OPTIMAL PRICING AND LOT-SIZING DECISIONS UNDER WEIBULL DISTRIBUTION DETERIORATION AND TRADE CREDIT POLICY

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1 Yugoslav Journal of Operations Researh Vol 8 (8), Number, -34 DOI:.98/YUJOR8M OPIMAL PRICING AND LO-SIZING DECISIONS UNDER WEIBULL DISRIBUION DEERIORAION AND RADE CREDI POLICY S. K. MANNA Department of Management Siene, National Chiao ung University Hsinhu, aiwan, R. O. C. skmanna_5@hotmail.om K. S. CHAUDHURI Department of Mathematis, Jadavpur University, Kolkata 7 3, India C. CHIANG Department of Management Siene, National Chiao ung University Hsinhu, aiwan, R. O. C. Reeived: June 8 / Aepted: Otober 8 Abstrat: In this paper, we onsider the problem of simultaneous determination of retail prie and lot-size (RPLS) under the assumption that the supplier offers a fixed redit period to the retailer. It is assumed that the item in stok deteriorates over time at a rate that follows a two-parameter Weibull distribution and that the prie-dependent demand is represented by a onstant-prie-elastiity funtion of retail prie. he RPLS deision model is developed and solved analytially. Results are illustrated with the help of a base example. Computational results show that the supplier earns more profits when the redit period is greater than the replenishment yle length. Sensitivity analysis of the solution to hanges in the value of input parameters of the base example is also disussed. Keywords: Retail prie, lot-size, inventory management.. INRODUCION Inventories of physial goods play a major role in the eonomy of any ountry beause a huge amount of money is tied up in inventories. Proper management of

2 S.K. Manna, K.S. Chaudhuri, C. Chiang / Optimal Priing and Lot-Sizing Deision inventories an be very worthwhile and profitable. Studies on inventory management started with the EOQ (eonomi order quantity) formula whih is derived based on the assumption that the retailer (buyer) has to pay fully for the items as soon as he reeives them from a supplier. However, as a standard pratie in the real markets, a supplier usually allows a ertain fixed period (known as the redit period) for settling the amount of money that the retailer owes to him for the items supplied. his C (trade redit) poliy plays an important role in the business of many produts and it serves the interests of both the supplier and retailer. he supplier usually expets the profit to inrease sine rising sales volumes ompensate the apital losses inurred during the redit period. Also, the supplier finds an efetive means of prie disrimination whih irumvents anti-trust measures. On the other hand, the retailer earns an interest by investing the sale proeeds earned during the redit period. In addition, a C poliy develops a relationship of mutual trust between the supplier and retailer. he benefit of C poliies is more pronouned when the demand of a produt depends on its retail prie. Under suh a marketing situation, the retailer is able to hoose the selling prie from a wider range of options existing in the market by utilizing the redit period ofered by the supplier. herefore, C poliies are a ommon and realisti industrial poliy usually followed by wholesalers. For these reasons, inventory modellers felt the need to take C poliies into onsideration. Beause of the marketing signifiane of the C poliy, inventory modellers like Haley and Higgins (973), Ben-Horim and Levy (98), Kingsman (983), Chapman et al. (984), Goyal (985), Ward and Chapman (987), Chung (989), Aggarwal and Jaggi (994), et. devoted their attention to it. One of the results of their studies is that the EOQ is independent of the length of the redit period. his result is unexpeted, perhaps due to their assumption of onstant demand. As for the RPLS problem, previous studies inlude Kunreuther and Rihard (97), Kunreuther and Shrage (973), Ladany and Sternlieb (974) and Shah and Jha (99) that assume the demand to be a dereasing funtion of retail prie. Abad (988) and Lee (993) studied the RPLS problem under an additional assumption that the demand is a dereasing funtion of prie and that the supplier ofers all-unit quantity disounts. Abad (988a) also onsidered the ase of inremental quantity disounts. Kim et al. (995) disussed the RPLS problem under the C poliy assuming the retailer's borrowing and lending rates of apital to be equal. If the buyer's borrowing and lending rate of apital are equal, then the redit period has no efet on the EOQ (Chapman et al. (984), Goyal (985), Haley and Higgins (973)). Also, this type of assumption seems quite restritive from the pratial point of view. Reent C related papers inlude eng (), Huang (3) and You (5), et. All the researh works mentioned above assume that inventory is depleted by onsumer's demand only. his assumption is valid only for non-deteriorating items. here are numerous inventory items whose utility does not remain onstant through out their life yle. he type of produts subjet to on-going deterioration an be broken down into those with a maximum usable lifetime (perishable produts) and those without (deaying produts). Blood, mediines and ertain foods are examples of perishable produts with a maximum usable lifetime. Volatile liquids suh as alohol and gasoline are produts whih deay without a maximum lifetime. For this type of produts, inventory is depleted not only by demand but also by deterioration. Ghare and Shrader (963) derived a revised form of the eonomi order quantity model assuming

