hence case IV is similar to case III. hus total relevant cost in (0,H) is Z 4 = s + cd + θ3 + I c cd + θ3 + chd 3 + θ5 0 M ( M) 1 θ( M) + 4 + I d PD M e r 1 (1) IV. HEOREICAL RESULS Since inflation rate r is very small. Using truncated taylor s series expansion for the exponential terms, we get the modified (approximated) values of Z i (),i=1,,3&4 as follows Z 1 = 1 r S + chdθ 0 4 + I c cd θ 8 3 + chd + cdθ + I ccd + cd erh 1 (17) Z = 1 r Z 3 = 1 r S + chdθ 0 4 + cdθ + chd s M M4 θ 4 + I dpdm + cdθ + chd I ccdθm + I d pd + (cd I dpdm) (18) 4 + I ccdθ θ 4 3 1 + chdθ 0 + cd I c cdm + M + θm3 + θm + I c cd 1 θm 19 4
Z 4 = 1 r s M M4 θ 4 + I dpdm + cdθ + chd I ccdθm 1 + chdθ 0 + cd I c cdm + M + θm3 4 + I ccdθ θ 4 3 + θm + I c cd 1 θm 0 he optimal solutions are obtained by taking the first and second order derivatives of Z i (),i=1,,3&4 with respect to, we obtain dz 1 d = 1 S + chdθ r 5 3 + 3I ccdθ 8 + chd + cdθ 3 + I ccd (1) dz d = 1 S + chdθ r 5 3 + cdθ + chd 3 + I dpd () dz 3 = 1 s + M + M4 θ I d PD M 1 + chd θ 3 + I c cdθ θ d r 4 5 8 + cdθ + chd I c CDθM + θm + 3 IccD 1 θm erh 1 (3) d Z 1 d = 1 r S + 3chDθ 3 5 + 3I ccdθ 4 + chd+cdθ 3 > 0 4 d Z d = 1 r S + 3chDθ 3 5 + cdθ +chd 3 > 0 (5) d Z 3 = 1 s M M4 θ + I d PD M + 3chD θ + I d r 4 3 5 c cdθ θ + cdθ + chd I 4 ccdθm + θm13erh 1>0 () For optimal (minimum) solution, put dz i d = 0, i = 1,,3,4, we obtain from 1 dz 1 = 0 d 4chDθ 5 + 45I c cdθ 4 + 40 cdθ + chd 3 + 0I c cd 10s = 0 (7) from dz = 0 d chdθ 5 + 10 cdθ + chd 3 + 15 I d pd 30s = 0 (8) from 3 dz = 0 d 4chDθ 5 + 0I c cdθ 15θ 4 + 40 cdθ + chd I c cdθm + θm 3 + (10I c cd 0 0θM ) 10s 0M 5M θ + 0I d pdm = 0 (9) V. EXAMPLES AND ABLES 1.NUMERICAL EXAMPLES: Case I, 0 < < d Example 1. Let s=$ 150/order, c= $ 5/units, h=$ /unit/year, I c =0.10/$/year, D=500 unit/year, p=$ 30 per unit, r=0.05 per unit, θ=0.0/unit/year, I d =0.05/$ /year, H=1year, Substituting these values in (7) and (3) and (10) we get the values 1 =0.3859 year and Q 1 =119.31733 units also Z 1 () = $ 13855.308 Case II, d < M 5
Example. let D= 100 units, θ=0.0, c= $ 30/units, p=$ 40 per unit, h=$ /unit/year, I d =0.05/$ /year, H=1year, θ=0.0/unit/year, s=$ 50/order, I c =0.08/$/year, r=0.05 per unit, M=110days, Substituting these values in (8) and (3) and (1) we get the values =0.75757 year and Q = 7.58897 units also Z () = $ 330.058708 Case III, d M Example 3. let D= 100 units, θ=0.0, c= $ 10/units, p=$ 0 per unit, h=$ /unit/year, I d =0.05/$ /year, H=1year, θ=0.0/unit/year, s=$ 100/order, I c =0.10/$/year, r=0.05 per unit, M=90days, Substituting these values in (9) and (3) and (15) we get the values 3 =0.489381 year and Q = 48.977179 units also Z () = $ 1343.47448. SENSIIVIY ANALYSIS: Sensitivity analysis has been performed by considering various values of the parameters like unit ordering cost (s), unit purchasing cost (c), holding cost (h) and credit period (M), the corresponding values obtained with respect to the changes in above parameters are replenishment cycle time (), economic order quantity Q and total relevant cost Z() by taking into consideration the following different cases. i. When 0 < < d [tables 1(a),1(b),1(c)] ii. When d < M [tables (a),(b),(c)] Iii When d