Peter Berling

Optimal Inventory Policies when Purchase Price and Demand Are Stochastic

In this paper we consider the problem of a firm that faces a stochastic (Poisson) demand and must replenish from a market in which prices fluctuate, such as a commodity market. We describe the price evolution as a continuous stochastic process and we focus on commonly used processes suggested by the financial literature, such as the geometric Brownian motion and the Ornstein-Uhlenbeck process. It is well known that under variable purchase price, a price-dependent base-stock policy is optimal. Using the single-unit decomposition approach, we explicitly characterize the optimal base-stock level using a series of threshold prices. We show that the base-stock level is first increasing and then decreasing in the current purchase price. We provide a procedure for calculating the thresholds, which yields closed-form solutions when price follows a geometric Brownian motion and implicit solutions under the Ornstein-Uhlenbeck price model. In addition, our numerical study shows that the optimal policy performs much better than inventory policies that ignore future price evolution, because it tends to place larger orders when prices are expected to increase.

The Effects of Financial Risks on Inventory Policy

The effect of financial risks on (R, Q) inventory policies is analyzed in a real options framework. Simple adjustments of the usual formulas for R and Q are suggested and tested. Stochastic demand and purchase costs are considered, both with known systematic (business-cycle-related) risk. The systematic risk of stochastic demand has typically a negligible effect on the optimal values of R and Q, although an improvement may be achieved by a simple adjustment of R. The systematic risk of the purchase price, c, has a significant effect on R and Q. The capital holding cost should be estimated as r i¾· c, where r is the sum of the risk-free interest rate, the expected price decrease, and the risk premium associated with the systematic risk of c. For goods quoted on commodity exchanges, r may be estimated directly from the prices on forward contracts. Its size (and sign) varies considerably for different commodities.

The capital cost of holding inventory with stochastically mean-reverting purchase price

Most models of inventory control assume that the per unit purchase price is constant. The capital cost of holding inventory can then be taken into account by adding a fixed interest rate, r, times the purchase price, C, to the out-of pocket holding cost. However, it is not uncommon that the purchase price varies over time. How the capital cost then should be calculated is the focus of the present paper. The paper studies the common single-item inventory model with a fixed set-up cost and assumes that the stochastic purchase price follows the mean-reverting Ornstein–Uhlenbeck process. Methods for computing an adjusted interest rate, r, are suggested along with modifications of well-known heuristics and formulas for lot-sizing. Simulation tests, where the optimal policy has been compared to policies obtained using modified versions of the Silver–Meal method, the Part Period algorithm and the EOQ formula, suggest that r should be estimated as the sum of the unadjusted interest rate and the average expected purchase price decrease, measured over a period between 1/3 and 2/3 of the length of the order cycle.

Multi-Echelon Inventory Control - A Normal Demand Model for Implementation in Practice

This paper presents an approximation model for coordinated control of one-warehouse multiple-retailer inventory systems, where all locations use continuous review (R, nQ) policies. The motivation stems from close collaboration with a supply chain management software company, Syncron International, and one of their customers. A core objective has been to develop an accurate method for determining near-optimal reorder points that can be directly applied to real-life systems. The approach is based on decomposing the complex multi-echelon problem into N + 1 single-echelon problems, using a near-optimal-induced backorder cost at the central warehouse. Important extensions made compared to earlier work include the addition of procedures to adjust for lead-time variability, and for undershooting the reorder point when customers’ order sizes vary. The result is a flexible model that is computationally and conceptually simple enough to be implemented in practice. A numerical study, including real data from the case company, illustrates that the new model outperforms existing methods in the literature. Compared to the current methods used by the case company, it offers significant improvements in both service-level fulfilment and system-wide inventory holding costs. Implementations of the model into the Syncron software are in progress.

