LOSS OF CUSTOMER GOODWILL IN THE SINGLE ITEM LOT-SIZING PROBLEM WITH IMMEDIATE LOST SALES

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1 Istanbul, July 6 th 23 EURO/INFORMS Joint International Meeting 23 LOSS OF CUSTOMER GOODWILL IN THE SINGLE ITEM LOT-SIZING PROBLEM WITH IMMEDIATE LOST SALES Deniz Aksen College of Administrative Sciences and Economics Koç University SarÕyer, ISTANBUL Kemal AltÕnkemer Krannert Graduate School of Management Purdue University West Lafayette, INDIANA Introduction: Lot-sizing Solution to the tradeoff between inventory holding costs and setup costs. Dynamic demand streams or rates bankrupt the nice formulae of EOQ 2 Scommon Daverage and EOQ with backlogging: hcommon + h b 2 S common Daverage hcommon bcommon common common 2 S common Daverage hcommon bcommon hcommon + bcommon When also cost data is changing over time, then lot-sizing techniques become indispensable. 2 1

2 Introduction: Lot-sizing Pioneering work: Wagner-Whitin Model W-W (18) Single-item, deterministic, uncapacitated, finite planning horizon, time-variant demand and setup costs, zero production (procurement) costs. Also time-variant production and inventory holding costs. c t, h t, S t,d t 3 Lot-sizing Literature Alternative Variants and Solution Techniques for the W-W Model without backlogging (w/o backordering) : Faster Myopic Lot-sizing Heuristics Silver and Meal (173) Groff (17 Faster Optimal Algorithms for the W-W model with uniform costs Federgruen and Tzur () : O(T log T ) or O(T ) Wagelmans, Van Hoesel, and Kolen (11) O(T log T ) or O(T ) 4 Forecasting demand beyond the planning horizon Stadtler (2) : rolling schedules 2

3 Lot-sizing Literature The version with backlogging: satisfy demand in a later period of the planning horizon. Zangwill (16, 186) Blackburn and Kunreuther (174) Pochet and Wolsey (188) Webster (18) Choo and Chan () Gupta and Brennan (12) Lot-sizing Literature Capacitated Multi-Item Version of Dynamic Lot-Sizing: Eisenhut (17) Lambrecht and Vanderveken (17) Dixon and Silver (181) Do ramacõ, Panayiotopoulos, and Adam (181) Karni and Roll (182) Thizy and Van Wassenhove (18) Fleischmann (188) KÕrca and Kökten (14) 6 3

4 Lot-sizing Literature Dynamic Lot-Sizing with Lost Sales : demand is not backlogged, but it does not have to be satisfied either. Sandbothe and Thompson (, 13) Loss of demand incurs a unit cost. There are uniform or period-dependent capacity constraints. All costs are time-invariant. p, c, h, S, d t 7 Lot-sizing Literature Excellent Surveys: Maes and Van Wassenhove (188) Multi-item single-level capacitated lot-sizing heuristics: a general review Baker (18) Journal of Operational Research Society 3 (11) 1-4. Lot-sizing procedures and a standard data set. A reconciliation of the literature. Journal of Manufacturing Operations Management Saydam and Evans () A comparative performance analysis of the Wagner-Whitin algorithm and lot-sizing heuristics. Computers & Industrial Engineering 18 (1)

5 Lot-sizing Literature Excellent Surveys: Salomon, Kroon, Kuik, and Van Wassenhove (11) Some extension of the discrete lotsizing and scheduling problem. Management Science 37 (7) Kuik, Salomon and Van Wassenhove (14) Batching decisions: Structure and models. European Journal of Operational Research 7 (2) Solving Dynamic Lot-sizing Models W-W model solved to optimality with a Forward- Recursive Dynamic Programming (DP) algorithm requiring O(T 2 ) CPU time and memory storage. Efficient Implementation by James R. Evans (18) k k 1 k M = S + c d + h d jk j j t t r t= j t= j r= t+ 1 F = minimum F + M, j = argmin F + M * k 1 j k j 1 jk k 1 j k j 1 jk

6 Solving Single-Item Dynamic Lot-sizing Models P Network Representations X 1 X 2 X 3 X T M T I 1 I 2 I 3 I T 1 d 1 L 1 d 2 L 2 d 3 L 3 d T L T d 1 d 2 d 3 d T 11 1 L 2 3 L 1 L 2 L 3 L T T Shortest Path (SP) network and Dijkstra s Algorithm For the W-W model w/o backlogging A concave cost network flow problem For the W-W model w/ backlogging For the W-W model w/ lost sales Modifying the model of Sandbothe Thompson () Aksen, AltÕnkemer, Chand (23) p, c, h, S, d t p t, c t, h t, S t, d t Demand &DSDFLW\FRQVWUDLQWVOLIWHG 14, 1, 8, 6, 4, Spring Summer Fall Winter Demand 3, 11,7 4, Price $1 $21 $12 $18 Season Seasonal demand levels vs. prices $21 $18 $1 $12 $ $6 $3 $ Price 12 6