M [tables 3(a),3(b),3(c),3(d)] able 1. (case 1: When 0 < < d ) able 1(a): Sensitivity analysis on s s 1 Q 1 Z 1 150 0.3859 119.31733 13855.308 10 0.44 1.135775 13897.75343 170 0.49 14.83895 13939.51 180 0.5483 17.4390781 13979.877 190 0.59831 19.94473 14019.935 00 0.48 13.34899 14058.781 able 1(b): Sensitivity analysis on c c 1 Q 1 Z 1 5 0.3859 119.31733 13855.308 0.357 117.35195 14383.574 7 0.3033 11.037308 14911.477 8 0.8993 114.515131 15439.051 9 0.095 113.079 159.3504 30 0.338 111.8543 1493.30019 able 1(c): Sensitivity analysis on h h 1 Q 1 Z 1.0 0.3859 119.31733 13855.308.1 0.3541 117.734 1387.30. 0.341 11.31455 13878.9909.3 0.9573 114.805 13890.39504.4 0.85 113.451904 13901.514.5 0.485 11.11304 1391.39143
able. (case : When d < M ) : able (a): Sensitivity analysis on s s Q Z 50 0.75757 7.58897 330.058708 55 0.8519 8.508 334.3745 0 0.9395 9.403441 334.00743 5 0.3095 30.3870813 3359.050 70 0.310 31.031518 3375.93983 75 0.317781 31.78879701 339.4809 able (b): Sensitivity analysis on c c Q Z 30 0.75757 7.58897 330.058708 3 0.70499 7.0549744 351.0885 34 0.53.59475 37.19845 3 0.1101.1103341 3935.958455 38 0.587 5.984975 4145.58917s able (c): Sensitivity analysis on h h Q Z.0 0.75757 7.58897 330.058708.1 0.71813 7.1879940 3309.90188. 0.8091.81558 3313.38198.3 0.457.431730 3317.7494.4 0.131.19048 330.81851.5 0.58058 5.8115837 334.7495 able 3. (case 3: When d M ) able 3(a): Sensitivity analysis on s s 3 Q 3 Z 3 100 0.489381 48.977179 1343.47448 110 0.50598 50.4178115 134.30333 10 0.5141 5.114144 1384.3149 130 0.53471 53.98555 1404.00535 140 0.55057 55.11337 143.04035 150 0.5404 5.4381477 1441.58911 able 3(b): Sensitivity analysis on c c 3 Q 3 Z 3 10 0.489381 48.977179 1343.47448 1 0.458144 45.8445419 158.079531 14 0.43319 43.34993 1790.41548 1 0.4104 41.8381419 010.9385 18 0.3950 39.5407748 30.0454 able 3(c): Sensitivity analysis on h h 3 Q 3 Z 3.0 0.489381 48.977179 1343.47448. 0.4778 47.7039039 1351.48941.4 0.4543 4.5578745 1358.4407 7
. 0.45487 45.5180717 135.7054.8 0.445388 44.58501 137.4185 3.0 0.437 43.9535599 1378.951003 VI. CONCLUSION ANALYSIS OF HE RESULS SHOWN IN ABLES 1 O 3: It is observed from the computational results shown in table 1(a) that for higher values of ordering cost s, the corresponding values of replenishment cycle time 1, order quantity Q 1 and total relevant cost Z 1 also go higher as per expectations and the table 1(b) indicate that with the increasing of unit purchasing cost c, the corresponding values of replenishment cycle time 1, order quantity Q 1 are decreasing while the total relevant cost Z 1 is increasing with the increasing values of unit purchasing cost c and the table 1(c) imply that the higher values of holding cost h imply lower values of replenishment cycle time 1 and order quantity Q 1 but higher values of total relevant cost Z 1, the tendency of these results is the same as those shown in table 1(b). he computational results obtained in table (a) indicate that ordering cost s is directly proportional to the replenishment cycle time, economic order quantity Q and total relevant cost Z i.e. an increase in s implies the proportional increase in, Q, Z and in table (b) indicate that purchasing cost c