3 S.K. Manna, K.S. Chaudhuri, C. Chiang / Optimal Priing and Lot-Sizing Deision 3 exponential deterioration of inventory and onstant demand. Cohen (977) onsidered the RPLS problem for an exponentially deteriorating items. Hariga (995) examined the inventory lot-sizing problem for deaying produts. Hwang and Shinn (997) dealt with the joint prie and lot-size determination model for an exponentially deteriorating produts under the ondition of permissible delay in payments. Jamal et al. () and Sarker et al. () addressed the optimal payment time under permissible delay in payment for deaying items. Reently, Quyang et al. (5) established an EOQ model for deteriorating produts, in whih the supplier provides not only a ash disount but also a delay in payments to ustomers. Barrotoni (96) observed, while disussing the diffiulties of fitting empirial data to mathematial distributions, that both leakage failure of dry batteries and life expetany of ethial drugs ould be expressed in terms of Weibull distribution. hus deterioration of physial goods modeled by a two-parameter Weibull distribution is muh more realisti and generalized than the onstant deterioration rate onsidered by the above researhers. his paper disusses the RPLS problem for a two-parameter Weibull distribution deteriorating items as in Covert and Philip (973). But unlike Covert and Philip's model, we assume the prie-dependent demand represented by a onstant prieelastiity funtion of retail prie. In addition, this paper inorporates C poliies into our model. We relax the assumption of equal interest rates (as in Kim et al. (995)) and allow the retailer's borrowing rate to be greater than or equal to his lending rate. he rest of this paper is organized as follows. he model is first developed and solved analytially. Next, results are illustrated with the help of a base example. Sensitivity analysis of the solution to hanges in the value of input parameters of the base example is also disussed.. ASSUMPIONS AND NOAIONS We adopt the following assumptions for the model to be disussed: Replenishments are instantaneous with a known and onstant lead time. he inventory system involves only one item. Shortages are not allowed. Annual demand rate is represented by a onstant prie elastiity funtion of retail prie. he distribution of the time to deterioration follows a two parameter Weibull distribution. Notations used in this paper are as follows: C unit purhase ost. S ordering ost. H inventory arrying ost, exluding the apital opportunity ost. R apital opportunity ost (as a perentage). I earned interest rate (as a perentage). t redit period set by the supplier. Q lot size. replenishment yle time. P unit retail prie for the produt.

