Stochastic Lot-sizing Problem Research Articles

Replenishment policy based on modified ant colony optimisation and statistical analysis under the pre-order penetration point

In the supply chain, most businesses in the pre-order penetration point (pre-OPP) operate under the forecast-driven mode, so that the decisions regarding inventory are made in accordance with the forecast and replenishment planning. This paper considers the stochastic dynamic lot-sizing problem of the two-phased transportation cost, service level constraint, and cash flow under a non-deterministic demand. This problem includes a nonlinear integer programming sub-problem. Therefore, this paper proposes an optimisation replenishment policy method based on modified ant colony optimisation (ACO) and response surface methodology. The main differences between the modified ACO and the traditional ACO lie in the modified update of pheromone intensity and the dynamic mutation operator. The experimental result shows that when the demand is normal distribution, the proposed approach, successfully finds the stationary point of minimum response. Besides, in the test of the algorithm solution quality, the modified ACO is better than the traditional ACO in all scenarios.

On stochastic lot-sizing problems with random lead times

We give multi-stage stochastic programming formulations for lot-sizing problems where costs, demands and order lead times follow a general discrete-time stochastic process with finite support. We characterize the properties of an optimal solution and give a dynamic programming algorithm, polynomial in the scenario tree size, when orders do not cross in time.

Approximation Algorithms for Stochastic Inventory Control Models

We consider two classical stochastic inventory control models, the periodic-review stochastic inventory control problem and the stochastic lot-sizing problem. The goal is to coordinate a sequence of orders of a single commodity, aiming to supply stochastic demands over a discrete, finite horizon with minimum expected overall ordering, holding, and backlogging costs. In this paper, we address the important problem of finding computationally efficient and provably good inventory control policies for these models in the presence of correlated, nonstationary (time-dependent), and evolving stochastic demands. This problem arises in many domains and has many practical applications in supply chain management. Our approach is based on a new marginal cost accounting scheme for stochastic inventory control models combined with novel cost-balancing techniques. Specifically, in each period, we balance the expected cost of overordering (i.e., costs incurred by excess inventory) against the expected cost of underordering (i.e., costs incurred by not satisfying demand on time). This leads to what we believe to be the first computationally efficient policies with constant worst-case performance guarantees for a general class of important stochastic inventory control models. That is, there exists a constant C such that, for any instance of the problem, the expected cost of the policy is at most C times the expected cost of an optimal policy. In particular, we provide a worst-case guarantee of two for the periodic-review stochastic inventory control problem and a worst-case guarantee of three for the stochastic lot-sizing problem. Our results are valid for all of the currently known approaches in the literature to model correlation and nonstationarity of demands over time.

The Mixing Set with Flows

We consider the mixing set with flows: $s+x_t \geq b_t, x_t \leq y_t {\rm for} 1 \leq t \leq n; s \in \R^1_+, x \in \R^n_+, y \in \Z^n_+.$ It models a “flow version” of the basic mixing set introduced and studied by Gu¨nlu¨k and Pochet [Math. Program., 90 (2001), pp. 429-457], as well as the most simple stochastic lot-sizing problem with recourse. More generally it is a relaxation of certain mixed integer sets that arise in the study of production planning problems. We study the polyhedron defined as the convex hull of the above set. Specifically we provide an inequality description, and we also characterize its vertices and rays.

A Fully Polynomial Time Approximation Scheme for Single-Item Stochastic Lot-Sizing Problems with Discrete Demand

The single-item stochastic lot-sizing problem is to find an inventory replenishment policy in the presence of a stochastic demand under periodic review and finite time horizon. The computational intractability of computing an optimal policy is widely believed and therefore approximation algorithms should be considered. To the best of our knowledge, this is the first work that develops a fully polynomial time approximation scheme for this problem. In other words, we design a tractable polynomial time algorithm that finds a policy that is arbitrarily close in the relative sense to the value of an optimal policy. In addition, we formally prove that finding an optimal policy is intractable in the standard sense.

The Mixing Set with Flows

We consider here the mixing set with flows: s + xt ≥ bt, xt ≤ yt for 1 ≤ t ≤ n; s E IR, x E IR, y E Z. It models the flow version of the basic mixing set introduced and studied by Gunluk and Pochet, as well as the most simple stochastic lot-sizing problem with recourse, and more generally is a relaxation of certain mixed integer sets that arise in the study of production planning problems. We study the polyhedron obtained by convexifying the above set. Specifically we provide a system of inequalities that gives its external description and characterize its vertices and rays.

Approximate solutions for a stochastic lot-sizing problem with partial customer-order information

We address a stochastic single product manufacturing system in a make-to-stock environment with partial knowledge on future demand resulting from customers ordering in advance of their actual needs. The problem consists of determining the optimal size of a production lot to replenish inventory, so that delivery promises are met on time at the expense of minimal average costs. A Markov decision model is formulated for finding the optimal policy. However, since the optimal policy is likely to be too complex in most practical situations, we present approximate strategies for obtaining good production lot sizes. The well-known ( R, S) inventory policy is compared to two rules where production decisions take into account the available information on future customer requirements and the probabilistic characterization of orders yet to be placed. Furthermore, it is shown that the ( s, S) inventory policy is a special case of one of the rules. An extensive numerical study reveals that the newly developed strategies outperform classical inventory policies.

