Dynamic inventory management with cash flow constraints

In this article, we consider a classic dynamic inventory control problem of a self-financing retailer who periodically replenishes its stock from a supplier and sells it to the market. The replenishment decisions of the retailer are constrained by cash flow, which is updated periodically following purchasing and sales in each period. Excess demand in each period is lost when insufficient inventory is in stock. The retailer's objective is to maximize its expected terminal wealth at the end of the planning horizon. We characterize the optimal inventory control policy and present a simple algorithm for computing the optimal policies for each period. Conditions are identified under which the optimal control policies are identical across periods. We also present comparative statics results on the optimal control policy. © 2008 Wiley Periodicals, Inc. Naval Research Logistics 2008Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/61323/1/20322_ftp.pd

Log-concavity of compound distributions with applications in operational and actuarial models

We establish that a random sum of independent and identically distributed (i.i.d.) random quantities has a log-concave cumulative distribution function (cdf) if (i) the random number of terms in the sum has a log-concave probability mass function (pmf) and (ii) the distribution of the i.i.d. terms has a non-increasing density function (when continuous) or a non-increasing pmf (when discrete). We illustrate the usefulness of this result using a standard actuarial risk model and a replacement model.We apply this fundamental result to establish that a compound renewal process observed during a random time interval has a log-concave cdf if the observation time interval and the inter-renewal time distribution have log-concave densities, while the compounding distribution has a decreasing density or pmf. We use this second result to establish the optimality of a so-called (s, S) policy for various inventory models with a stock-out cost coefficient of dimension [$/unit], significantly generalizing the conditions for the demand and leadtime processes, in conjunction with the cost structure in these models. We also identify the implications of our results for various algorithmic approaches to compute optimal policy parameters. Copyrigh

Denumerable state semi-Markov decision processes with unbounded costs average cost criterion

AbstractThis paper establishes a rather complete optimality theory for the average cost semi-Markov decision model with a denumerable state space, compact metric action sets and unbounded one-step costs for the case where the underlying Markov chains have a single ergotic set. Under a condition which, roughly speaking, requires the existence of a finite set such that the supremum over all stationary policies of the expected time and the total expected absolute cost incurred until the first return to this set are finite for any starting state, we shall verify the existence of a finite solution to the average costs optimality equation and the existence of an average cost optimal stationary policy

A simple forward algorithm to solve general dynamic lot sizing models with n periods in O(nlogn) or O(n) time

This paper is concerned with the general dynamic lot size model, or (generalized) WagnerWhitin model. Let n denote the number of periods into which the planning horizon is divided. We describe a simple forward algorithm which solves the general model in 0(n log n) time and 0(n) space, as opposed to the well-known shortest path algorithm advocated over the last 30 years with 0 (n 2) time. A linear, i.e., 0(n)-time and space algorithm is obtained for two important special cases: (a) models without speculative motives for carrying stock, i.e., where in each interval of time the per unit order cost increases by less than the cost of carrying a unit in stock; (b) models with nondecreasing setup costs. We also derive conditions for the existence of monotone optimal policies and relate these to known (planning horizon and other) results from the literature. (DYNAMIC LOT SIZING MODELS; DYNAMIC PROGRAMMING; COMPLEXITY) This paper is concerned with the dynamic lot size model, one of the most frequently employed deterministic single-item inventory planning models. This model was introduced by Wagner and Whitin (1958) and is therefore often referred to as the WagnerWhitin model ( W-W model): it specifies a horizon divided into finitely many (say n) periods each with a known demand which must be satisfied. An unlimited amount may be ordered (produced) in each period. The cost structure consists of fixed-plus-linear order (or production) costs and holding costs assumed to be proportional with the endof-the-period inventory levels. All parameters, i.e., demands, setup costs, variable replenishment and holding cost rates, may differ from period to period. Two distinct rationales prevail for maintaining inventories in systems with deterministic demands and unlimited replenishment opportunities: (I) the cycle stock motive: economies of scale in the replenishment costs provide an incentive for order quantities to cover more than a single period's demand; (II) the speculative motive (see Chand and Morton 1986): even in the absence of economies of scale, it may be advantageous to order some future period's demand in the current period, if the future cost of ordering a unit exceeds the cost of ordering this unit now and carrying it until the future period. In this paper we describe a simple algorithm which solves the general dynamic lot size model in 0(n log n) time and 0(n) space, as opposed to the well-known shortest path algorithm advocated over the last 30 years with 0 (n2) time. A linear, i.e., 0 (n)-time and space algorithm is obtained for two important special cases: (a) models without speculative motives for carrying stock, i.e., instances in which in each interval of time, the per unit order cost increases by less than the cost of carrying a unit in stock over this interval (constant variable order cost rates represent a special case of such models; Wagner and Whitin, for example, originally confined themselves to this case)

Optimal capacity in a coordinated supply chain

We consider a supply chain in which a retailer faces a stochastic demand, incurs backorder and inventory holding costs and uses a periodic review system to place orders from a manufacturer. The manufacturer must fill the entire order. The manufacturer incurs costs of overtime and undertime if the order deviates from the planned production capacity. We determine the optimal capacity for the manufacturer in case there is no coordination with the retailer as well as in case there is full coordination with the retailer. When there is no coordination the optimal capacity for the manufacturer is found by solving a newsvendor problem. When there is coordination, we present a dynamic programming formulation and establish that the optimal ordering policy for the retailer is characterized by two parameters. The optimal coordinated capacity for the manufacturer can then be obtained by solving a nonlinear programming problem. We present an efficient exact algorithm and a heuristic algorithm for computing the manufacturer's capacity. We discuss the impact of coordination on the supply chain cost as well as on the manufacturer's capacity. We also identify the situations in which coordination is most beneficial. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/58030/1/20271_ftp.pd

Wholesale price contracts for reliable supply

Firms can enhance the reliability of their supply through process improvement and overproduction. In decentralized supply chains, however, these mitigating actions may be the supplier's responsibility yet are often not contractible. We show that wholesale price contracts, despite their simplicity, can perform well in inducing reliable supply, and we identify when and why they perform well. This could explain the widespread use of wholesale price contracts in business settings with unreliable supply. In particular, we investigate how the performance of wholesale price contracts depends on the interplay between the nature of supply risk and the type of procurement process. Supply risk is classified as random capacity when events such as labor strike disrupt the firm's ability to produce, or as random yield when manufacturing defects result in yield losses. The procurement process is classified as control when the buyer determines the production quantity, or as delegation when instead the supplier does. Analyzing the four possible combinations, we find that for random capacity, irrespective of the procurement process type, contract performance monotonically increases with the supplier's bargaining power; thus, wholesale price contracts perform well when the supplier is powerful. However, this monotonic trend is reversed for random yield with control: in that case, wholesale price contracts perform well when instead the buyer is powerful. For random yield with delegation, wholesale price contracts perform well when either party is powerful