A Periodic Review Inventory Model in a Random Environment

We consider a single product, periodic review inventory model with variable cqacity, random yield, and uncertain demand in a random environment. All model parameters and distributions depend on environmental fluctuations. It is assuumed that the environmental process follows a discrete-time Markov chain. The optimd inventory policy to minimize the total discounted expected cost is derived via dynamic programming. For the finite-horizon model, we show that the objective function is quasi-convex and that the structure of the optimal policy is characterized by a single environmental-dependent critical number for the initial inventory level at each period. Expressions for solving the critical number and the optimal planned ordering are obtained. We further show that the solution for the finite-horizon model converges to that of the infinite-horizon model.

Robust Stochastic Lot-Sizing by Means of Histograms

Traditional approaches in inventory control first estimate the demand distribution among a predefined family of distributions based on data fitting of historical demand observations, and then optimize the inventory control using the estimated distributions. These approaches often lead to fragile solutions whenever the preselected family of distributions was inadequate. In this article, we propose a minimax robust model that integrates data fitting and inventory optimization for the single-item multi-period periodic review stochastic lot-sizing problem. In contrast with the standard assumption of given distributions, we assume that histograms are part of the input. The robust model generalizes the Bayesian model, and it can be interpreted as minimizing history-dependent risk measures. We prove that the optimal inventory control policies of the robust model share the same structure as the traditional stochastic dynamic programming counterpart. In particular, we analyze the robust model based on the chi-square goodness-of-fit test. If demand samples are obtained from a known distribution, the robust model converges to the stochastic model with true distribution under generous conditions. Its effectiveness is also validated by numerical experiments.National Science Foundation (U.S.) (Contract CMMI-0758069

Supply Chain and Revenue Management for Online Retailing

This dissertation focuses on optimizing inventory and pricing decisions in the online retail industry. Motivated by the importance of great customer service quality in the online retail marketplace, we investigate service-level-constrained inventory control problems in both static and dynamic settings. The first essay studies multi-period production planning problems (with or without pricing options) under stochastic demand. A joint service-level constraint is enforced to restrict the joint probability of having backorders in any period. We use the Sample Average Approximation (SAA) approach to reformulate both chance-constrained models as mixed-integer linear programs (MILPs). Via computations of diverse instances, we demonstrate the effectiveness of the SAA approach, analyze the solution feasibility and objective bounds, and conduct sensitivity analysis. The approaches can be generalized to a wide variety of production planning problems. The second essay investigates the dynamic versions of the service-level-constrained inventory control problems, in which retailers have the flexibility to adjust their inventory policies in each period. We formulate two periodic-review stochastic inventory models (backlogging model and remanufacturing model) via Dynamic Programs (DP), and establish the optimality of generalized base-stock policies. We also propose 2-approximation algorithms for both models, which is computationally more efficient than the brute-force DP. The core concept developed in our algorithms is called the delayed marginal cost, which is proven effective in dealing with service-level-constrained inventory systems. The third essay is motivated by the exploding use of sales rank information in today's internet-based e-commerce marketplace. The sales rank affects consumers' shopping preference and therefore, is critical for retailers to utilize when making pricing decisions. We study periodic-review dynamic pricing problems in presence of sales rank, in which customers' demand is a function of both prices and sales rank. We propose rank-based pricing models and characterize the structure and monotonicity of optimal pricing policies. Our numerical experiments illustrate the potential of revenue increases when strategic cyclic policy is used.PHDIndustrial & Operations EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/144159/1/ycjiang_1.pd

A NOTE ON A PERIODIC REVIEW INVENTORY MODEL WITH UNCERTAIN DEMAND IN A RANDOM ENVIRONMENT

Abstract. This paper is on the analysis of a single product, periodic review inventory model, where the distributions of demands vary with the state of the environment variable. The state of the environment is assumed to follow a discrete-time Markov chain. The optimal inventory policy to minimize the total discounted expected cost is derived via dynamic programming. For the finite-horizon model, we show that an environmental-dependent base-stock policy is optimal, and derive some characteristics of the optimal policy. Under additional conditions, we further derive the monotonicity of the optimal policy. Introduction The inventory control has long focused on managing certain specific types of probability in the demand for the products. But, on the other hand, consumer's liking becomes variously in the real-life. The demand is fluctuated by the economic climate, weather condition, trend of public opinion, and so forth. Mere including a purely random component in the demand process will be impossible to express such situations. So, in this paper, under the assumption that the environmental process follows a discretetime Markov chain, we model a single product inventory system of which the distributions of demands depend on environmental fluctuations, and discuss the management policy. We further investigate the effect of the environmental fluctuations on the optimal policy. The main advantage of the Markov chain approach is that it provides a national and flexible framework for formulating various changes described above. The effect of a randomly changing environment in inventory model received only limited attention in the earlier paper. Kalymon[12] studies a multiple-period inventory model in which the unit cost of the product is determined by a Markov process, and the distribution of demand in each period depends on the current cost. Feldman[8] models the demand environment as a continuous-time Markov chain. The demand is modulated by a compound Poisson process where the parameters are determined by the state of the environment. But he studies only the steady-state distribution of the inventory position. Song and Zipkin[18] present a continuous-review inventory model where the demand process is a Markov modulated Poisson process, and they derive some basic characteristics of the optimal policy and algorithms for computing the optimal policy. In recent articles,Özekici and Parlar The purpose of this paper is to show that the environmental-dependent base-stock policy is optimal, and that the optimal policy have the monotonicity for review periods by analyzing finite-horizon periodic-review inventory model where the demand distribution depen

