Dynamic knapsack sets and capacitated lot-sizing

 A dynamic knapsack set is a natural generalization of the 0-1 knapsack set with a continuous variable studied recently. For dynamic knapsack sets a large family of facet-defining inequalities, called dynamic knapsack inequalities, are derived by fixing variables to one and then lifting. Surprisingly such inequalities have the simultaneous lifting property, and for small instances provide a significant proportion of all the facet-defining inequalities. We then consider single-item capacitated lot-sizing problems, and propose the joint study of three related sets. The first models the discrete lot-sizing problem, the second the continuous lot-sizing problem with Wagner-Whitin costs, and the third the continuous lot-sizing problem with arbitrary costs. The first set that arises is precisely a dynamic knapsack set, the second an intersection of dynamic knapsack sets, and the unrestricted problem can be viewed as both a relaxation and a restriction of the second. It follows that the dynamic knapsack inequalities and their generalizations provide strong valid inequalities for all three sets. Some limited computation results are reported as an initial test of the effectiveness of these inequalities on capacitated lot-sizing problems. Received: March 28, 2001 / Accepted: November 9, 2001 Published online: September 27, 2002 RID="★" ID="★" Research carried out with financial support of the project TMR-DONET nr. ERB FMRX–CT98–0202 of the European Union. Present address: Electrabel, Louvain-la-Neuve, B-1348 Belgium. Present address: Electrabel, Louvain-la-Neuve, B-1348 Belgium. Key words. knapsack sets – valid inequalities – simultaneous lifting – lot-sizing – Wagner-Whitin costs     

Lower bound on size of branch-and-bound trees for solving lot-sizing problem

We show that there exists a family of instances of the lot-sizing problem, such that any branch-and-bound tree that solves them requires an exponential number of nodes, even in the case when the branchings are performed on general split disjunctions. This result is of interest since there exists dynamic programming algorithm that solves lot-sizing in polynomial-time. To the best of our knowledge, this is the first study that shows that dynamic programming can be exponentially faster than branch-and-bound.     

Sequential-analysis Based Randomized-regret-methods for Lot-sizing and Scheduling

Lot-sizing and scheduling comprises activities that have to be done repeatedly within MRP-systems. We consider the proportional multi-item, capacitated, dynamic lot-sizing and scheduling problem that is more general than the discrete lot-sizing and scheduling problem, as well as the continuous set-up lot-sizing problem. A greedy randomized algorithm with regret-based biased sampling is presented. We partition the parameter space of the stochastic algorithm and choose subspaces via sequential analysis based on hypothesis testing. The new methods provided in this paper, i.e. the randomized-regret-based backward algorithm, as well as the controlled search via sequential analysis, have three important properties: they are simple, effective and rather general. Computational results are also presented.     

Dynamic programming approximation algorithms for the capacitated lot-sizing problem

This paper provides a new idea for approximating the inventory cost function to be used in a truncated dynamic program for solving the capacitated lot-sizing problem. The proposed method combines dynamic programming with regression, data fitting, and approximation techniques to estimate the inventory cost function at each stage of the dynamic program. The effectiveness of the proposed method is analyzed on various types of the capacitated lot-sizing problem instances with different cost and capacity characteristics. Computational results show that approximation approaches could significantly decrease the computational time required by the dynamic program and the integer program for solving different types of the capacitated lot-sizing problem instances. Furthermore, in most cases, the proposed approximate dynamic programming approaches can accurately capture the optimal solution of the problem with consistent computational performance over different instances.     

An FPTAS for a single-item capacitated economic lot-sizing problem with monotone cost structure

We present a fully polynomial time approximation scheme (FPTAS) for a capacitated economic lot-sizing problem with a monotone cost structure. An FPTAS delivers a solution with a given relative error ɛ in time polynomial in the problem size and in 1/ɛ. Such a scheme was developed by van Hoesel and Wagelmans [8] for a capacitated economic lot-sizing problem with monotone concave (convex) production and backlogging cost functions. We omit concavity and convexity restrictions. Furthermore, we take advantage of a straightforward dynamic programming algorithm applied to a rounded problem.     

