Power-of-Two Policies for Single- Warehouse Multi-Retailer Inventory Systems with Order Frequency Discounts

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1 Power-of-wo Polces for Sngle- Warehouse Mult-Retaler Inventory Systems wth Order Frequency Dscounts José A. Ventura Pennsylvana State Unversty (USA) Yale. Herer echnon Israel Insttute of echnology (Israel) Bran Q. Reksts Insttute for Defense Analyss (USA)

2 Overvew Introducton Seral wo-echelon System Optmal soluton Motvaton for Order Frequency Dscounts Sngle-Warehouse Mult-Retaler Inventory System All-unt order frequency dscount 94%-effectve polcy for a varable base perod Incremental order frequency dscount 94%-effectve polcy for a fxed base perod 98%-effectve polcy for a varable base perod

3 Introducton A supply chan s a network of facltes that procure raw materals, transform them nto ntermedate goods and then fnal products, and delver the products to customers through a dstrbuton system. Multechelon nventory systems are used to manage the nventores of a supply chan, whch spans procurement, manufacturng, and dstrbuton. Key objectves of a supply chan are to mnmze the cost assocated wth the entre multechelon nventory system as well as to wn the compettve battle n brngng the product to the customers as promptly as possble. Suppler 1 Mfg. 1 Assembly Faclty 1 Retaler 1 Customer 1 Suppler Mfg. DC Retaler Customer Suppler 3 Mfg. 3 Assembly Faclty Retaler 3 Customer 3

4 Seral wo-echelon System Centralzed control Sngle product. Suppler Manufacturer Dstrbuton Center Retaler Assumptons: Installaton 1 Installaton he assumptons of the basc EOQ model hold at nstallaton, where d = constant demand rate, Q = order quantty provded by nstallaton 1 when the nventory level drops to zero. At nstallaton 1, Q 1 = order quantty provded by suppler placed n tme to replensh the nventory at nstallaton 1when the nventory level s zero. Relevant costs at nstallatons 1 and : K = setup cost, h () = unt holdng cost, = 1,. Unts ncrease n value from nstallaton 1 to nstallaton. So, h (1) < h (). Lead tmes are assumed to be zero (or constant).

5 Non-synchronzed nventory levels (Q 1 =4, Q =3) Inventory level at Installaton 1 4 Q me - Inventory level at Installaton Q me

6 Synchronzed nventory levels (Q 1 = 3 Q ) Inventory level at Installaton 1 Q 1 Echelon nventory (system nventory), tem 1 Installaton nventory, tem 1 Q 1 -Q Q 1 -Q me Inventory level at Installaton Installaton nventory = Echelon nventory, tem Q me

7 Propertes of an Optmal Polcy Nested If nstallaton 1 orders, nstallaton also orders,.e., Q 1 = n Q, where n s a fxed postve nteger. Zero-Inventory Orderng Orders placed only when nventory s zero. Statonary Constant order ntervals and order quanttes.

8 Varable Cost per me Unt Varable cost per tme unt at each nstallaton: d K1 (1) Q1 Q1 d K1 n 1 h Q1 Z1 = + h = + Q1 n Q1 n () d K h Q Z = + Q Snce Q 1 = n Q, K1 d Q Z = Z1 + Z = + K + [( n 1) h1 + h] n Q Defne h 1 = h (1) : echelon unt holdng cost for nstallaton 1, h = h () h (1) : echelon unt holdng cost for nstallaton. d K h Q K1 d Q Z Z K + ( n h1 + h ) 1 1 Q = = + = + = = n Q (1)

9 Echelon Holdng Cost h (1) Stage 1 Inventory h 1 System Inventory h () h Stage Inventory

10 Optmal Polcy Order quanttes: K d 1 + K n Q* = n h + h Varable cost per tme unt: Multplcaton factor: Let K1 h n ' = K h 1 K Z * = d + n If n < 1, n* = 1; f n s nteger, n* = n. ( n h ) 1 + K 1 h 1 n ' n' + 1 If, n* = n' ; otherwse, n* = n' + 1. n' n'

11 Inventory Control n Supply Chans Complex nventory structures wth multple stages Dffcult to coordnate nventory levels Power-of-two polces: ractable for networks Effcent Easy to mplement We develop power-of-two nventory polces for order frequency dscounts

12 Motvaton for Order Frequency Dscounts Optmal polces may be non-nested and nonstatonary (order quanttes). Munson and Rosenblatt (1998) survey: 7% companes offer quantty dscounts to decrease order frequency. Reasons to reduce order frequency: Longer producton runs. Reduced transportaton costs. Fewer manufacturng setups.

