# Optimal lateral transshipment policies in spare parts inventory models

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1 Optimal lateral transshipment policies in spare parts inventory models A N F 2 E 2 E Sandra van Wijk A N F Joint work with Ivo Adan and Geert-Jan van Houtum Beta Conference 2010 Eindhoven, November 3, 2010

2 Introduction 2/12 Spare Parts Inventory System Technically advanced machines: down-times extremely expensive Breakdown: demand for spare part Ready-for-use spare parts are kept on stock: repair-by-replacement strategy Broken parts returned and repaired No back-orders: emergency repair procedure J? F E J F = H J I 4 A F = E H I A N F, A I 2 E / H K F B =? D E A I

3 Introduction (II) 3/12 Two stock points: pooling of inventory F = H J I 2 E A N F F = H J I = J A H = J H = I I D E F 2 E A J I A N F Lateral transshipment costs are small (compared to emergency procedure costs), hence costs can be saved.

4 Introduction (III) 4/12 PhD project: "Creation of Pooling in Queueing and Inventory Systems" E F = H J I 2 E I A H L A H I A N F A N F F = H J I = J A H = J H = I I D E F 2 E A J I 2 E A N F 2-2 I A H L A H I A N F

5 Lateral Transshipment Problem /12 Two location lateral transshipment problem How to route the demands? Directly from stock; Via lateral transshipment (penalty costs P LTi ); Via emergency procedure (penalty costs P E Pi ). Characterize the optimal lateral transshipment policy. Derive conditions under which simple policies are optimal.

6 Lateral Transshipment Problem (II) 6/12 Simple policies Complete pooling (always hand out parts in case of demands, LTs) Optimal if: { PLT1 + λ 2 λ 2 +µ P E P 2 P E P1, P LT2 + λ 1 λ 1 +µ P E P 1 P E P2. Hold back policy (always hand out parts in case of demands, hold back parts in case of LTs: hold back levels T 1, T 2 ) Optimal if: ) P E P2 P LT2 + (1 + µ λ2 ) P E P1 P LT1 + (1 + µ λ1 P E P1, P E P2.

7 Lateral Transshipment Problem (III) 7/12 Symmetric Parameters Hold back policy is optimal Complete pooling optimal if P LT P E P µ λ/µ. P LT P EP T=2 T=4 T=3 T= (no pooling) 0.2 T=1 (complete pooling) ΛΜ

8 N N Lateral Transshipment Problem (IV) 8/12 Model and proof.. Markov Decision Problem (MDP) states (stock levels): (x 1, x 2 ) Dynamic Programming value function is multi-modular (provided equal repair rates) N 6, 1, A = J? = J E 6, 1, E H A? J O B H I J? = J A H = J H = I I D E F A J - A H C A? O F H? K H A N - 2 N B E N - 2 N N B E N - 2 6, 1 N

9 Lateral Transshipment Problem (V) 9/12 Model extensions: Consumables (holding costs) LTs in one direction Asymmetric repair rates Limited repair capacity More than two stock points?? Approximation algorithm Related model

10 Quick response warehouse 10/12 Related problem, same techniques: Inventory model with a quick response warehouse? = M = H A D K I A I 3 K E? H A I F I A M = H A D K I A

11 Quick response warehouse (II) 11/12 Characterized optimal policy. Conditions under which simple policies are optimal: Always quick response at local warehouse if: J k=1 λ k (P E P k P Q R k ( ) P0 E P µ 0 + J λ k ). k=1 Always hand out a part to own demand stream at QR warehouse if: J ( J λ 0 P0 E P + λ k (Pk E P P Q R k ) (Pj E P P Q R j ) µ 0 + λ k ). k=1 k=0

12 Summary 12/12 Summary Two location lateral transshipment model Quick response warehouse model optimal policy structures & conditions simple policies A.C.C. van Wijk, I.J.B.F. Adan and G.-J. van Houtum, Optimal Lateral Transshipment Policy for a Two Location Inventory Problem (Eurandom report # ). Approximate Evaluation of Multi-Location Inventory Models with Lateral Transshipments and Hold Back Levels (in preperation). Optimal Policy for a Multi-location Inventory System with a Quick Response Warehouse (in preperation).

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