is inversely proportional to replenishment cycle time and economic order quantity Q directly proportional to the total relevant cost Z i.e. an increase in c shows proportional decrease in andq while as increase in Z and in table (c) indicate that higher values of holding cost h are associated with the lower values of the replenishment cycle time and economic order quantity Q and higher values of total relevant cost Z. he computational results obtained in table 3(a) indicate that unit ordering cost s is directly proportional to all the three values i.e. replenishment cycle time 3 and economic order quantity Q 3 and total relevant cost Z 3 and 1n table 3(b) show that the value of replenishment cycle time 3 and economic order quantity Q 3 decrease with the increasing of unit purchasing cost c while total relevant cost Z 3 increase with the increasing values of unit purchasing cost c and in table 3(c) indicate that higher values of holding cost h imply the lower values of the replenishment cycle time 3 and economic order quantity Q 3 and higher values of total relevant cost Z 3. PROPOSED MODEL he proposed model can be extended in many more ways such as, we can consider the demand rate in quadratic time dependent form. We can also consider the demand as a function of quantity or selling price. Further the shortages may also be taken in to account to generalize the model thus this paper can be useful developed as a wholesaler and retailer system model. REFERENCES [1]. Ghare, P.M., Schrader, G.P.(193). A model for an exponentially decaying inventory journal of industrial engineering 14, 38-43. []. Covest, R.B., Philip, G.S (1973). An EOQ model with weibull distribution deterioration. AIIE ransactions 5, 33-3. [3]. Misra, R.B. (1975). Optimal production lot-size model for a system with deteriorating inventory. International journal of production research, 13 (3), 495-505. [4]. Shah, Y.K.(1977). An order level lot-size inventory model for deteriorating items. AIIE ransaction, 9(1), 108-11. [5]. Hollier, R.H., and Mak, K.L.(1983). Inventory replenishment policies for deteriorating items is a declaring market. International journal of production research, 1(4), 813-8. []. Heng, K.J., Labban, J., and Linn, R.J.(1991). An order level lot- size inventory model for deteriorating items with finite replenishment rate. Computers and industrial engineering 0(1), 187-197. [7]. Goyal, S.K., and Giri, B.C.(001). Recent trends in modeling of deteriorating inventory European journal of operational research, 134, 1-. [8]. Raafat, F. (1991). Survey of literature on continuously deterioration inventory models. Journal of the operational research society, 40, 7-37. [9]. Misra, U.K., Sahu, S.K., Bhaula, B., and Raju, L.K. (011). An inventory model for weibull deteriorating items with permissible delay in payments under inflation. [10]. Shah, Y.K.,and Jaiswal, M.C. (1977). An order level inventory model for a system with constant rate of deterioration. Opsearch 14, 174-184. [11]. Aggarwal, S.P. (1978). A note on an order level inventory model for a system with constant rate of deterioration. Opsearch, 15, 184-187. 8
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