P. Manju Priya 1, M.Phil Scholar. G. Michael Rosario 2, Associate Professor , Tamil Nadu, INDIA)

Transcription

1 International Journal of Computational an Applie Mathematics. ISSN Volume, Number (07 Research Inia Publications AN ORDERING POLICY UNDER WO-LEVEL RADE CREDI POLICY WIH ALLOWABLE SHORAGE AND PERMISSIBLE DELAY IN PAYMENS P. Manju Priya, M.Phil Scholar G. Michael Rosario, Associate Professor, (Research centre of Mathematics, Jayaraj Annapaciam College for Women (Autonomous, Periyaulam- Abstract 6560, amil Nau, INDIA In this paper, a moel is evelope to etermine retailer's inventory policy uner two-level trae creit system. It is assume that the retailer has a powerful position so that he can obtain the full trae creit offere by supplier whereas customer can obtain just the partial trae creit from the retailer. he propose moel also allows fully baclogge shortages. his robust Mathematical moel consiers price epenent eman an is evelope for various cases in which the length of the cycle is ivie into two parts, i.e., length of the perio with positive stocs an length of the perio with negative stocs. he ultimate aim of the stuy is to minimize the total cost of the retailer by fining the optimum orer quantity. Numerical examples an Sensitivity analysis are presente to prove the valiity of the moel. Keywors: Inventory, Price epenent eman, Shortages, wo-level trae creit policy. Introuction Inventory theory eals with the management of stoc levels of goos with the aim of ensuring that eman for these goos is met. Most moels are esigne to aress two funamental ecision issues: when a replenishment orer shoul be place, an what the orer quantity shoul be. heir complexity epens on the assumptions mae about eman, the cost structure an physical characteristics of the system. he costs involve inclue orering/prouction costs, setup cost, costs for holing the prouct in stoc an penalty costs for not being able to satisfy eman when it occurs.[] manju.80394@gmail.com tony87rio@gmail.com 708

2 International Journal of Computational an Applie Mathematics. ISSN Volume, Number (07 Research Inia Publications rae creit is an essential tool for financing growth. In most business transactions, the supplier woul allow a specifie creit perio to the retailer for payment without penalty to stimulate the eman of his/her proucts. he number of ays for which a creit is given is etermine by the company allowing the creit, an is agree upon by both the company allowing the creit an the company receiving it. Before en of the trae creit perio, retailer can sell goos an accumulate revenue an earn interest. A higher interest is charge if the payment is not settle by en of the trae creit perio. Fisman [] inicates that trae creit i.e. permissible elay in payment is an important form of financing for businesses in a broa range of inustries. Goyal [3] presente a single-item inventory moel uner trae creit. Chung [4] establishe an alternative approach to etermine the economic orer quantity uner conition of trae creit. Jamal et al. [5] consiere an orering policy for eteriorating items with allowable shortage an permissible elay in payment. able in Jamal et al. [5] reveal that they are first to incorporate both concepts of allowable shortage an permissible elay in payment to establish a new inventory moel. Furthermore, Sarer et al. [6] aress a moel to etermine an optimal orering policy for eteriorating items uner inflation, permissible elay of payment an allowable shortage. Optimal payment time Jamal et al. [7] was aresse uner permissible elay in payment with commoity eterioration. eng [8] assume that the selling price is not equal to the purchasing price to moify the Goyal s moel [3]. Chung et al. [9] iscusse that the selling price is not equal to the purchasing price an ifferent payment rules are allowe. Shinn & Hwang [0] etermine the retailer s optimal price an orer size simultaneously uner the conition of orer-size-epenent elay in payments. Chung & Huang [] extene this type of problem-solving within the EPQ framewor an evelope an efficient proceure to etermine the retailer s optimal orering policy. Huang & Chung [] extene Goyal s moel [3] to allow for cash iscount for early payment. Salameh et al. [3] extene this issue to inventory ecision uner continuous review. Huang [4] assume that the retailer will aopt a similar trae creit policy to stimulate eman from customer to evelop the retailer s replenishment moel. Huang et al. [5] etermine orering ecision-maing uner two-level trae creit policy for a retailer with a powerful position. Recently, Chung & Huang [6] iscusse orering policy with allowable shortage an permissible elay in payments. Present stuy extens Huang s moel [5] an Chung & Huang s moel [6] to investigate that the retailer can obtain the full trae creit offere by the suppliers an the retailer just offers the partial trae creit to customers with shortage an price-epenent eman. his paper presents a moel on retailer s inventory ecision-maing as a cost minimization problem to etermine the retailer s optimal orering policies. 709