4 4 S.K. Manna, K.S. Chaudhuri, C. Chiang / Optimal Priing and Lot-Sizing Deision K saling onstant (> ). β the onstant prie elastiity (> ). In addition, we make the following assumptions and notations: (i) he annual demand rate D( P) = KP β, is a onstant prie elastiity funtion. Now, dd( P) ( β ) = KβP + < implies demand is a downward sloping funtion of prie. dp his is a ommon result in the real market. (ii) he inventory deteriorating rate θ() t = αβt β where α ( > ) and β ( > ) are respetively sale and shape parameters. Deterioration rate is inreasing with t for β > and dereasing for β <. he ase of onstant deterioration first onsidered by Ghare and Shrader (963) is also a speial ase when β =. It is lear that if the initial deterioration rate is extremely high, then the two-parameter Weibull distribution is appropriate for an item with dereasing rate of deterioration. his distribution is appliable to inreasing rate of deterioration if the initial deterioration rate is approximately zero. (iii) During the redit period proposed by supplier, sales revenue is deposited in an interest bearing aount with rate I. At the end of the period, the redit is settled and the retailer starts paying the apital opportunity ost for items in stok with rate RR ( I). 3. FORMULAION OF HE MODEL We shall onsider a ontinuous-review, deterministi-demand model with twoparameter Weibull distribution deterioration. Let qt () be the inventory position of the system at any time t. Demand rate DP ( ) is assumed to be positive. Under ontinuous review it is logial to assume that depletion due to deterioration and depletion due to meeting demand will our simultaneously. Aordingly, the differential equation desribing the time behavior of the inventory system is dq() t β + αβt q() t = D( P), t () dt with the onditions q() = Q, q( ) =. Solution of equation () is n nβ + β α t qt () = exp( α t ) q() DP ( ), t. + ) n! () Consequently, Z() t, the inventory loss due to deterioration in the time interval [,t ] is given by n nβ + β β α t Z( t) = q()( exp( αt )) + D( P)exp( αt ) D( P) t + ) n!

5 S.K. Manna, K.S. Chaudhuri, C. Chiang / Optimal Priing and Lot-Sizing Deision 5 herefore, the quantity ordered per yle is Q = Z( ) + D( P) β = q() exp( α ) q() D( P) n nβ + α + ) n! Using the onditions q () = Q and q ( ) =, we have (3) n nβ + α Q = D( P) + ) n! Using (4), we have from () n nβ+ n nβ+ β α α t qt () = DP ( )exp( α t ) + ) n! + ) n!, t. he retailer's objetive is to maximize the annual net profit Π( P, ) produts. he inventory ost an be expressed as C (, P) (4) (5) from the sales of the in the first yle onsisting of four omponents C (, P ) = Ordering ost + Purhasing ost + Inventory arrying ost + Capital opportunity ost. S We have the annual ordering ost =. he annual purhasing ost CQ = n nβ α = CD( P). + ) n! he annual inventory arrying ost H = qt () n n n n ( ) β+ β+ HD P α α t β = exp( t ) dt α + ) n! + ) n! HD P t t = n n n n m n n n n m m m ( ) β+ β α + α β ( ) α ( β + )! ( β + )! m=! m n+ m ( n+ m) β+ m n+ m ( n+ m) β+ HD( P) ( ) α ( ) α = m= + )( mβ + ) n! m! m= + )(( n+ m) β + ) n! m! m n+ m ( n+ m) β + HD( P) ( ) α =. m= ( mβ + )(( n+ m) β + ) n! m! dt

6 6 S.K. Manna, K.S. Chaudhuri, C. Chiang / Optimal Priing and Lot-Sizing Deision Figure : Credit period vs. replenishment yle time Annual apital opportunity ost: (i) Case ( t ) :(see Fig. (a)) During the redit period t, the sales revenue is used to earn interest with annual rate I. Average number of produts in stok earning DPt ( ) interest during time (, t ) is and the interest earned per order beomes DPtCI ( ). After settling the redit, the produt still in stok is assumed to be finaned with annual rate R. Interest payable per order an be expressed as CR t q() t dt. herefore, the annual apital opportunity ost = CR q() t dt CID( P) t t DP ( ) = CR m= + )( mβ + ) n! m! m= m n+ m nβ+ mβ+ mβ+ ( ) α ( t ) α CIt + )(( n+ m) β + ) n! m! m n+ m ( n+ m) β+ ( n+ m) β+ ( ) ( t ) ] (ii) Case ( t ):( seefig. ( b)) In the ase t >, all the sales revenue is used to > earn interest with annual rate I during the redit period t. he average number of

