An inventory model with temporary price discount when lead time links to order quantity

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1 80 Journal of Scientific & Indutrial Reearch J SCI IN RES VOL 69 MARCH 00 Vol. 69 March 00 pp An inventory model with temporary price dicount when lead time link to order quantity Chih-Te Yang Liang-Yuh Ouyang Kun-Shan Wu 3 and Hiu-Feng Yen 4 epartment of Indutrial Engineering & Management Ching Yun Unverity Jung-Li Taoyuan 30 Taiwan epartment of Management Science and eciion Making 3 epartment of Buine Adminitration 4 Graduate Intitute of Management Science Tamkang Univerity Tamui Taipei 5 Taiwan Received 3 June 009; revied 3 ecember 009; accepted 04 January 00 Thi paper preent effect of a temporary price dicount offered by upplier on a retailer replenihment policy when lead time i linked to order quantity. A deciion proce for retailer i developed in deciding whether to adopt a regular or pecial order policy during a temporary ale period. Optimal pecial order quantity i determined by maximizing total cot aving between pecial and regular order and illutrated by everal numerical example along with enitivity analyi of optimal olution. Keyword: Inventory Lead time Temporary price dicount Introduction Variou inventory model have been propoed to gain deeper inight into relationhip between price dicount and order policy. Naddor propoed inventory model of repone to a temporary price dicount where pecial ale coincide with replenihment time. Barker & Vilcaim tudied ituation where pecial ale coincide with retailer ale period. Ardalan 3 developed optimal order policie with poible combination of replenihment time and ale period. Terine & Schwarzkopf 4 relaxed aumption for length of pecial ale by permitting more than one replenihment cycle before ale expiration. Goyal 5 dicued temporary price dicount model for calculating average inventory of a regular policy. Aull-Hyde 6 preented a temporary price dicount model for backlog permiible. Variou tudie 7-0 are available of repone to a temporary ale price. However in thee tudie almot all the lead time (LT) wa aumed to be contant and zero. In reality LT i non-zero and variable due to time to fill order at ource hipping time and time required to carry out paperwork. Several tudie - 5 are available on inventory problem with variable LT. Author for correpondence Tel: #467; Fax: ctyang@cyu.edu.tw Thi tudy invetigate effect of a temporary price dicount offered by upplier on retailer replenihment policy with LT linked to order quantity. Model Formulation LT of a pecial order policy L i alway longer than that of regular order policy L when retailer order large quantitie. Thi may lead to an inventory hortage before pecial order quantity arrive implying that a hortage cot i incurred. Therefore retailer may place a large pecial order of quantity (adopt L ) or alternatively adopt L. Temporary rice icount Occur at Retailer Reorder Time t t ) oint ( r In thi ituation retailer may place a large pecial order of quantity or alternatively may place a regular order of quantity at dicounted price ( δ ) c at retailer replenihment time followed by EO policy with original price c for next replenihment (Fig. ). ue to length of LT of L being greater than length of LT for L a hortage occur and backorder may be placed if retailer decide to order for. Therefore total cot ( TCS ) coniting of cot for ordering purchaing hortage and holding i expreed a

2 YANG et al: TEMORARY RICE ISCOUNT EFFECT WHEN LEA TIME LINKS TO ORER UANTITY 8... Fig. Special v regular order policie when temporary price dicount occur at retailer' reorder time point ( t t ) r b ( ) TCS A δ c ( ) ( ) δ hc () If retailer adopt a regular EO policy intead of placing a large pecial order quantity policy during the period / then total cot ( TCN ) i a TCN A ( ) c δ () From Eq () and () total cot aving between pecial and regular order during time period can be obtained a g ( TCN TCS ) ( δ ( ) A hc c ) hc / A hc ( δ ) hc b δ c ( δ ) hc ( ) (3) Temporary rice icount oe Not Occur At Retailer Reorder t t ) Time oint ( r When upplier offer a temporary price dicount at time point t if retailer decide to place a pecial order quantity then retailer receive item after time length L. Conequently inventory level q before pecial order quantity arrive i given by q t. With regard to total cot aving two poible cae are found baed on value of q : () q 0 and () q < 0 (Fig. ). Cae : q 0 When q 0 it mean that inventory level either or 0 before arrive. Total cot of pecial order during time period ( q) / i given a TCS hc A ( δ ) c ( δ ) hc t t (4)

