Uncapacitated Lot-sizing Problem Research Articles

The single item uncapacitated lot-sizing problem with time-dependent batch sizes: NP-hard and polynomial cases

This paper considers the uncapacitated lot sizing problem with batch delivery, focusing on the general case of time-dependent batch sizes. We study the complexity of the problem, depending on the other cost parameters, namely the setup cost, the fixed cost per batch, the unit procurement cost and the unit holding cost. We establish that if any one of the cost parameters is allowed to be time-dependent, the problem is NP-hard. On the contrary, if all the cost parameters are stationary, and assuming no unit holding cost, we show that the problem is polynomially solvable in time O(T3), where T denotes the number of periods of the horizon. We also show that, in the case of divisible batch sizes, the problem with time varying setup costs, a stationary fixed cost per batch and no unit procurement nor holding cost can be solved in time O(T3 logT).

Lot sizing with carbon emission constraints

This paper introduces new environmental constraints, namely carbon emission constraints, in multi-sourcing lot-sizing problems. These constraints aim at limiting the carbon emission per unit of product supplied with different modes. A mode corresponds to the combination of a production facility and a transportation mode and is characterized by its economical costs and its unitary carbon emission. Four types of constraints are proposed and analyzed in the single-item uncapacitated lot-sizing problem. The periodic case is shown to be polynomially solvable, while the cumulative, global and rolling cases are NP-hard. Perspectives to extend this work are discussed.

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.

Heuristics for the multi-item capacitated lot-sizing problem with lost sales

This paper deals with the multi-item capacitated lot-sizing problem with setup times and lost sales. Because of lost sales, demands can be partially or totally lost. To find a good lower bound, we use a Lagrangian relaxation of the capacity constraints, when single-item uncapacitated lot-sizing problems with lost sales have to be solved. Each subproblem is solved using an adaptation of the O(T2) dynamic programming algorithm of Aksen et al. [5]. To find feasible solutions, we propose a non-myopic heuristic based on a probing strategy and a refining procedure. We also propose a metaheuristic based on the adaptive large neighborhood search principle to improve solutions. Some computational experiments showing the effectiveness and limitation of each approach are presented.

Pricing for production and delivery flexibility in single-item lot-sizing

Adjusting prices to influence demand so as to increase revenue has become common practice. We investigate adjusting prices to influence demand so as to reduce cost. More specifically, we consider offering price discounts in return for production and delivery flexibility. We do so in the context of the single-item, single-level uncapacitated lot-sizing problem. We show that even though the resulting optimization problem has a nonlinear objective function it can still be solved in polynomial time. Furthermore, we report results of a computational study analyzing the benefits of offering price discounts in return for production and delivery flexibility in various settings.

A Note on “A MAX-MIN Ant System for Unconstrained Multi-Level Lot-Sizing Problems”

Pitakaso et al.[1] presented an ant systems based on random cumulative Wagner-Whitin (RCWW) (RCWW-STVS) for uncapacitated multilevel lot-sizing (MLLS) problems and gave out a computational result for Dellaert's instance [2] with a random variable r = 0.43. The result is not quite right. This paper highlighted the error and presented a revision to the result.

A Polyhedral Study of Multiechelon Lot Sizing with Intermediate Demands

In this paper, we study a multiechelon uncapacitated lot-sizing problem in series (m-ULS), where the output of the intermediate echelons has its own external demand and is also an input to the next echelon. We propose a polynomial-time dynamic programming algorithm, which gives a tight, compact extended formulation for the two-echelon case (2-ULS). Next, we present a family of valid inequalities for m-ULS, show its strength, and give a polynomial-time separation algorithm. We establish a hierarchy between the alternative formulations for 2-ULS. In particular, we show that our valid inequalities can be obtained from the projection of the multicommodity formulation. Our computational results show that this extended formulation is very effective in solving our uncapacitated multi-item two-echelon test problems. In addition, for capacitated multi-item, multiechelon problems, we demonstrate the effectiveness of a branch-and-cut algorithm using the proposed inequalities.