13 Lterature Revew R. Roundy. 98%-Effectve Integer-Rato Lot-Szng for One-Warehouse Mult- Retaler Systems, Management Scence 31 (1985) A. Federgruen and Y. Zheng Optmal Power-of-wo Replenshment Strateges n Capactated General Producton/Dstrbuton Networks, Management Scence D. Sun. Exstence and Propertes of Optmal Producton and Inventory Polces, Mathematcs of Operatons Research, to appear. L. Lu and Y. Qu. Worst-case Performance of a Power-of-wo Polcy for the Quantty Dscount Model, Journal of the Operatons Research Socety 45 (1994) F. Chen, A. Federgruen, and Y.-S. Zheng. Near-Optmal Prcng and Replenshment Strateges for a Retal/Dstrbuton System, Operatons Research 49 (1)

14 One-Warehouse Mult-Retaler System Centralzed control. Sngle product. Warehouse receves order frequency dscounts. Retaler 1 (Faclty 1) External Suppler Warehouse (Faclty ) Retaler (Faclty ) Retaler 3 (Faclty 3)

15 Assumptons An EOQsh envronment: Contnuous constant demand No backloggng Neglgble lead tmes Intal nventory s zero. Unt holdng costs are non-decreasng between stages. Order frequency dscounts: All-unt Incremental Power-of-two polcy Order ntervals are a power-of-two of the base plannng perod.

16 wo Retaler Example Warehouse Retaler 1 Retaler me

17 Another wo Retaler Example Warehouse Retaler 1 me Retaler

18 Notaton B K h d D = Order nterval for faclty, =,, n = Base plannng perod = Setup cost for faclty, =,, n = Echelon holdng cost for faclty, =,,n = Demand rate at faclty, = 1,, n = otal demand rate of all facltes, = n d = 1

19 All-Unt Dscount at Warehouse β j = j th breakpont n the tme doman for order frequency dscount, j = 1,, m. c j = per unt cost for all tems, for β j < β j+1. C U ( ) = Average purchasng cost for a warehouse order nterval = c j D, for β j < β j+1.

20 All-Unt Order Frequency Dscount Purchasng Cost ($) β o β 1 β β 3 β 4 Order Interval

21 Average Inventory Holdng Cost If, the product s not held n nventory at the warehouse. In the other case wth <, the nventory at the warehouse replenshes the orders at the retaler every tme unts. Average holdng cost for the system: N \{} h d + h d max{, }

22 Power-of-wo Formulaton Mn Z U n K K h d hd max{, } U ( ) = + + C ( ) + + = 1 subject to B = for =,..., n,, B nteger for =,..., n, where C U ( ) = c j D, for β j < β. j+1

23 Lower Bound Formulaton Relaxed problem R U : Mn subject to Z U (), for N. U hs formulaton s decomposed nto several formulatons, R j, j M, for specfc ranges of the warehouse order nterval: Mn Z ju () = K + N \{} K + h d + h d max{, } + c j D subject to β j β j+1, for N.