3 International Journal of Computational an Applie Mathematics. ISSN Volume, Number (07 Research Inia Publications he remainer of this paper is organize as follows. In Section, notations an assumptions of the mathematical moel is escribe. Section 3 gives the formulation of the mathematical moel. An optimal solution of the moel is etermine in Section 4. In Section 5, a numerical example is given to illustrate the moel an sensitivity analysis has been mae for ifferent changes in the parameter values.. Notations an Assumptions.. Notations Dp ( Price epenent eman per year. D p= a bp where a an b are positive constants s h A c p Shortage cost per item per year Unit stoc-holing cost per year excluing interest charges Orering cost per year Unit purchase cost Unit selling price Customer s fraction of the total amount payable at a time of placing Mar-up rate, an orer within the elay perio to retailer, 0 I Rate of interest earne per year I Rate of interest payable per year to the supplier M N retailer Retailer s trae creit perio as measure by years offere by supplier Customer s trae creit perio as measure by years offere by Length of the perio with positive stoc of item 70

4 International Journal of Computational an Applie Mathematics. ISSN Volume, Number (07 Research Inia Publications Length of the perio with negative stoc of item Length of the inventory cycle.. RC, Annual total relevant cost, function of Optimal cycle time of, * ij, RC, i,,3; j, * Q Optimal orer quantity, Q D * *.. Assumptions Deman rate is assume to be a function of selling price i.e. D p a bp which is a function of selling price p, where a, b are positive constants an 0< b a / p. Unsatisfie eman / Shortages are allowe an completely baclogge. ime horizon of the inventory system is infinite. I I, M N. Replenishments are instantaneous an lea-time is negligible. Supplier offers the full trae creit to retailer whereas retailer offers the partial trae creit to customers. Up to the time the account is not settle, generate sales revenue is eposite to retailer s interest bearing account. At the en of this perio, the account is settle an we start paying for the interest charges on the items in stoc 3. Moel Formulation Base on the above assumptions, the consiere inventory system goes lie this: he total annual relevant cost consists of the following elements: a Cost of placing orers = A A 7

5 International Journal of Computational an Applie Mathematics. ISSN Volume, Number (07 Research Inia Publications b Stoc-holing cost Dh Dh ( D (Since, the average stoc = c Shortage cost Ds Ds ( D (Since, the average shortage = Cost of interest charges for the items ept in stoc: As the items are sol an before the replenishment account is settle, the sales revenue is use to earn interest. When the replenishment account is settle, the situation is reverse an effectively the items in the stoc have to be charge at interest rate I. here are two cases to be consiere in annual interest payable. When M, the account is settle at M an retailer starts paying for the interest charges on the items in stoc with rate I. When M, all items are sol at an retailer can pay the amount of purchasing cost to supplier at M. Hence, retailer oes not nee to pay any interest charge when Case ( M : M. Annual interest pai = ci D( M ci ( = D M ( Case ( M : Annual interest pai = 0 e Cost of interest earne uring the permissible settlement perio: here are three cases to be consiere in annual interest earne. At the beginning of the time interval, the bacorere quantity D shoul be replenishe first. he maximum accumulate amount earning interest at the beginning of the time interval equals Dp. he maximum amount earning interest within the settlement perio equals DpM if M an Dp if M. Retailer can accumulate interest from customer s partial payment uring 0, N an from the total amount of payment on, N M with interest I. Case ( M : 7