Optimizing Inventory for Profitability and Order Fulfillment Improvement

Despite the extensive research on inventory management, few studies have investigated the optimization of inventory classification and control policies for maximizing the net present value of profit and order fulfillment performance. This dissertation aims to fill the gaps, and consists of two main essays. Essay One (Chapter 1) presents a new multi-period optimization model to explicitly address nonstationary demand, arbitrary review periods, and SKU-specific lead times, with the objective of maximizing the net present value of profit. A real-world application and computational experiments show that the optimal dynamic inventory classification and control decisions obtained from the model significantly reduce both safety stock and base stock levels compared to a multi-criteria inventory classification scheme and the traditional ABC approach. Essay Two (Chapter 2) examines two order-based fulfillment performance measures: the order fill rate, defined as the percentage of orders that are completely filled from available inventory; and the average customer-order fill rate, defined as the mean percentage of total units in a customer order that can be filled from on-hand inventory. Novel optimization models are developed to maximize the order fulfillment performance. Computational results indicate that a commonly used item-based measure in general does not adequately indicate order-based performance, and the tradeoffs between profit and order-based measures vary with inventory investment. This research contributes to the existing literature by providing new approaches to optimize inventory classification and control policies with various performance criteria. It also provides practitioners with a viable way to manage inventory with nonstationary demand, general review periods and lead times, and further allows companies to quantity the tradeoffs of different performance measures

Joint Replenishment Problem in Drug Inventory Management of Pharmacies under Stochastic Demand

In today’s fast-paced and competitive market place, organizations need every edge available to them to ensure success in planning and managing inventory of items with demand uncertainty. In such an effort, cost effective methods in determining optimal replenishment policies are paramount. In this paper, a mathematical model is proposed that optimize inventory replenishment policies of a periodic review inventory system under stochastic demand; with particular focus on malaria drugs in Ugandan pharmacies. Adopting a Markov decision process approach, the states of a Markov chain represent possible states of demand for drugs that treat malaria. Using weekly equal intervals, the decisions of whether or not to replenish additional units of drugs were made using discrete time Markov chains and dynamic programming over a finite period planning horizon. Empirical data was collected from two pharmacies in Uganda. The customer transactions of drugs were taken on a weekly basis; where data collected was analyzed and tested to establish the optimal replenishment policy and inventory costs of drugs. Results from the study indicated the existence of an optimal state-dependent replenishment policy and inventory costs of drugs at the respective pharmacies