3 8 J SCI IN RES VOL 69 MARCH 00 Fig. Special v regular order policie when temporary price dicount doen't occur at retailer' reorder time point ( t t ) r Total cot of time period policy i repreented by TCN hc t ( q) / under EO A hc c (5) Therefore total cot aving during time period ( q) / can be obtained a g ( TCN TCS ) A hc (δ) hc δc t A (6) Cae : q < 0 When q < 0 it mean that hortage occur before pecial order quantity arrive. Total cot of adopting a pecial order policy i given a

4 YANG et al: TEMORARY RICE ISCOUNT EFFECT WHEN LEA TIME LINKS TO ORER UANTITY 83 TCS A ( δ) c ( δ ) hc Total cot of time period given a TCN b A hc c t t (7) / under EO policy i (8) Therefore total cot aving during time period / can be obtained a g ( TCN TCS ) A hc δ c ( δ ) hc b A t t (9) Theoretical Reult It i worth placing a pecial order when total cot aving i poitive. Otherwie retailer hould adopt regular order policy. Temporary rice icount Occur at Retailer Reorder Time t t ) oint ( r Taking firt and econd order derivative of g( ) in Eq (3) with repect to one get dg ( d and d d g ( ) A hc δ c b ( δ ) hc ( ) (0) ) b ( δ ) hc () Given that d g( ) / d < 0 then g ( ) i a concave function of. Therefore a unique value exit of (ay ) which maximize g ( ). Solving Eq. dg ( ) / d 0 give ( A/ δc hc /) b ( δ) ) b ( δ) ) () Subtituting Eq. () into Eq. (3) correponding maximum total cot aving can be obtained a A hc g ( ) δ c [ b ( δ ) ) ] ( δ) hc ( ) > 0. Thu when temporary price dicount occur at retailer reorder point optimal pecial quantity. Temporary rice icount oe Not Occur at Retailer Reorder Time oint ( t t ) r Cae : q 0 [ ( t ) / ] Taking firt order derivative of g ( ) in Eq. (6) with repect to and letting reult equal zero give dg ( d ) A hc δ c ( ) hc δ δ ) hc ( t By olving Eq. (3) a unique olution of ) i obtained a [( A / 0 δc hc / ) ( δ ) ( δ ) ) (3) (ay t )] (4) and econd order derivative of g ( ) in Eq. (6) with repect to i

5 84 J SCI IN RES VOL 69 MARCH 00 d g ( ) ( δ ) hc d (5) Subtituting Eq. (4) into Eq. (6) correponding maximum total cot aving for Cae i g ( ( δ ) hc ) A A hc δ c ( δ) t) A ( δ) ) (6) It i worthwhile noting that when then g ( ) < 0 which mean that retailer total cot aving i negative. In thi ituation optimal deciion for retailer i not to order until next replenihment time at which point they hould order regular quantity. For convenience aume that ( t )/ g( ) and i optimal pecial order quantity. Then Theorem Optimal pecial order quantity for Cae a follow: if > 0 and > 0 0 otherwie. Remark When 0 it mean that optimal policy for retailer i not to order until next replenihment time and at that time order regular order quantity. Cae : q < 0 [ ( t ) > / ] Taking firt and econd order derivative of g ( ) in Eq. (9) with repect to give dg ( ) A hc b δ c t d and d ( δ ) hc t g ( ) b ( δ ) hc ( ) d (7) (8) Given that d g( ) / d < 0 g ( ) i a concave function of a unique value of (ay ) which maximize g ( ) can be found. Solving Eq. dg ( ) / d 0 give b ( δ) hc ( ) ( t ) b ( δ A hc δ c ) hc ( ) (9) Subtituting Eq. (9) into Eq. (9) correponding maximum total cot aving for Cae i g ( b (δ) ) ) b ( δ) hc b ( δ ) hc ( ) [ b ( δ ) hc ( ) ] A hc δc b ( δ ) hc ( t ) ( t ) A b ( δ) ) ( t ) A (0) Similarly to obtain optimal pecial order quantity ( ) for Cae let 3 ( t ) / and g ( ) then 4 Theorem Optimal pecial order quantity for Cae a follow: if 3 > 0 and 4 > 0 0 otherwie. Summarizing above reult an algorithm i developed to obtain optimal olution for two ituation. Algorithm Step If t t. Go to Step 6. r then optimal pecial order quantity i