6 International Journal of Computational an Applie Mathematics. ISSN Volume, Number (07 Research Inia Publications Annual interest earne PI = DN ( DN DM ( M N [ ] D pmi = pi D M N M [ ( ] Case ( N M : Annual interest earne DN ( DN D( N PI[ D ( M ] D pmi = = pdi [ M ( N ] Case 3 ( N : Annual interest earne D PI[ D ( N D ( M N] D pmi = = pdi [M ( N ] Note that the interest earne shoul be subtracte from other variable costs in orer to get the total variable cost per year. From the above arguments, annual total relevant cost for retailer can be expresse as, RC(, = Orering cost+ Stoc-holing cost+ Shortage cost+ Interest payable - Interest earne. herefore, 73

7 International Journal of Computational an Applie Mathematics. ISSN Volume, Number (07 Research Inia Publications RC (, if M RC(, RC (, if N M RC (, if 0 N 3 Where A Dh Ds ci D( M RC (, ( ( ( pdi M N M ( [ ( ] ( A Dh Ds RC(, ( ( pdi[ M ( N ] ( ( A Dh Ds RC3(, ( ( pdi [ M ( N ] ( (3 From now on, for convenience, we treat RC (,, RC (, an RC3 (, all are efine on 0, 0 an 0. he following heorem has been propose as heorem of Chung an Huang [6]. heorem i RC (, is convex on 0, 0, 0 ii RC (, is convex on 0, 0, 0 iii RC3 (, is convex on 0, 0, 0 4. Determination of optimal solutions of RC(, 74

8 International Journal of Computational an Applie Mathematics. ISSN Volume, Number (07 Research Inia Publications Simplifying equations (, ( an (3, we get RC Z (, i i(,, i,,3 ( Where Z (, A Dh Ds CI D( M pdi ( M ( N M Z (, ADh Ds pdi (M M ( N Z (, A Dh Ds pdi (M M ( N 3 he optimal solutions (,,,,3 * * i i i of, RC as etermine by equations, i RCi (, 0 RCi (, an 0,i=,, 3. are ( h ci K Mc( I IK s Mc( I I h s ci As M c I I D ( ( ( ( h pi s ( ( ( ( ( spi N M c I I s c I I h s ci h ci h s ci / As pi ( N s D ( h pi( h pi s / 3 3( h pi pi( N s 75

9 International Journal of Computational an Applie Mathematics. ISSN Volume, Number (07 Research Inia Publications As p I ( N ( ( ( pi ( ( N p I N D 3 h s pi h s pi h pi h s pi / Also, we have 0 if i RC i( 0 if i 0 if i i RC is ecreasing on an increasing on 0, i Let RC( if M RC( RC ( if N M RC 3( 0 if N. i, heorem Let DM ( h s pi ( h pi spdi ( N A an s D( h ci [ N ( h s ci MNc( I I ] spi ( N M c( I I ( s c( I I A, then s Proof: i M if an only if 0, 0. ii N M if an only if 0, 0. iii i M N (since M N N if an only if 0, 0. 76

10 International Journal of Computational an Applie Mathematics. ISSN Volume, Number (07 Research Inia Publications If M Mc( I I h s ci / As ( ( ( ( ( spi N M c I I s c I I h s ci h ci h s ci M c I I D M ( ( ( As M c I I D ( ( ( ( ( ( ( ( spi N M c I I s c I I h s ci h ci h s ci M ( h s pi h s ci As M ( h s pi ( h pi spi ( N D DM ( h s pi ( h pi spdi ( N 0 A s 0 If N Mc( I I h s ci As M c ( I ( ( ( ( I spi N M c I I s c I I D N ( h s ci ( h ci ( h s ci / As M c I I D ( ( ( ( ( ( ( ( spi N M c I I s c I I h s ci h ci h s ci Mc( I I N( h s ci h s ci 77