7 S.K. Manna, K.S. Chaudhuri, C. Chiang / Optimal Priing and Lot-Sizing Deision 7 produts in stok earning interest during time (, ) and ( t, ) beome and D ( ) respetively. herefore, the annual apital opportunity ost DP ( ) = ( DPCI ( ) + DPt ( ) ( CI ) ) = CID( P) CID( P) t he annual inventory ost C (, P) = he annual net profit C (, P) Π ( P, ) =Annual sales revenue - Annual inventory ost = PD( P) CP (, ). (6) Let Π ( P, ) Π ( P, ) and be the annual net profits when t and respetively. Using (6), we have t > Π = β ( P, ) KP P C n nβ α + ) n! m n+ m ( n+ m) β + ( ) α H m= ( mβ + )(( n+ m) β + ) n! m! CR m= m n+ m nβ mβ+ mβ+ ( ) α ( t ) + )( mβ + ) n! m! m n+ m ( n+ m) β+ ( n+ m) β+ ( ) α t S + + ( CIt )]], (7) m= + )(( n+ m) β + ) n! m! Π = β ( P, ) KP P C H m= n nβ α + ) n! m n+ m ( n+ m) β + ( ) α S CI + CIt ]. (8) ( mβ + )(( n+ m) β + ) n! m! he neessary onditions to maximize Π (, ) and Π (, ) are respetively given by the equations

8 8 S.K. Manna, K.S. Chaudhuri, C. Chiang / Optimal Priing and Lot-Sizing Deision Π (, ) P = P Π (, ) P and =, (9) Π (, ) P Π = ( P and, ) =. () P Equation (9) yields the results β P [ n nβ α P C + ) n! m n+ m ( n+ m) β + ( ) α H m= ( mβ + )(( n+ m) β + ) n! m! CR m= m n+ m nβ mβ+ mβ+ ( ) α ( t ) + )( mβ + ) n! m! and m n+ m ( n+ m) β+ ( n+ m) β+ ( ) α ( t ) + + CIt m= + )(( n+ m) β + ) n! m! =. () ] KP β C n nβα + ) n! nβ m n+ m ( n+ m) β ( ) α (( n+ m) β + ) H m= ( mβ + )(( n+ m) β + ) n! m! CR m= ( ) α (( n+ m) β + ) ) t + )( mβ + ) n! m! m n+ m ( n+ m) β nβ mβ+ m n+ m ( n+ m) β + ( ) α + m= + ) n! m! + )(( n+ m) β + ) n! m! m= + CIt ] + Similarly equation () gives m n+ m ( n+ m) β+ ( n+ m) β+ ( ) α ( t ) S =. ()

9 S.K. Manna, K.S. Chaudhuri, C. Chiang / Optimal Priing and Lot-Sizing Deision 9 And n nβ β α P C P + ) n! H + = KP m n+ m ( n+ m) β + ( ) α CI CIt m= ( mβ + )(( n+ m) β + ) n! m! β C n nβα + ) n! nβ m n+ m ( n+ m) β ( ) α (( n+ m) β + ) S H CI + =. ( mβ + )(( n+ m) β + ) n! m! m= (3) (4) he solutions of the equations () and () give the optimal values P = P and whih maximize Π ( P, ), provided they satisfy the suffiient onditions Π Π Π Π P P P ( ) >, < Π < = and. (5) Also for Π ( P, ), optimal solution P, of the equations (3) and (4) must satisfy the suffiient onditions Π Π Π Π P P P ( ) >, < Π < and. (6) 4. COMPUAIONAL RESULS Profit funtions for the two ases are the funtions of P and. Our objetive is to determine P and whih maximize the profit funtions. Using Seond-Order Newton-Raphson method in programming language Fortran-77, we solve the highly nonlinear equations () and () to obtain P and for given input parameters. Optimal profit and order size for ase are alulated from expressions (7) and (4) respetively. Using the same programming language, we solve equations (3) and (4) for P and. hen optimal profit and order size for ase are determined from expressions (8) and (4) respetively. o illustrate, onsider the base example: S = 9, K = 6, C = 5, H =.5, R =.4, I =., β =.5, β =, t =.4 and α =. in appropriate units. he optimal solution for ase is P =.34, X =.47, and orresponding optimal profit Π ( P, ) = and optimal order size Q = 7.99 For t =.8 in base example, the optimal solution for ase is P = 9.9, =.46, Π ( P, ) = 88.4 and Q = Comparing the results of the two numerial examples, we find that for a ent perent inrease in t, the profit inreases by 4.38%, order quantity inreases by 6.45%, prie dereases by 4.4% and yle time dereases by.69%.