6 YANG et al: TEMORARY RICE ISCOUNT EFFECT WHEN LEA TIME LINKS TO ORER UANTITY 85 t r Table Optimal olution of Example under different value of t TCN TCS g TCN and TCS denote minimum total cot for regular and pecial order policy repectively; g i correponding maximum total cot aving; indicate retailer will not order until next replenihment time and adopt regular EO policy. Step etermine value of and from Eq. (4) and Eq. (9) repectively. Step 3 Step 4 Step 5 Calculate value of ( t ) / and g( ) in Eq. (6). If > 0 and > 0 then et and g ( ) ; otherwie et 0 and g ( ) 0. Calculate value of 3 ( t )/ and 4 g( ) in Eq. (0). If 0 3 > and 4 > 0 then et and g ( ) 4 ; otherwie et 0 and g ( ) 0. Find max { g ( ) g ( )}. If g( ) g( ) then ; otherwie. Step 6 Stop. Once optimal olution i obtained correponding minimum total cot can be found for regular and pecial order and ubequently determining maximum cot aving. Numerical Example To illutrate olution proce for two ituation following example are preented Example An inventory ytem with following data i conidered: c $0 / unit 000 unit/ year A $50 / order b $ 5 /unit/year δ and h 0.3. From thee data it can be found that Set of value of and t are aumed a { } and t { t r 0.0} where t r ( / ) L ( / ) ( / ). From algorithm optimal order policie depending on different value of t r and t can be obtained (Table ). From Table it i oberved that retailer will determine optimal pecial order quantity by trading off benefit of price dicount againt additional holding cot and hortage cot if neceary. For example for

7 86 J SCI IN RES VOL 69 MARCH 00 t 0.0 and 3000 or 3500 in Table optimal value of 0 which implie that retailer hould ignore temporary price dicount offered by upplier and will not order until next replenihment time and then adopt regular EO policy. For other cae optimal policy of retailer i to order a pecial quantity. In addition when upplier production rate increae pecial order quantity increae at the beginning and decreae later. However total cot aving decreae. Example In thi example a enitivity analyi i conducted demontrating robutne of model wherein effect of change in ytem parameter ( c δ A h and b ) on optimal pecial order quantity and optimal total cot aving are examined. ata ued are ame a thoe in Example. For convenience conidering the cae when 000 and t Senitivity analyi i conducted by changing each of the parameter by 50% 5% 5% and 50% and imultaneouly altering one parameter at a time while all other parameter remain unchanged (Table ). Baed on reult in Table following managerial inight are gained: ) When purchae cot c increae pecial order quantity decreae. However total cot for regular order policy TCN total cot for pecial order policy TCS and total cot aving g increae. Thu a purchae cot increae it i more beneficial for retailer to adopt pecial order policy. Neverthele pecial order quantity will decreae; ) A TCN TCS and g increae δ increae. Thi implie that greater the price dicount larger the pecial order quantity total cot of regular order total cot of pecial order and total cot aving for retailer. Reult are conitent with baic common theory. Moreover δ i enitive to total cot aving. From upplier viewpoint offering retailer a high temporary dicount rate will lead to increaed ale volume; 3) TCN TCS and g increae when and A increae. It implie that when either market demand rate or ordering cot or both increae retailer will order a larger quantity. Moreover when market demand rate i ufficiently low (value of reduce to 50% in Table ) it i not worthwhile for retailer to place a pecial order; and 4) Accordingly when either h or b or both increae TCN TCS and g decreae. It implie that when either holding cot or hortage cot or both are higher retailer will order a lower quantity and total cot aving will decreae. Additionally holding cot i more enitive to optimal olution than hortage cot. Table Effect of change in variou parameter of Example arameter Change % % c δ A h b Concluion Thi tudy preented effect of a temporary price dicount offered by upplier on a retailer replenihment policy. An algorithm wa etablihed to determine optimal order police. Numerical example were preented to illutrate olution procedure. Acknowledgement Author greatly appreciate referee for their valuable and helpful uggetion. Notation and Aumption Notation production rate of the upplier. c δ A h market demand rate. unit purchaing price. price dicounted fraction from the upplier ( 0 < δ < ). ordering cot per regular or pecial order. holding cot rate. TCN Change % TCS g