Progressive hedging as a meta-heuristic applied to stochastic lot-sizing

In a great many situations, the data for optimization problems cannot be known with certainty and furthermore the decision process will take place in multiple time stages as the uncertainties are resolved. This gives rise to a need for stochastic programming (SP) methods that create solutions that are hedged against future uncertainty. The progressive hedging algorithm (PHA) of Rockafellar and Wets is a general method for SP. We cast the PHA in a meta-heuristic framework with the sub-problems generated for each scenario solved heuristically. Rather than using an approximate search algorithm for the exact problem as is typically the case in the meta-heuristic literature, we use an algorithm for sub-problems that is exact in its usual context but serves as a heuristic for our meta-heuristic. Computational results reported for stochastic lot-sizing problems demonstrate that the method is effective.

Make-to-order policies for a stochastic lot-sizing problem using overtime

We address a stochastic single item production system in a make-to-order environment where customer orders receive a promised delivery date upon arrival. Capacity is reserved for the production of the item but can be extended by working overtime. The problem consists of determining the optimal size of a production lot, so that delivery promises are met on time at the expense of minimal average costs. These include setup costs, holding costs for orders that are finished before their promised delivery date and penalty costs for orders that are not satisfied on time and are therefore backordered. Also, the use of capacity beyond the regularly available incurs overtime costs. Given that the optimal policy is likely to be too complex in most practical situations, we propose four different strategies for obtaining production lot sizes. Three of the rules are based on the well-known ( s, S) and ( R, S) policies for make-to-stock problems while the fourth strategy is motivated by a rule originally derived for a similar problem with unlimited production capacity. An extensive numerical study indicates the conditions under which each lot-sizing rule shows the best performance.

Production strategies for a stochastic lot-sizing problem with constant capacity

This paper presents a single item capacitated stochastic lot-sizing problem motibated by a Dutch company operating in a Make-To-Order environment. Due to a highly fluctuating and unpredictable demand, it is not possible to keep any finished goods inventory. In response to a customer's order, a fixed delivery date is quoted by the company. The objective is to determine in each period of the planning horizon the optimal size of production lots so that delivery dates are met as closely as possible at the expense of minimal average costs. These include set-up costs, holding costs for orders that are finished before their promised delivery date and penalty costs for orders that are not satisfied on time and are therefore backordered. Given that the optimal production policy is likely to be too complex in this situation, attention is focused on the development of heuristic procedures. In this paper two heuristics are proposed. The first one is an extension of a simple production strategy derived by Dellaert [5] for the uncapacitated version of the problem. The second heuristic is based on the well-known Silver-Meal algorithm for the case of deterministic time-varying demand. Experimental results suggest that the first heuristic gives low average costs especially when the demand variability is low and there are large differences in the cost parameters. The Silver-Meal approach is usually outperformed by the first heuristic in situations where the available production capacity is tight and the demand variability is low.

Heuristic procedures for a stochastic lot-sizing problem in make-to-order manufacturing

We consider a single item, uncapacitated stochastic lot-sizing problem motivated by a Dutch make-to-order company producing steel pipes. Since no finished goods inventory is kept, a delivery date is fixed upon arrival of each order. The objective is to determine the optimal size of production lots so that delivery dates are met as closely as possible with a limited number of set-ups. Orders that are not satisfied on time are backordered and a penalty cost is incurred in those cases. We formulate the problem as a Markov Decision Process and determine the optimal production policy by dynamic programming. Since this approach can only be applied to very small examples, attention is given to the development of three simple lot-sizing rules. The first strategy consists of producing the orders for a fixed numberT of periods whenever the demand for the current period reaches a pre-specified limitx. A simple set of tests is proposed leading to cost improvements in situations where the best combination for the decision variablesx andT deviates from the optimal policy. The second lot-sizing rule is based on the well-known Silver-Meal heuristic for the case of deterministic time-varying demand. A fixed cycle production strategy is also derived. Numerical examples taking into account different demand patterns are provided. The analysis of the results suggests that the first heuristic is particularly suitable for the problem under consideration. Finally, the model is incorporated in the operations control level of the hierarchical production planning system of the Dutch company and assists the management in the evaluation of the quality of the aggregate decisions. A consequence of this feedback mechanism is the modification of the aggregate plans.

Bounds on the Solution of the Lagged Optimal Inventory Equation with No Demand Backlogging and Proportional Costs

These equations depict the following inventory situation. A nonnegative order for a single product is to be placed at the beginning of each of an infinite sequence of equally spaced periods into the future, labeled 0, 1, 2, .. Demands depleting the inventory form a nonnegative sequence of independent identically distributed random variables 40, 1, '2, * . For analytic convenience the distribution is assumed to possess a continuous positive density function 0(4) on [0, oo). An order zi placed at the beginning of period i arrives after a fixed Aperiod lag at the beginning of period i + A, and is relabeled Y(i + ), which explains (4). The pipeline vector qi at the beginning of period i consists of the inventory on hand at that point xi together with the sequence of outstanding orders to arrive in succeeding periods Y(i+ 1), Y(i+ 2)' ... , Y(i+ -1) If demand in a given period exceeds inventory on hand, two extreme cases are considered. In model I, also called the no backlogging or the lost sales case, no customer will wait, so the inventory level is conveniently thought of as being nonnegative; it is recursively generated by (5a). In model II, also called the backlogging case, all customers