10 3 S.K. Manna, K.S. Chaudhuri, C. Chiang / Optimal Priing and Lot-Sizing Deision P,, Π, Π he sensitivity of eah of the deision variables and Q to hanges in eah of the parameters SKCHRI,,,,,, β, t and α are examined in ables -. he sensitivity analysis is performed by hanging eah of the parameters by -5%, -%, % and 5%, taking one parameter at a time and keeping the remaining parameters unhanged. It is diffiult to exhibit the sensitivity results analytially in respet of even one parameter beause of the ompliated nature of the profit funtions Π and Π. From ables -, we see that the optimal profit is insensitive to hanges in R and β. he optimal profit is slightly sensitive to hanges in the values of the parameters SHI,,, β, t and α in both the ases t and t >. It is quite sensitive to hanges in the parameter K. It is also sensitive to larger hanges in C. he optimal order quantity is almost insensitive to hanges in the parameters H, I, R, β and slightly sensitive to hanges in β, t andα. It has low sensitivity to S and moderate sensitivity to K. It is quite sensitive to larger hanges in C. able : Sensitivity analysis for ase ( t ) (%) (%) hanging (%) Π hange hange parameters P Q hange in in S K C H R I Π Q

11 S.K. Manna, K.S. Chaudhuri, C. Chiang / Optimal Priing and Lot-Sizing Deision 3 able (ontinued): Sensitivity analysis for ase ( t ) hanging parameters (%) hange P Π Q (%) hange in Π (%) hange β β t α able : Sensitivity analysis for ase ( t > ) hanging parameters (%) hange P Π Q (%) hange in Π in Q (%) hange S K C H R I in Q

12 3 S.K. Manna, K.S. Chaudhuri, C. Chiang / Optimal Priing and Lot-Sizing Deision able (ontinued): Sensitivity analysis for ase ( t > ) hanging parameters (%) (%) (%) Π hange hange P Q hange in in β β t α Π Q 5. CONCLUSIONS his paper deals with the RPLS problem under a C poliy when the item in stok deteriorates over time. he deterioration rate is time-dependent and follows a twoparameter Weibull distribution. his type of deterioration, first onsidered by Covert and Philip (973), is more realisti than the exponential deay in the RPLS models of Cohen (977). In the present paper, introdution of the two-parameter Weibull distribution deterioration reates muh mathematial and omputational diffiulties. he expression of the profit funtion involves double-summation infinite series whose onvergene is ritially dependent on the hoie of the values of the parameters α and β, and < α <, β > are appropriate hoies. β > represents an inreasing rate of deterioration when the initial rate is approximately zero. For < α <, β >, the ontribution of the infinite series beomes negligible after a few terms. Computational results show that supplier earns more profits when the redit period is greater than the replenishment yle length. he proposed model an be extended in several ways. For instane, a threeparameter (instead of two-parameter) Weibull distribution an be assumed. Also, demand an be a funtion of both selling prie and time. Finally, quantity disounts and inflation may be inorporated into the model.