11 International Journal of Computational an Applie Mathematics. ISSN Volume, Number (07 Research Inia Publications As spi N M c I I s c I I h s ci D ( ( ( ( ( M c ( I I ( h ci ( h ci [ N( h s ci Mc( I I ] K As spi N M c I I s c I I D ( ( ( ( ( h ci [ N ( h s ci K MNc( I I ] D( h ci [ N ( h s ci MNc( I I ] spi ( N M c( I I ( s c( I I 0 A s 0 herefore, (i is prove. Similarly, (ii an (iii can be prove. he following heorem has been propose as heorem of Chung an Huang [6]. heorem 3 solution. ( h ci Mc( I I i If, 0, 0 then. Hence, s will be the optimal ( h pi ii If 0, 0 then. Hence, s will be the optimal solution. solution. ( h pi pi( N iii If 0, 0 then 3. Hence, s will be the optimal 5. Sensitivity Analysis o illustrate the results obtaine in this paper, let us apply the propose metho to efficiently solve the numerical examples. For convenience, the values of the parameters are selecte at ranom. 78

12 International Journal of Computational an Applie Mathematics. ISSN Volume, Number (07 Research Inia Publications For A 000, c 400, I 0.3, I 0.09, h 50, s 00, M / 4, a 000, b, 0.5, N 0.5 p D heor em able Q D >0 <0 3-(ii >0 <0 3-(ii >0 <0 3-(ii >0 <0 3-(ii RC, >0 <0 3-(ii >0 <0 3-(ii >0 <0 3-(ii >0 <0 3-(ii >0 <0 3-(ii >0 <0 3-(ii >0 <0 3-(ii

13 International Journal of Computational an Applie Mathematics. ISSN Volume, Number (07 Research Inia Publications For A 000, I 0.3, I 0.09, h 50, s 00, M / 4, a 000, b,.5. able N c heorem Q D RC, >0 <0 3-(ii >0 <0 3-(ii <0 <0 3-(i >0 <0 3-(ii >0 <0 3-(ii <0 <0 3-(i >0 >0 3-(iii >0 >0 3-(iii <0 <0 3-(i >0 <0 3-(ii >0 <0 3-(ii <0 <0 3-(i >0 <0 3-(ii >0 <0 3-(ii <0 <0 3-(i >0 >0 3-(iii >0 >0 3-(iii <0 <0 3-(i >0 <0 3-(ii >0 <0 3-(ii <0 <0 3-(i >0 <0 3-(ii >0 <0 3-(ii <0 <0 3-(i >0 >0 3-(iii >0 >0 3-(iii <0 <0 3-(i

14 International Journal of Computational an Applie Mathematics. ISSN Volume, Number (07 Research Inia Publications For A=000, I=0.3, I=0.09, h=50, s=00, M=/4, N=0.5, c=400, α=0.5, μ=.5 able 3 a b heorem Q D RC, 00.5 >0 <0 3-(ii >0 >0 3-(iii >0 >0 3-(iii <0 <0 3-(i >0 <0 3-(ii >0 >0 3-(iii <0 <0 3-(i >0 <0 3-(ii From able, we can observe the optimal cycle time with various parameters of μ. he following inference can be mae base on able.when the value of μ (i.e. Mar-up rate is increasing, the optimal cycle time will increase. But, the eman per year will ecrease when the retailer increases the μ value. From able, we can observe the optimal cycle time with various parameters of α, N, c. he following inferences can be mae base on able. When the value of α (i.e. Customer s fraction of the total amount payable at a time of placing an orer within elay perio to retailer is increasing, the optimal cycle time will ecrease. he Orering quantity an the total relevant cost ecreases. When the value of N (i.e. Customer s trae creit perio as measure by years offere by retailer is increasing, the optimal cycle time will increase. he Orering quantity an the total relevant cost increases. When the value of c (i.e. Unit purchase cost is increasing, the optimal cycle time will increase. he Orering quantity an the total relevant cost ecreases. From able 3, we can observe the optimal cycle time with various parameters of a, b. he following inferences can be mae base on able 3. 7