24 Solvng Problem he costs ncurred at the retalers may be separated as Roundy (1985) has shown that the value, b ( ), that mnmzes z (, ) for a gven s where and. U R j }, max{ ), ( d h d h K z + + = < < + = = >. f / /, f / ), f ) / ( ), ( nf ) ( h h K d h (h d / K h h K d z b τ τ τ τ h d K / = τ ) ( / h h d K + = τ

25 Solvng Problem U R j (cont.) Note that the functonal form of the value of b ( ) changes as crosses ether and, so these values wll be termed the thresholds. he objectve functon of wthout the constant purchasng cost c j D can be rewrtten as follows: K B( ) = b ( ) Accordng to Roundy (1985), B( ) s strctly convex and goes to as approaches and. hs mples that B( ) acheves a unque mnmum for some fnte postve value of. Roundy s algorthm between each set of consecutve breakponts can be appled to solve the relaxed problem. + N

26 Relaxed Problem B( ) + C U ( ) *

27 Power-of-wo Polcy Solve the relaxed problem: Lower bound on optmal cost. May not be feasble. Base plannng perod s B = *. Fnd 1 such that * * B, for =,, n. Set = for =,, n. B Power-of-two polcy s feasble. Cost wthn 6% of the lower bound.

28 Average Holdng and Setup Cost f( ) f * ( *) = f ( *) = f * * * Order Interval for Retaler ( )

29 β j c j Incremental Order Frequency Notaton = j th breakpont n the tme doman = margnal cost for addtonal unts for β j < β j+1 C j ( ) = c j D( β j ) + C j 1 (β j ) when β j < β j+1 = total purchasng cost C( ) = C j ( ) when β j < β j+1

30 Incremental Order Frequency Dscounts Purchasng Cost ($) β o β 1 β β 3 β 4 Order Interval

31 Lower Bound Proof We defne a relaxed problem R j for each functon C j ( ). he doman of each R j s relaxed from β j < β j+1 to >. A lower bound on the cost of any feasble polcy s the mnmum cost polcy among all R j. hs polcy wll be wthn the approprate range for C j ( ).

32 Lower Bound for Incremental Case B( ) + C ( ) R R1 R

33 Power-of-wo Polces 94%-effectve polcy Consder relaxed problem R j correspondng to the optmal warehouse order nterval Compute Roundy s fxed base perod power-oftwo polcy for R j Effectveness Order nterval at warehouse may be outsde doman of R j Cost for the actual objectve wll stll be 94%-effectve 98%-effectve polcy A smlar result s obtaned by applyng Roundy s varable base perod algorthm to R j

34 Concludng Remarks We have analyzed an nventory system wth order frequency dscounts Effcent Power-of-wo polces have been developed for All-unt order frequency dscounts Incremental order frequency dscounts Addtonal result n the paper: Extenson to purchasng prce dependent holdng costs

35 Research Productvty Journal Artcles: B.Q. Reksts, J.A. Ventura, Y.. Herer, and D. Sun. Worst-Case Performance of Power-of- wo Polces for Seral Inventory Systems wth Incremental Quantty Dscounts. Naval Research Logstcs, to appear. B.Q. Reksts, J.A. Ventura, Y.. Herer, and D. Sun. Power-of-wo Polces for Sngle- Warehouse Mult-Retaler Inventory Systems wth Order Frequency Dscounts. Computers and Operatons Research, submtted. Conference Presentatons/Proceedngs : J.A. Ventura, B.Q. Reksts, and Y.. Herer. An Effectve Inventory Polcy wth Order Frequency Dscounts. Proceedngs of the Annual Industral Engneerng Research Conference, Orlando, FL, May 6. Y.. Herer, J.A. Ventura, and B.Q. Reksts. Power-of-wo Polces for Sngle-Warehouse Mult-Retaler Inventory Systems wth Order Frequency Dscounts. Proceedngs of the Fourteenth Israel Industral Engneerng and Management Conference, el Avv, Israel, March 6. Addtonal Conference Presentatons/Proceedngs: Annual INFORMS (Operatons Research) Meetng, Pttsburgh, PA, November 6. Eleventh MSOM Conference, INFORMS Socety on Manufacturng and Servce Operatons Management, Atlanta, GA, June 6. Annual INFORMS (Operatons Research) Meetng, Atlanta, GA, November 3. EURO/INFORMS Jont Internatonal Meetng, Istanbul, urkey, July 3. Awards: Best Paper Award, Logstcs and Inventory, Annual Industral Engneerng Research Conference, Orlando, FL, May 6.

36 hank You Questons Comments

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