Two-stage minimax regret robust uncapacitated lot-sizing problems with demand uncertainty

In this paper, we consider the two-stage minimax robust uncapacitated lot-sizing problem with interval uncertain demands. A mixed integer programming formulation is proposed. Even though the robust uncapacitated lot-sizing problem with discrete scenarios is an NP -hard problem, we show that it is polynomial solvable under the interval uncertain demand set.

Neighborhood search techniques for solving uncapacitated multilevel lot-sizing problems

In this paper, several neighborhood search techniques for solving uncapacitated multilevel lot-sizing problems are investigated. We introduce three indexes: distance, changing range, and changing level that have great influence on the searching efficacy of neighborhood search techniques. These insights can help develop more efficient heuristic algorithms. As a result, we have developed an iterated neighborhood search (INS) algorithm that is very simple but that demonstrates good performance when tested against 176 benchmark instances under different scales (small, medium, and large), with 25 instances having been updated with new best known solutions in the computing experiments.

Uncapacitated lot-sizing problem with production time windows, early productions, backlogs and lost sales

We consider the single item uncapacitated lot-sizing problem with production time windows, lost sales, early productions and backlogs over a planning horizon of T periods. In this context, a demand not processed within its time window can either be lost (lost sale), satisfied from a production that is processed before the release period of the demand (early production) or satisfied from a production that occurs after the demand due period (backlog). We present several properties of the optimal solution for different variants of the problem when production time windows are non-customer specific. We propose dynamic programming algorithms to solve the examined problems in O(T 2).

A Note on “Lot-sizing with fixed charges on stocks: The convex hull”

We update and complete the proof of Proposition 7 in Van Vyve and Ortega (2004) [1], which states that the projection of a facility location reformulation of an uncapacitated lot-sizing problem with fixed charges on stocks (ULSW) to the original space is equivalent to that of the tight shortest path reformulation of ULSW. Their proof is interesting and consists of two cases, only first of which is analyzed in detail. We show that the second case exhibits several challenges not present in the first one and necessitates an updated proof.

A dynamic uncapacitated lot-sizing problem with co-production

We consider an extension of the dynamic uncapacitated lot-sizing problem to account for co-production, where multiple products are produced simultaneously in a single production run. We formulate the problem as a mixed-integer linear programming problem. We then show that a variant of the well-known zero-inventory property holds for this problem, and use this property to extend a dynamic program given for the single-item lot-sizing to solve the problem with co-production. Finally, we provide an illustrative example for our approach.

A Single-Item Uncapacitated Lot-Sizing Problem with Remanufacturing and Outsourcing

This paper addresses the single-item, dynamic lot-sizing problem for systems with remanufacturing and outsourcing. Therein, demand and return amounts are deterministic over a finite planning horizon. Demand may be satisfied by the manufacturing of new items, remanufactured items, or outsourcing, but it cannot be backlogged. The objective of this study is to determine the lot sizes for manufacturing, remanufacturing, and outsourcing that minimise the total cost, which consists of the holding costs for returns and manufactured/remanufactured products, setup costs, and outsourcing costs. The problems addressed in this paper are an extension of those addressed by Richter et al. (2000,and 2001), Teunter et al. (2006), and Aksen et al. (2003). In this paper, the separate setup costs scheme is considered, we propose a dynamic programming approach to derive the optimal solution in the case of large quantities of returned product. The complexity of this dynamic programming approach is O(T2), wherein T is the number of periods in the planning horizon.

An [formula omitted]-time algorithm for the stochastic uncapacitated lot-sizing problem with random lead times

In this paper, we develop a dynamic programming algorithm for the scenario-tree-based stochastic uncapacitated lot-sizing problem with random lead times. Our algorithm runs in O ( N 2 ) time, where N is the input size of the scenario tree, and improves the recently developed algorithm that runs in O ( N 3 ) time.