A model for heuristic coordination of real life distribution inventory systems with lumpy demand

This paper presents an approximation model for optimizing reorder points in one-warehouse N-retailer inventory systems subject to highly variable lumpy demand. The motivation for this work stems from close cooperation with a supply chain management software company, Syncron International, and one of their customers, a global spare parts provider. The model heuristically coordinates the inventory system using a near optimal induced backorder cost at the central warehouse. This induced backorder cost captures the impact that a reorder point decision at the warehouse has on the retailers’ costs, and decomposes the multi-echelon problem into solving N+1 single-echelon problems. The decomposition framework renders a flexible model that is computationally and conceptually simple enough to be implemented in practice.

Approximation algorithms for optimal purchase/inventory policy when purchase price and demand are stochastic

We consider a purchase/inventory control problem in which the purchase price and demand are stochastic, a common situation encountered by firms that replenish in a foreign currency or from commodity markets. More specifically, we assume that the demand follows a Poisson arrival process and that the log-price evolves according to a general Wiener process. Under these circumstances, the optimal policy is a state dependent base-stock policy that can be described as a series of threshold prices. An iterative procedure for determining the optimal thresholds has been derived earlier but, even for the simplest price process, the solution quickly becomes numerically intractable. To deal with this, we propose an approximation that allows us to derive simple heuristics for finding thresholds that are close to optimal. For certain price processes the heuristics are just a series of closed-form expressions. The computational complexity is reduced significantly, and the numerical study shows that the new heuristics perform considerably better than earlier suggested heuristics.

Green Inventory Management

Managing inventories, and thereby material flows, is of key importance for achieving efficient and sustainable supply chains. Green inventory management is characterized by complementing the traditional economic (cost) focus with environmental (emissions) considerations. In this chapter we identify and discuss key questions and challenges for green inventory management research. We do so by categorizing the costs and emissions of operating an inventory system into those associated with: ordering and transporting items, holding items in stock, and not satisfying customer demand on time. A literature overview illustrates what issues have been addressed so far in this emerging field. We conclude that there is a promising potential for green inventory management practices to reduce both costs and emissions, but much remains to be done. Not least in terms of developing more general green inventory models for practical use.

Dynamic Speed Optimization in Supply Chains with Stochastic Demand

In this paper, we analyze how to continuously adjust the speed in a supply chain with stochastic demand. For each unit e.g., truckload, shipping container in the chain, one must decide at which speed it should be moved downstream, given the state of the system, to minimize total supply chain costs. We decompose the problem into a set of one-dimensional subproblems that can be easily solved and characterize the optimal variable speed policy: under some assumptions, we show that it is optimal to set a speed that is first increasing in the distance to the market, and then decreasing. As a result, at optimality a given unit will experience an accelerating speed and then it will be slowed down, unless a demand occurs, in which case, the speed will be adjusted upward. We finally provide a transportation case study where we estimate the benefits of a variable-speed compared to a fixed-speed policy and show them to be significant both financially and from a CO2-emissions perspective.

Environmental implications of transport contract choice - capacity investment and pricing under volume and capacity contracts

Inspired by the observation that capacity contracts are used by some retailers to increase their transport provider’s investments in green transport solutions, we investigate and compare a service provider’s optimal investment, and its environmental implications under a volume and a capacity contract respectively. We solve the service provider’s investment problem under the assumption that the retailer uses the service to replenish a warehouse with storable goods. We then show that a capacity contract leads to more green transports, but not necessarily a larger investment in green transport solutions. At the same time, the optimal solution involves heavy investment in inventory at the retailer. The investment in inventory is non-decreasing in the cost benefit of the green transports, which may have a significant negative environmental impact. The implication is that a capacity contract will lead to better environmental performance than a volume contract only when the green transports’ cost benefit is within a given interval. Whether the capacity contract is the more profitable option for the service provider within this interval depends on inventory related costs and the relative environmental costs from transportation and inventory. Interestingly, owing to this, regulation that target the price of the conventional vehicles, such as a carbon tax, may lead to both an increase or a decrease in environmental performance.