15 International Journal of Computational an Applie Mathematics. ISSN Volume, Number (07 Research Inia Publications When the value of a is increasing, the optimal cycle time will increase. he Orering quantity ecreases an the total relevant cost increases. When the value of b is increasing, the optimal cycle time will ecrease. he Orering quantity increases an the total relevant cost ecreases. Conclusion his paper incorporate both Huang s moel [5] an Chung s moel [6] by consiering the price epenent eman. We provie three theorems to help the retailer in quicly etermining the optimal cycle time. heorem proves the convexities of the functions RC (,, RC (, an RC3 (, when 0, 0, +>0. heorem 3 helps retailer accurately an speeily etermine the optimal orering policy after computing for the numbers Δ an Δ. Finally, numerical examples are given to illustrate the result evelope in this paper. here are several managerial insights as follows: ( When the Mar-up rate is increasing, the optimal cycle time will increase. But, the eman per year will ecrease when the retailer increases the μ value. In the long run, the retailer has a isavantage if the eman eeps on ecreasing. he retailer must not only focus on ecreasing the total relevant cost but shoul also increase the annual eman. So, the retailer shoul be very careful in fixing the Mar-up rate; ( When the customer s fraction of the total amount ue at the time of placing an orer to the retailer is increasing, the retailer will orer a smaller quantity. he retailer can save a larger amount of cost when customer s fraction of the total amount ue at the time of placing an orer within the elay perio offere by retailer is small; (3. When a longer trae creit perio offere to customer, the retailer will orer a larger quantity to save interest payments pai to suppliers to compensate the loss of interest earne pai by his/her customers; an (4. When the unit purchasing price is increasing, retailer will orer a smaller quantity to enjoy the benefits of the trae creit more frequently; (5. When the annual eman is increase, the optimal cycle time ecreases so that the retailer can enjoy the interest earne in sales revenue until the next replenishment time. References [] Walmann, K. H., Inventory moels, Opti Opns Res. [] Fisman, R., 00, rae creit an prouctive efficiency in eveloping countries, Worl Dev, 9: 3 3 [3] Goyal, S. K., 985, Economic orer quantity uner conitions of permissible elay in payments, J Opl Res Soc, 36:

16 International Journal of Computational an Applie Mathematics. ISSN Volume, Number (07 Research Inia Publications [4] Chung, K. J., 998, A theorem on the etermination of economic orer quantity uner conitions of permissible elay in payments, Comp Opns Res, 5: [5] Jamal, A. M. M., Sarer, B. R., Wang, S., 997, An orering policy for eteriorating items with allowable shortage an permissible elay in payment, J Opl Res Soc, 48: [6] Sarer, B. R., Jamal, A. M. M., Wang, S., 000, Supply chain moels for perishable proucts uner inflation an permissible elay in payment, Comp Opns Res, 7: [7] Jamal, A. M. M., Sarer, B. R., Wang, S., 000, Optimal payment time for a retailer uner permitte elay of payment by the wholesaler, Int J Pro Econ, 66: [8] eng, J.., 00, On the economic orer quantity uner conitions of permissible elay in payments, J Opl Res Soc, 53: [9] Chung, K. J., Huang, Y. F., Huang, C. K., 00, he replenishment ecision for EOQ inventory moel uner permissible elay in payments, Opsearch, 39: [0] Shinn, S.W., Hwang, H., 003, Optimal pricing an orering policies for retailers uner orer-size-epenent elay in payments, Comp Opns Res, 30: [] Chung, K. J., Huang, Y. F., 003, he optimal cycle time for EPQ inventory moel uner permissible elay in payments, Int J Pro Econ, 84: [] Huang, Y. F., Chung, K. J., 000, Optimal replenishment an payment policies in the EOQ moel uner cash iscount an trae creit, Asia-Pacific J Opl Res, 0: [3] Salameh, M. K., Abbou, N. E., El-Kassar, A. N., Ghattas, R.E., 003, Continuous review inventory moel with elay in payments, Int J Pro Econ, 85: [4] Huang, Y. F., 003, Optimal retailer s orering policies in the EOQ moel uner trae creit financing, J Opl Res Soc, 54: [5] Huang, Y. F., Yen, G., Hsu, K.H., Orering ecision-maing uner two-level trae creit policy for a retailer with a powerful position, JSIR- 40. [6] Chung, K. J., Huang, C. K., 009, An orering policy with allowable shortage an permissible elay in payments, Applie Mathematical Moelling, 33: