Competition and Collaboration in a Service Parts Management System

Meriçcan Usta Dr. SeçilSavaşaneril, Dr. Yasemin Serin Industrial Engineering Middle East Technical University (METU) 06531, Ankara-TR Competition and Collaboration in a Service Parts Management System

Outline • Motivation and Relevant Literature • Problem Definition and Modeling • Proposed Solution Algorithm • Numerical Experiments • Ongoing/Anticipated Work

Motivation • Service Centers (SRC) with expensive spare parts • automobile/plane spare parts, machine parts etc. • Centralized • Independent/Decentralized • Hybrid • High profit potential • Difficult to operate • Highly variable demand, high inventory costs

Motivation (cont’d) • Global Competition: Equipment Service Centers (SRC) forced to maintain profitability subject to high service levels • An opportunity to quantitatively analyze its effects: Lateral Transshipment • Satisfy demand from another SRC subject to a comission. • Revenue sharing  collaboration & competition

Relevant Literature • Sheerbrooke (1968) and Muckstadt (1973) • METRIC and MODMETRIC Models • Axsäter (1990) • Random inventory pooling • Kukreja et al. (2001), Grahovac and Chakravarty (2001) • Zhao et al. (2005) • Simple Lateral Transshipment Policies • Zhao et al. (2006) • Characterization of best policies

Research Questions • How can we find an SRCs best response subject to fixed policy of another? • How can we find the equilibrium policy of the system? • To what extent would independent SRCs cooperate? • Subject to different cost structures, how would the equilibrium • policies, • profitabilities • service performances be effected?

Modeled Problem Context • Single echelon, single product, two SRC, memoryless system • Poisson, one item demand arrivals • Exponential, one item production times • SRCs: independent profit maximizers • A dedicated production line for each SRC • SRCs act like a memoryless, single server queue • Policy of one SRC is fixed • We seek to find the best response of another

What is this “Fixed SRC Policy”? • S2,K2,Z2,T2 ; S2≥K2 ≥ Z2 ≥ T2 policy

Modeled Decision Problem • Markov property: Markov Decision Process • State Def’n: Inv./Queue status of the two SRCs, (i,j) • Actions for each state: • Customer Arrival: accept, DoS, send transshipment request • Receive Transshipment Request: accept, reject • New product request, stop production

Modeled Decision Problem (cont’d) • Relevant Revenues: • Sales Revenue, R (per unit) • Relevant Costs: • Commission: r (per unit) • Transportation from neighbor SRC, tr (per unit) • Neighbor SRC gets r , receiver gets R-r-tr • Inv. Holding cost Ch, backorder costCl(per unit-time) • DoS penalty, py (per unit)

Decision Problem Definition • Uniformization/discretization of the process • Lipman (1975). Uniform rate, β, Sum of arrival rates to the SRC system • λ: demand rate, μ: prod. rate • Discount rate, α=0.05 • v*: discounted optimality equation for infinite horizon

Decision Problem Definition (cont’d) • When the operators are expanded, problem expressed as a recursion Ex. • Φ1:Actions associated with customer arrivals to SRC-1 (accept, ask from SRC-2,DoS)

Characterization of Optimal Policy • It can be shown that the SRC-1 has aS1(j) ≥ K1(j) ≥ Z1(j)≥ T1(j) policy • In other words, the optimal policy is S,K,Z,T levels as a function of other SRCs inventory level, j • Level definitions (indifference equations between actions):

Characterization of Optimal Policy • In most cost structures, stairs that descend with j are observed: • We propose to work with inventory level independent S,K,Z,T levels

Algorithm to Find an Equilibrium • Policy Iteration Phase • Starts with any S1,K1,Z1,T1 S2,K2,Z2,T2 policy. • Hence, v(i,j) can be simultaneously solved • Output: S(j)-K(j)-Z(j)-T(j) stairs implied from v(i,j)’s • S,K,Z,T values from most probable j values • Steepest Ascent Phase • Search the whole neighborhood of the policy (+1,0,-1 combinations for all) where S ≥ K ≥ Z ≥ T • Go to the policy with most improvements. • Switch SRCs