13 S.K. Manna, K.S. Chaudhuri, C. Chiang / Optimal Priing and Lot-Sizing Deision 33 Aknowledgements: he authors express their sinerest thanks to Jadavpur University, Kolkata for providing infrastrutural support to arry out this work. Revision of this researh paper is done while the first author stayed at National Chiao ung University, aiwan (researh grant no H-9-3 by National Siene Counil). REFERENCES [] Abad, P.L.,, Determining optimal selling prie and lot-size when the supplier offers all-unit quantity disounts, Deision Siene, 9 (988) [] Abad, P.L., Joint prie and lot-size determination when the supplier offers inremental quantity disounts. Journal of the Operational Researh Soiety, 39 ( 988) [3] Aggarwal, S.P., and Jaggi, C.K., Credit finaning in eonomi ordering poliies of deteriorating items. International Journal of Prodution Eonomis, 34 (994) [4] Barrotoni, J.N., Pratial appliations of Weibull distribution, ASQC ehnial Conferene ransations, 96, [5] Ben-Horim, M., and Levy, H., Inflation and a trade redit period. Management Siene, 8(6) (98) [6] Chapman, C.B., Ward, S.C., Cooper, D.F., and Page, M.J., Credit poliy and inventory ontrol. Journal of the Operational Researh Soiety, 35 (984) [7] Chung, K.H., Inventory ontrol and trade-redit revisited, Journal of the Operational Researh Soiety, 4(5) (989) [8] Cohen, M.A., Joint priing and ordering poliy for exponentially deaying inventory with known demand. Naval Researh Logistis Quarterly, 4 (977) [9] Covert, R.P., and Philip, G.C., An EOQ model for items with Weibull distribution deterioration. AIIE ransations, 5 (973) [] Ghare, P.M., and Shrader, G.F., A model for an exponential deaying inventory, Journal of Industrial Engineering, 4 (963) [] Goyal, S.K., Eonomi order quantity under onditions of permissible delay in payments. Journal of the Operational Researh Soiety, 36 (985) [] Haley, C.W., and Higgins, R.C., Inventory poliy and trade redit finaning. Management Siene, (973) [3] Hariga, M.A., Lot-sizing models for deteriorating items with time-dependent demand. International Journal of Systems Siene, 6 (995) [4] Huang, Y.F., Optimal retailer's ordering poliies in the EOQ model under trade redit finaning. Journal of the Operational Researh Soiety, 54 (3) -5. [5] Hwang, H., and Shinn, S.H., Retailer's priing and lot-sizing poliy for exponentially deteriorating produts under the ondition of permissible delay in payments. Computers and Operations Researh, 4(6) (997) [6] Jamal, A.M.M., Sarker, B.R., and Wang, S., Optimal payment time for a retailer under permitted delay of payment by the wholesaler, International Journal of Prodution Eonomis, 66 () [7] Kim, J.S., Hwang, H., and Shinn, S.W., An optimal redit poliy to inrease supplier's profits with prie dependent demand funtions. Prodution Planning and Control, 6 (995) [8] Kingsman, B.G., he effet of payment rule on ordering and stok holding in purhasing. Journal of the Operational Researh Soiety, 34 (983) [9] Kunreuther, H. and Rihard, J.F., Optimal priing and inventory deisions for non-seasonal items. Eonometria, 39, (97), [] Kunreuther, H., and Shrage, L., Joint priing and inventory deisions for onstant pried items, Management Siene, 9 (973)

14 34 S.K. Manna, K.S. Chaudhuri, C. Chiang / Optimal Priing and Lot-Sizing Deision [] Ladany, S. and Sternlieb, A., he interation of eonomi ordering quantities and marketing poliies, AIIE ransitions, 6 (974) [] Lee, W.J., Determining order quantity and selling prie by geometri programming: optimal solution, bounds, and sensitivity, Deision Siene, 4 (993) [3] Ouyang, L.Y., Chang, C.., and eng, J.., An EOQ model for deteriorating items under trade redits. Journal of the Operational Researh Soiety, 56 (5) [4] Sarker, B. R., Jamal, A.M.M., and Wang, S., Optimal payment time under permissible delay in payment for produts with deterioration, Prodution Planning and Control, () [5] Shah, Y.K., and Jha, P.J., A single period stohasti inventory model under the influene of marketing poliies. Journal of the Operational Researh Soiety, 4 (99) [6] eng, J.., On the eonomi order quantity under onditions of permissible delay in payments, Journal of the Operational Researh Soiety, 53 () [7] Ward, S.C., and Chapman, C.B., Inventory ontrol and trade redit - A reply to Daellenbah, Journal of the Operational Researh Soiety, 38 (987) [8] You, P.S., Inventory poliy for produts with prie and time-dependent demands, Journal of the Operational Researh Soiety, 56 (5)