Retail inventory management with lost sales

The inventory control problem of traditional store-based grocery retailers has several challenging features. Demand for products is stochastic, and is typically lost when no inventory is available on the shelves. As the consumer behavior studies reveal, only a small percentage of customers are willing to wait when confronted with an out-of-stock situation, whereas the remaining majority will either buy a different product, visit another store, or entirely drop their demand. A store orders inventory on a periodic basis, and receives replenishment according to a fixed schedule. The ordered stock is typically delivered before the next ordering moment, which results in lead times shorter than the review period length. Order sizes are often constrained to integer multiples of a fixed batch size, the case packs, generally dictated by the manufacturer. Upon order receipt at the store, the stock is manually stacked on the shelves, to serve customer demand. Shelf space allocation of many products is limited, dictated by marketing constraints. Hence, surplus stock, which does not fit on the regular shelf, is temporarily stored in the store’s backroom, often a small place, poorly organized. The focus of this dissertation is on developing quantitative models and designing solution approaches for managing the inventory of a single item, under periodic review, when some or all of the following characteristics are taken into account: ?? Lost sales. Demand that occurs when no inventory is available is lost, rather than backordered. ?? Fractional lead time. Time between order placement and order delivery is shorter than the review period length. ?? Batch ordering. Order sizes are constrained to integer multiples of a fixed batch size. ?? Limited shelf space. Shelf space allocation is predetermined. The retailer’s inventory is split between the sales floor and the backroom, which is used to temporarily store surplus inventory not accommodated by the regular shelves. We consider optimal, as well as easy-to-understand inventory replenishment policies, where the objective is to minimize the long-run average cost of the system. Two types of costs are primarily recognized in the inventory models developed in this dissertation: ?? inventory related costs: for ordering, for holding products on stock, and penalty costs for not being able to satisfy end-customer demand, and ?? handling related costs: for shelf stacking, and for handling backroom stock. Despite empirical evidence on the dominance of handling costs in the store, remarkably little is reported in the academic literature on how to manage inventory in the presence of handling costs. A reason for this is that formal models of handling operations are still scarce. In this dissertation, we first formalize a model of shelf stacking costs, using insights from an empirical study. Then, we extend the traditional single-item lost-sales periodic-review inventory control model with several realistic dimensions of the replenishment practices of grocery retailers: batch ordering, handling costs, shelf space and backroom operations. The models we consider are too complex to lend themselves to straightforward analytical tractability. As a result, numerical solution methods based on stochastic dynamic programming are proposed in this dissertation, and near-optimal alternative replenishment policies are investigated. Chapter 2 addresses operational concerns regarding the shelf stacking process in grocery retail stores, and the key factors that influence the execution time of this common store operation. Shelf stacking represents the regular store process of manually refilling the shelves with products from new deliveries, which is typically time consuming and costly. We focus on products that are replenished in pre-packed form but presented to the end-customer in individual units. A motion and time study is executed, and the complete shelf stacking process is broken down into several sub-activities. The main time drivers for each activity are identified, relationships are established, tested and validated using real-life data collected at two European grocery retailers. A simple prediction model of the total stacking time per order line is then inferred, in terms of the number of case packs and consumer units. The model can be applied to estimate the workload and potential time savings in the stacking process. Implications of our empirical findings for inventory replenishment decisions are illustrated by a lot-sizing analysis in Chapter 2, and further explored in Chapter 3. Chapter 3 defines a single item stochastic lost sales inventory control model under periodic review, which is designed to handle fractional lead times, batch ordering and handling costs. We study the settings in which replenishment costs reflect shelf stacking costs and have an additive form with fixed and linear components, depending on the number of batches and units in the replenishment order. We explore the structure of optimal policies under the long-run average cost criterion and propose a new policy, referred to as the (s;QjS; nq) policy, which partially captures the optimal policy structure and shows close-to-optimal performance in many settings. In a numerical study, we compare the performance of the policy against the best (s; Q; nq) and (s; S; nq) policies, and demonstrate the relative improvements. Sensitivity analyses illustrate the impact of the different problem parameters, in particular the batch size and the handling cost parameters, on the optimal solutions and associated average costs. Managerial insights into the effect of ignoring handling costs in the optimization of replenishment decisions are also discussed. Chapter 4 extends the retail setting from Chapter 3 to situations in which there is a limited shelf space to display goods on the sales floor, and the retailer uses the store’s backroom to temporarily store surplus stock. As a result, the back stock is regularly transferred from the backroom to the sales floor to satisfy end-customer demand, which results in additional handling costs for the retailer. We investigate the effect of using the backroom on the inventory system performance, where performance is measured with respect to the optimal ordering decisions, and the long-run average cost of ordering, holding, lost-sales and merchandise handling. Two extensions of the inventory model with ample shelf space are proposed in Chapter 4, which include a (i) linear or (ii) fixed cost structure for additional handling operations. In a numerical study, we discuss several qualitative properties of the optimal solutions, illustrate the additional complexities of the second model, and compare the findings with those of the previous chapter. Furthermore, we build several managerial insights into the effect of problem parameters, in particular the shelf space capacity, on the system’s performance. Finally, we quantify the expected cost penalty the retailer may face by ignoring the additional handling costs in the optimization of inventory decisions, and illustrate the trade-off between the different cost components. Chapter 5 studies a variant of the traditional infinite-horizon, periodic-review, singleitem inventory system with random demands and lost sales, where we assume fractional lead times and batch ordering, and allow for ??xed non-negative ordering costs. We present a comparison of four situations: zero vs. positive setup costs, and unit vs. non-unit batch sizes. For all cases, the optimal policy structure is only partially known in general. We show in a numerical study that the optimal policy structure of the most general model is usually more complex than that of the models with positive setup cost, or batch ordering only. Based on the gained insights, we further test the performance of the near-optimal (s;QjS; nq) heuristic policy in the different cases, and demonstrate its effectiveness. Also, well-known inventory control policies of base-stock, or (s; S) type are extended to the case of batch ordering and studied in comparison with the new heuristic under several conditions

Inventory control for a non-stationary demand perishable product: comparing policies and solution methods

This paper summarizes our findings with respect to order policies for an inventory control problem for a perishable product with a maximum fixed shelf life in a periodic review system, where chance constraints play a role. A Stochastic Programming (SP) problem is presented which models a practical production planning problem over a finite horizon. Perishability, non-stationary demand, fixed ordering cost and a service level (chance) constraint make this problem complex. Inventory control handles this type of models with so-called order policies. We compare three different policies: a) production timing is fixed in advance combined with an order up-to level, b) production timing is fixed in advance and the production quantity takes the age distribution into account and c) the decision of the order quantity depends on the age-distribution of the items in stock. Several theoretical properties for the optimal solutions of the policies are presented. In this paper, four different solution approaches from earlier studies are used to derive parameter values for the order policies. For policy a), we use MILP approximations and alternatively the so-called Smoothed Monte Carlo method with sampled demand to optimize values. For policy b), we outline a sample based approach to determine the order quantities. The flexible policy c) is derived by SDP. All policies are compared on feasibility regarding the α-service level, computation time and ease of implementation to support management in the choice for an order policy.National project TIN2015-66680-C2-2-R, in part financed by the European Regional Development Fund (ERDF)