A generic coordination mechanism for lot-sizing in supply chains

A new generic mechanism to coordinate decentral planning of a group of independent and self-interested decision makers, who are searching for an agreeable contract regarding multiple interdependent issues, in the case of asymmetric information is presented. The basic idea of the mechanism is that the group members cooperatively carry out an evolutionary search in the contract space. Therefore the (1,?)-selection procedure, which is used in many evolutionary strategies, is combined with the Borda maximin voting rule, which has been applied successfully in group decision making. The proposed mechanism is realized, applied and evaluated for production coordination in a supply chain. A decentralized variant of the multi-level uncapacitated lot-sizing problem (MLULSP) is taken as the production model. For the evaluation 95 problem instances are generated based on MLULSP instances taken from the literature, with problem sizes varying from 5 to 500 items, from 12 to 52 periods. Experimental results show that the proposed mechanism is effective to determine fair cost distributions.

Stochastic lot-sizing problem with deterministic demands and Wagner–Whitin costs

In this paper, we consider a two-stage stochastic uncapacitated lot-sizing problem with deterministic demands and Wagner–Whitin costs. We develop an extended formulation in the higher dimensional space that provides integral solutions by showing that its constraint matrix is totally unimodular. We also provide the integral polyhedron of the problem in the original space by projecting the extended formulation to the original space.

Decentralized multi-level uncapacitated lot-sizing by automated negotiation

An automated negotiation mechanism for decentralized production coordination is presented and evaluated. The coordination problem contains a set of self-interested software agents, representing the production facilities of a supply chain, searching for a mutually agreeable production plan, while taking private information into account. The negotiation mechanism is applied and evaluated using a multi-facility production coordination problem, which is a reformulation of the well-known multi-level uncapacitated lot-sizing problem (MLULSP). The basic element of the mechanism is a decentralized simulated annealing method, consisting of a transition rule carried out by a neutral mediator agent and a cooperative acceptance rule carried out by negotiating agents. We use 176 benchmark problems from relevant literature for the evaluation. Experimental results show that the proposed negotiation mechanism comes close to those results which are obtained by centralized planning. Furthermore, the developed simulated annealing method applied in a single, centralized planning task is competitive with the best known solution methods for the MLULSP. It was possible to compute new best solutions for 24 of the benchmark problems.

Polyhedral analysis for the two-item uncapacitated lot-sizing problem with one-way substitution

We consider a production planning problem for two items where the high quality item can substitute the demand for the low quality item. Given the number of periods, the demands, the production, inventory holding, setup and substitution costs, the problem is to find a minimum cost production and substitution plan. This problem generalizes the well-known uncapacitated lot-sizing problem. We study the projection of the feasible set onto the space of production and setup variables and derive a family of facet defining inequalities for the associated convex hull. We prove that these inequalities together with the trivial facet defining inequalities describe the convex hull of the projection if the number of periods is two. We present the results of a computational study and discuss the quality of the bounds given by the linear programming relaxation of the model strengthened with these facet defining inequalities for larger number of periods.

Two-echelon requirements planning with pricing decisions

We consider a two-level uncapacitated lot-sizing problem where production, inventory carrying, transportation, and pricing decisions are integrated to maximize total profits. We show how this problem, under many different revenue functions and production, inventory holding, and transportation cost structures can be solved in polynomial time. As a byproduct, we develop polynomial-time algorithms for generalizations of single-level lot-sizing problems with pricing as well.

Polynomial-Time Algorithms for Stochastic Uncapacitated Lot-Sizing Problems

In 1958, Wagner and Whitin published a seminal paper on the deterministic uncapacitated lot-sizing problem, a fundamental model that is embedded in many practical production planning problems. In this paper, we consider a basic version of this model in which problem parameters are stochastic: the stochastic uncapacitated lot-sizing problem. We define the production-path property of an optimal solution for our model and use this property to develop a backward dynamic programming recursion. This approach allows us to show that the value function is piecewise linear and right continuous. We then use these results to show that a full characterization of the optimal value function can be obtained by a dynamic programming algorithm in polynomial time for the case that each nonleaf node contains at least two children. Moreover, we show that our approach leads to a polynomial-time algorithm to obtain an optimal solution to any instance of the stochastic uncapacitated lot-sizing problem, regardless of the structure of the scenario tree. We also show that the value function for the problem without setup costs is continuous, piecewise linear, and convex, and therefore an even more efficient dynamic programming algorithm can be developed for this special case.