Stochastic dynamic lot-sizing problem using bi-level programming base on artificial intelligence techniques

Simulation is generally used to study non-deterministic problems in industry. When a simulation process finds the solution to an NP-hard problem, its efficiency is lowered, and computational costs increase. This paper proposes a stochastic dynamic lot-sizing problem with asymmetric deteriorating commodity, in which the optimal unit cost of material and unit holding cost would be determined. This problem covers a sub-problem of replenishment planning, which is NP-hard in the computational complexity theory. Therefore, this paper applies a decision system, based on an artificial neural network (ANN) and modified ant colony optimization (ACO) to solve this stochastic dynamic lot-sizing problem. In the methodology, ANN is used to learn the simulation results, followed by the application of a real-valued modified ACO algorithm to find the optimal decision variables. The test results show that the intelligent system is applicable to the proposed problem, and its performance is better than response surface methodology.     

Multi-period lot-sizing with supplier selection: Structural results,complexity and algorithms

We consider a multi-period lot-sizing problem with multiple products and multiple suppliers. Demand is deterministic and time-varying. The objective is to determine order quantities to minimize the total cost over a finite planning horizon. This problem is strongly NP-hard. For a special case, we extend the classical zero-inventory-ordering principle and solve it by dynamic programming. Based on this new extension, we also develop a heuristic algorithm for the general problem and computationally show that it works well.     

Multiple lot-sizing decisions in a two-stage production with an interrupted geometric yield and non-rigid demand

In a production system with random yield, it may be more cost effective to release lots multiple times towards fulfilling a customer order. Such a decision, called the multiple lot-sizing problem, has been investigated in various contexts. This paper proposes an efficient algorithm for solving a new multiple lot-sizing problem defined in the context of a two-stage production system with non-rigid demand when its process yields are governed by interrupted geometric distributions. We formulate this problem as a dynamic program (DP) and develop lemmas to solve it. However, solving such a DP may be computationally extensive, particularly for large-scale cases with a high yield. Therefore, this study proposes an efficient algorithm for resolving computational issues. This algorithm is designed to reduce the DP network into a much simpler algorithm by combining a group of DP branches into a single one. Extensive experiments were carried out. Results indicate that the proposed reduction algorithm is quite helpful for practitioners dealing with large-scale cases characterized by high-yield.     

Outbound shipment mode considerations for integrated inventory and delivery lot-sizing decisions

We present a two-echelon dynamic lot-sizing model with two outbound delivery modes where one mode has a fixed set-up cost structure while the other has a container-based cost structure. Studying the optimality properties of the problem, we provide a polynomial solution algorithm based on a dynamic programming approach.     

The single-item lot-sizing problem with two production modes,inventory bounds,and periodic carbon emissions capacity

We consider the single-item lot-sizing problem with inventory bounds under a carbon emissions constraint with two options for producing items: regular or green. We wish to find the optimal production plan so that the total carbon emissions from production cannot exceed the carbon emissions capacity in each period. Extending a problem without fixed carbon emissions and inventory bounds, we show that the extended problem is polynomially solvable by a dynamic programming algorithm.     

Single item lot-sizing problems with backlogging on a single machine at a finite production rate

In this paper we consider a single item lot-sizing problem with backlogging on a single machine at a finite production rate. The objective is to minimize the total cost of setup, stockholding and backlogging to satisfy a sequence of discrete demands. Both varying demands over a finite planning horizon and fixed demands at regular intervals over an infinite planning horizon are considered. We have characterized the structure of an optimal production schedule for both cases. As a consequence of this characterization, a dynamic programming algorithm is proposed for the computation of an optimal production schedule for the varying demands case and a simpler one for the fixed demands case.     

SOS: A Quantile Estimation Procedure for Dynamic Lot-sizing Problems

This paper reviews some of the current approaches available for computing the demand quantiles required to plan the procurement of items with stochastic non-stationary demands. The paper first describes the stochastic single-item lot-sizing problem considered and then presents a practical solution approach based on a dynamic lot-sizing model. Three methods available to compute demand quantiles are then reviewed and a new procedure based on smoothed order statistics (SOS) is proposed. Finally, the behaviour of these estimation methods, when used to solve single-item lot-sizing problems with non-stationary stochastic demands, is studied by simulation.     