8 YANG et al: TEMORARY RICE ISCOUNT EFFECT WHEN LEA TIME LINKS TO ORER UANTITY 87 b q L L r t r t hortage cot per unit per unit time. inventory level before the pecial order quantity arrive. the length of lead time of the regular order quantity. the length of lead time of the pecial order quantity. the reorder point of regular order policy (in term of inventory level). the reorder time at reorder point of regular order policy. the time at which the upplier offer a temporary price dicount. order quantity under regular order policy. optimal order quantity under regular order policy order quantity at dicount price. optimal order quantity at dicount price. Aumption () eterioration of item i not conidered in the model. () Supplier offer retailer a temporary price dicount at dicount rate δ. Temporary dicount occur at only one intant in time. (3) Inventory i continuouly reviewed and it i replenihed whenever inventory level fall to r under regular order policy. (4) rior to and after temporary price dicount retailer adopt an economic order quantity (EO) policy and optimal order quantity i A ( hc ). (5) Length of lead time depend on order quantity. (6) Shortage are allowed and item can be backordered under pecial order policy. (7) None of the price dicount i paed on to cutomer. Reference Naddor E Inventory Sytem (Wiley New York) Barker R C & Vilcaim N Continuou review price change inventory model roduction & Inventory Mgmt 4 (983) Ardalan A Optimal ordering policie in repone to a ale IIE Tran 0 (988) Terine R J & Schwarzkopf A B Optimal tock replenihment trategie in repone to a temporary price reduction J Buine Logitic 0 (989) Goyal S K A comment on Martin : Note on an EO model with a temporary ale price Int J rod Econ 43 (996) Aull-Hyde R L A backlog inventory model during retricted ale period J Operat Re Soc 47 (996) 9-00B. 7 Bhavin J S EO model for time-dependent deterioration rate with a temporary price dicount Aia-acific J Operat Re (005) ye C Y Chang H J & Wu C H urchae-inventory deciion model for deteriorating item with a temporary ale price Int J Inform & Mgmt Sci 8 (007) Bhaba R S & Mahmood A K Optimal ordering policie in repone to a dicount offer Int J rod Econ 00 (006) Abad L Buyer repone to a temporary price reduction incorporating freight cot Eur J Operat Re 8 (007) Bramon M J The variable lead-time problem in inventory control Operat Re ual 3 (96) 4. Kaplan R S A dynamic inventory model with tochatic leadtime Mgmt Soc 6 (970) Ouyang L Y Chen C K & Chang H C uality improvement etup cot and lead-time reduction in lot ize reorder point model with an imperfect production proce Compu & Operat Re 9 (00) an J C-H Lo M C & Hiao Y C Optimal reorder point inventory model with variable lead time and backorder dicount conideration Eur J Operat Re 58 (004) Lee W C Wub J W & Lei C L Computational algorithmic procedure for optimal inventory policy involving ordering cot reduction and back-order dicount when lead time demand i controllable Appl Math & Compu89 (007)