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PRESENTERS:AKSHAY BHARDWAJ

KSHITIJ PADHYESRIKANTH CHADA

On the Interactions Between Routing and

Inventory-Management Policies in a One-Warehouse

N-Retailer Distribution System

Leroy B. Schwarz, James E. Ward, Xin ZhaiKrannert Graduate School of Management, Purdue University, West Lafayette, Indiana MANUFACTURING & SERVICEOPERATIONS MANAGEMENTVol. 8, No. 3, Summer 2006, pp. 253–272issn 1523-4614 eissn 1526-5498 06 0803 025

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GOAL

This paper examines the interactions between routing and inventory-management decisions in a two-levelsupply chain consisting of a cross-docking warehouse and N retailers.

The authors’ goal is to determine a combined system inventory-replenishment, routing, and inventory allocationpolicy that minimizes the total expected cost/period of the system over an infinite time horizon.

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INTRODUCTION

Due to increasing competition in retailing, high availability and low-average inventory levels are imperative.

One of the ways suggested by the authors to achieve this is by:

-Dynamic Routing of Inventory-Dynamic Allocation of Inventory.

Thanks to advancements in technology (e.g. RFID at Wal-Mart and dynamic truck routing by Schneider Trucking), the above can be implemented.

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PROCEDUREThe authors demonstrated that the optimal static route

is not the shortest-total-distance (TSP) route, but depends on the variance of customer demands, and, if in-transit inventory-holding costs are charged, also on mean customer demands.

We then examine dynamic-routing policies, i.e., policies that can change the route from one system-replenishment-allocation cycle to another, based on the status of the retailers’ inventories.

Next, the performance of a change-revert heuristic policy is examined. Although its routing decisions are not fully dynamic, but determined and fixed for a given cycle at the time of each system replenishment, simulation tests with N = 2 and N = 6 retailers indicate that its use can substantially reduce system inventory-related costs even if most of the time the chosen route is the optimal static route.

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PROPOSED MODEL

The model examined by the authors consists of a single cross-docking warehouse, N retailers, one un-capacitated

vehicle, and a single item (SKU). This supply chain is centrally managed, using a periodic-review system.

Each retailer i experiences normally distributed customer demand each time period. Demand realizations are independent across time and across retailers. Excess demand is backordered at cost $p/unit-period, and a holding cost of $h/unit-period is charged on system (i.e., on-vehicle or at-retailer) inventory. Transportation times between system sites are fixed and known.

i i

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ASSUMPTIONSThree major assumptions are made to facilitate our Analysis.

• The Allocation Assumption

• The Returns-Without-Penalty-Assumption

• The Last-Period Backorders Assumption

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MODEL

The total cost assigned to a single cycle that starts in period t0, and whose retailers’ allocation cycles have known arbitrary starting and ending periods, is

01

, [ ]1

( )( )o

m N

t i e iit

th h pt

SE

where Et is the system total net inventory, si t are thebackorders at retailer i in period t, and e[i ]is theend period in retailer i’s allocation cycle.

H is the holding cost and p is the penalty for back-orders.

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MODEL

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MODELWhile Y ∗, K∗, µ1

c and σ1c are identical under the two cost

structures,

Retailer-only holding costs: A sufficient condition to visit the closest retailer first is for the closest retailer to also have the largest variance. Otherwise, the choice of the optimal static route depends on a trade-offbetween travel times (i.e., a TSP-oriented parameter) and per-period demand variance (i.e., an inventory managementparameter).System based holding costs: Hence, when holding costs are also charged on the vehicle inventory, the retailers’ average demands affect the choice of the route.

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DYNAMIC ROUTING POLICIES

Dynamic Routing introduces two types of uncertainties, neither of which is present under a fixed routing policy: (1) “UAL uncertainty,” i.e., uncertainty in each retailer’s delivery lead time, and (2) “UPD uncertainty,” i.e., uncertainty in the number of periods of customer demand each retailer’s allocation is to supply.

A management decision to adopt a dynamic-routing policy should be viewed as a tradeoff between reducing expected inventory-related costs, in exchange for increased logistics-related (e.g., transportation) cost.

In a symmetric system for any given inventory-allocation policy, least-inventory first (LIF) is an optimal dynamic routing policy.

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The heuristic’s decision rule is applied before the vehicle leaves the warehouse, and yields one of two decisions: (a) use the default route in the current replenishment cycle or (b) choose another route.

The decision rule then chooses the route with the smallestvalue of C∗, whereC* =Z* +(p+h)SP

Where z* is the single-cycle costSp is the expected shortages of retailer in its last allocation

cycle.

CHANGE REVERT HEURISTIC

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CONCLUSION

The authors used simulation to examine the impact of the major assumptions used to develop the analytical model. The results indicate: that these assumptions are seldom violated; and, second, even when they are, the insights provided by the analytical model—in particular, the model’s estimates of cost/period and the desirability of changing routes—appear to be insensitive to them.

In particular, the simulation results provide evidence that the change revert heuristic provides statistically significant and managerially meaningful savings for multi retailer systems.

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SCOPE FOR RESEARCH

Finally, the authors deliberately leave two questions open for future research:

First is an examination of fully dynamic-routing policy. This form of dynamic routing would decide which retailer to visit next as part of the allocation decision made at every retailer (except the last) in each replenishment cycle.

Second is a closer examination of change-revert routingpolicies. Perhaps the most interesting question here is“What is the best default route?” We arbitrarily chosethe optimal static route to be the default route, but,given a change-revert policy, it is certainly possiblethat a different default route would provide even betterresults.