Approximation algorithms for deterministic continuous-review inventory lot-sizing problems with time-varying demand

This work deals with the continuous time lot-sizing inventory problem when demand and costs are time-dependent. We adapt a cost balancing technique developed for the periodic-review version of our problem to the continuous-review framework. We prove that the solution obtained costs at most twice the cost of an optimal solution. We study the numerical complexity of the algorithm and generalize the policy to several important extensions while preserving its performance guarantee of two. Finally, we propose a modified version of our algorithm for the lot-sizing model with some restricted settings that improves the worst-case bound.     

Single item lot-sizing with non-decreasing capacities

We consider the single item lot-sizing problem with capacities that are non-decreasing over time. When the cost function is (i) non-speculative or Wagner–Whitin (for instance, constant unit production costs and non-negative unit holding costs) and (ii) the production set-up costs are non-increasing over time, it is known that the minimum cost lot-sizing problem is polynomially solvable using dynamic programming. When the capacities are non-decreasing, we derive a compact mixed integer programming reformulation whose linear programming relaxation solves the lot-sizing problem to optimality when the objective function satisfies (i) and (ii). The formulation is based on mixing set relaxations and reduces to the (known) convex hull of solutions when the capacities are constant over time. We illustrate the use and potential effectiveness of this improved LP formulation on a few test instances, including instances with and without Wagner–Whitin costs, and with both non-decreasing and arbitrary capacities over time. This work was partly carried out within the framework of ADONET, a European network in Algorithmic Discrete Optimization, contract no. MRTN-CT-2003-504438. This text presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister’s Office, Science Policy Programming. The scientific responsibility is assumed by the authors.     

Exact solution approaches for bilevel lot-sizing

In this paper we propose exact solution methods for a bilevel uncapacitated lot-sizing problem with backlogs. This is an extension of the classical uncapacitated lot-sizing problem with backlogs, in which two autonomous and self-interested decision makers constitute a two-echelon supply chain. The leader buys items from the follower in order to meet external demand at lowest cost. The follower also tries to minimize its costs. Both parties may backlog. We study the leader’s problem, i.e., how to determine supply requests over time to minimize its costs in view of the possible actions of the follower. We develop two mixed-integer linear programming reformulations, as well as cutting planes to cut off feasible, but suboptimal solutions. We compare the reformulations on a series of benchmark instances.     

An integrated production and preventive maintenance planning model

We are given a set of items that must be produced in lots on a capacitated production system throughout a specified finite planning horizon. We assume that the production system is subject to random failures, and that any maintenance action carried out on the system, in a period, reduces the system’s available production capacity during that period. The objective is to find an integrated lot-sizing and preventive maintenance strategy of the system that satisfies the demand for all items over the entire horizon without backlogging, and which minimizes the expected sum of production and maintenance costs. We show how this problem can be formulated and solved as a multi-item capacitated lot-sizing problem on a system that is periodically renewed and minimally repaired at failure. We also provide an illustrative example that shows the steps to obtain an optimal integrated production and maintenance strategy.     

Single item lot-sizing problem for a warm/cold process with immediate lost sales

We consider the dynamic lot-sizing problem with finite capacity and possible lost sales for a process that could be kept warm at a unit variable cost for the next period t + 1 only if more than a threshold value Q_t has been produced and would be cold, otherwise. Production with a cold process incurs a fixed positive setup cost, K_t and setup time, S_t, which may be positive. Setup costs and times for a warm process are negligible. We develop a dynamic programming formulation of the problem, establish theoretical results on the structure of the optimal production plan in the presence of zero and positive setup times with Wagner–Whitin-type cost structures. We also show that the solution to the dynamic lot-sizing problem with lost sales are generated from the full commitment production series improved via lost sales decisions in the presence of a warm/cold process.     

Profit maximization in simultaneous lot-sizing and scheduling problem

This paper extends the simultaneous lot-sizing and scheduling problem, to include demand choice flexibility. The basic assumption in most research about lot-sizing and scheduling problems is that all the demands should be satisfied. However, in a business with a goal of maximizing profit, meeting all demands may not be an optimum decision. In the profit maximization simultaneous lot-sizing and scheduling problem with demand choice flexibility, the accepted demand in each period, lot-sizing and scheduling are three problems which are considered simultaneously. In other words the decisions pertaining to mid-term planning and short-term planning are considered as one problem and not hierarchically. According to this assumption, the objective function of traditional models changes from minimizing costs to maximizing profits.