7 min j, t The AAC model of dynamic lot-sizing Maximize the net profit from realized sales. An efficient algorithm with O(T 2 ) CPU time and memory storage. Exploit the structural properties of an optimal solution. Inspiration from the DP Algorithm of the W-W Model. Gross Marginal Profit is never negative! Lemma 1. L * t X* t = t {1, 2,, T} Zero-Inventory Ordering Policy of W-W W solution Lemma 2. I * t 1 X * t = t {1, 2,, T} Lemma 3. L * t (d t L* t )= t {1, 2,, T} 13 min j, t The AAC model of dynamic lot-sizing Definition 1a. Period t is a regeneration period if (I t = ) and (L t = ). Definition 1b. Period t is a production period if (X t > ). Definition 1c. Period t is a loss period if (I t > ). Definition 1d. Period t is a stockout period if (I t > ) and (I t = ). Definition 1e. Period t is a conservation period if (L t > ) and (I t > ). X t+1 I t+1 > I t+2 > I l 1 > I I k = l > I k 1 > L j+1 L j+2 t+1 t+2 l 1 L l l k 1 k j j+1 j+2 t I j = I j+1 = I j+2 = I t = d d t+2 d l 1 d k 1 d t+1 k 14 7

8 Optimally solving the AAC lot-sizing model Forward Recursive DP Formulation with cumulative best cost of handling definitions Tradeoff MC jt = minimum{ p t, c j + H t 1 H j 1 } /LQNHG/LVW. j = {k j+1, T : c j + H k 1 H j 1 > p k } Minimum Cost of handling periods 1, t with a last production made in period j C j (t)= C j (t 1) + MC jt d t C t (t)= F t 1 + S t + c t d t * { ()}, { ()} F = minimum C t j = argmin C t t j t j j, t j, t 1 Expanding the AAC model: Loss of Customer Goodwill Cost of Losing Demand > Gross Profit of Lost Demand Diminishing impact on the next demand : Goodwill Loss Extends to exactly one period following the current Effective Demand = Original Demand Diminishing Impact e t = d t LG t Diminishing impact in period t depends ONLY on the amount of effective demand PURPOSELY NOT SATISFIED in period (t 1) e t = max{, d t β LU t 1 } Total Lost Demand in period t : (LG t + LU t ) Needs a pretty good reformulation 16 Total Sales in period t : (e t LU t ) = d t (LG t + LU t ) 8

9 The AAC-GWL model: Loss of Customer Goodwill The intentional loss of demand in period t cannot exceed the effective demand of that period. LU t e t A logical constraint: e 1 d 1 A new inventory balance constraint in period t I t 1 X t (e t LU t ) I t 17 Indication of procurement activity in period t X t T r = t d y r t The AAC-GWL model: Loss of Customer Goodwill Lost goodwill effects of different number of loss periods Period Original Demand Effective Demand Satisfied Demand Lost Effective Demand Total Lost Demand May June 1, 1, July 3, 2, 2, 3, August 4, 2,7 2,7 1,2 May June July 3, 3, 3, 3, August 4, 2, 2, 1, 18

10 Solving the AAC-GWL model 13FRPSOHWH problem? An optimal procedure to solve? PROPOSAL Use the optimal procurement / loss schedule of the associated AAC model w/o goodwill considerations. Exhaustively Search and Restore the pseudo-optimality of that solution for the Goodwill Loss (GWL) case. Update the 1 st and 3 rd lemmae as follows (use Lemma 2 as is. ) Lemma 1. LU * t X * t = t {1, 2,, T} 1 Lemma 3. LU * t (e t LU * t )= t {1, 2,, T} Solving the AAC-GWL model Outline of the SEARCH-AND-RESTORATION Procedure From t = 1 towards t = T locate each (block of) loss period(s). Let t 1, t 2 denote such a period. When introduced, the goodwill loss hits the demand in period (t 2 +1) unless e t2 is zero. We either satisfy at least e t2 to prevent goodwill loss in period (t 2 +1) or lose at least LG t2 +1 therein. 2

11 Solving the AAC-GWL model Outline of the SEARCH-AND-RESTORATION Procedure Let periods u and q be the last production period preceding and the first such period following t 1, t 2, respectively. The solution space of handling loss periods t 1, t 2 is primarily characterized by these production periods There are four alternatives of meeting e t2. We choose the cheapest way of meeting e t2 and contrast it with 21 losing LG t2 +1 in period (t 2+1). Solving the AAC-GWL model Outline of the SEARCH-AND-RESTORATION Procedure Periods u and q which replace original production periods u and q. Original productions in u and q are entirely shifted to u and q. An extra production activity in some other period u. X u X q L t1 > L t1+1 > L t2 > L u = L u+1 = u u+1 = t 1 t 1 +1 t 2 q d u d u+1 I u 1 = I u > d t1 d t1+1 d dt2 t2+1 d d q+1 q I q > I q 1 = 22 11

12 Solving the AAC-GWL model Mutually exclusive and exhaustive cases of meeting the effective demand in t 2 Case 1 u = u Case 2 Case 4 t 1 u t u+1 q t 2 2 Case 3 u < u < t 1 23 Solving the AAC-GWL model Mutually exclusive and exhaustive cases of losing the effective demand in t 2 Case 1 q = q Case 3 X q = 24 Case 2 (q+1) q r The first regeneration period following q 12

13 A Numerical Example Data of the sample lot-sizing problem with lost goodwill Period t Demand Production Cost Holding Cost Selling Price Setup Cost 2 A Numerical Example solved β = % Period t Profit Π Optimal Solution w/o Goodwill Loss X t e t $1,68 L t (total) 12 Solution with Goodwill Loss X t e t $1,31 L t (total) Actual Optimal Solution with Goodwill Loss X t e t L t (total) $1,64 13

14 Remarks on AA-SR Heuristic Runs in O(T 2 ) CPU time with O(T 2 ) memory storage. Preprocessing via the O(T 2 ) algorithm AAC Insignificant rise in the CPU time of the AAC algorithm. Though NOT OPTIMAL! How come? Proof by a Counterexample PRESTO! 27 Questions & comments? 28 14