Numerical Experiments Parameter Combinations (Total: 750 exp.) • Symmetric • For both SRC’s: • Demand and production timing distributions iid • Cost and revenue structure same

Evaluation: Policy Iteration Phase • Greatly reduces the required # of evaluations, finishes up to 7 (avg. 3.01) iterations. • 345/750 (46%) of the experiments: steepest ascent did not change the solution • Improvement of the Steepest Ascent Phase • Average: 3.62% • Range: 0% - 60% • Holding Cost> 0.1 • Average: 0.10% • Range: 0% - 5.04% • Demand Rate< 0.6 • Average: 0.02% • Range: 0% - 0.83%

Results • Equilibrium Structure • 632/750 (84.2%) Single Eq. • 111/750 (14.8%) Multiple Eq. • 7/750 (1.0%) Cyclic Eq. • Benefit of Transshipment • Average: 3.16% • Range: 0%-16% • Competition dampens the benefit How does Benefit of Transshipment Change with Parameters?

Results (cont’d) • Equilibrium Structure • 632/750 (84.2%) Single Eq. • 111/750 (14.8%) Multiple Eq. • 7/750 (1.0%) Cyclic Eq. • Benefit of Transshipment • Average: 3.16% • Range: 0%-16% • Competition dampens the benefit How does Benefit of Transshipment Change with Parameters?

Results (cont’d) Effect of Transportation Cost on Performance Measures • Inventory increases • Profit decreases • Flow between SRCs decreases • Benefit of pooling decreases • Queue increases • DoS increases

Results (cont’d) Effect of Commission on Performance Measures • Same Trend: • Profit • Flow btw. SRC’s • Benefit of Transshipment • Inventory (increase, then decrease)

Results (cont’d) Effect of Inv. Holding Cost on Performance Measures • Inventory decreases • Profit decreases • Queue increases • DoS customers increases • Flow btw. SRCs and benefit of transshipment?

Ongoing/Anticipated Work • Benefit of transshipment under asymmetric conditions • First results: Since there is net flow, benefit of transshipment is larger • Larger experiment space required • Analyzing the benefits of centralization • Comparison with solution under competition (policy & performance trends) • A strong algorithm for the solution of the stated problem • Variable bounds for the S-T margin • Better methods to guess the S & T levels • Use Experimental Design/ Response Surface Methodologies to better express the effect of parameters

References • S. Axsater, (1990), “Modeling Emergency Lateral Transshipments in Inventory Systems”, Manage Sci, 36, 11, 1329-1338. • M.A. Cohen, C. Cull, H.L. Lee, D. Willen, (2000), “Saturn’s Supply-Chain Innovation: High Value in After-Sales Service”, Sloan Management Review, vol. 41, no. 4. • J. Grahovac and A.Chakravarty (2001), “Sharing and Lateral Transshipment of Inventory in a Supply Chain with Expensive Low Demand Items”, Manage Sci, 47, 4, 579-594. • A. Kukreja, C P. Schmidt and D. M. Miller (2001), “Stocking Decisions for Low-Usage Items in a Multi-location Inventory System, Manage Sci, 47, 10, 1371-1383. • S.A. Lippman, (1975), “Applying a New Device in the Optimization of Exponential Queuing Systems”, Operations Research, vol. 23, no. 4, pp 687-710 • J. Muckstadt (1973), “A Model For A Multi-Item, Multi-Echelon, Multi-Indenture Inventory System”, Manage Sci, 20, 472-481. • C. Sherbrooke (1968), “METRIC: A Multi-Echelon Technique for Recoverable Item Control”, 1968, Oper Res, 16, 122-141. • H.C. Tijms, (1994), “Stochastic Models: An Algorithmic Approach”, John Wiley and Sons, West Sussex • H. Zhao, V.Deshpande , J. K. Ryan (2005) “Inventory Sharing and Rationing in Decentralized Dealer Networks “, Manage Sci, 51, 4, 531-547 • H. Zhao, V. Deshpande, J. K. Ryan (2006), “Emergency Transshipment in Decentralized Dealer Networks When to Send and Accept Transshipment Requests”, Naval Research Logistics, 53, 547-567.

Competition and Collaboration in a Service Parts Management System