A holding cost bound for the economic lot-sizing problem with time-invariant cost parameters

We show that in an optimal solution of the economic lot-sizing problem the total holding cost in an order interval is bounded from above by a quantity proportional to the setup cost and the logarithm of the number of periods in the interval. We present two applications of this result.     

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.     

Robust formulations for economic lot-sizing problem with remanufacturing

In this paper, we consider a lot-sizing problem with the remanufacturing option under parameter uncertainties imposed on demands and returns. Remanufacturing has recently been a fast growing area of interest for many researchers due to increasing awareness on reducing waste in production environments, and in particular studies involving remanufacturing and parameter uncertainties simultaneously are very scarce in the literature. We first present a min-max decomposition approach for this problem, where decision maker’s problem and adversarial problem are treated iteratively. Then, we propose two novel extended reformulations for the decision maker’s problem, addressing some of the computational challenges. An original aspect of the reformulations is that they are applied only to the latest scenario added to the decision maker’s problem. Then, we present an extensive computational analysis, which provides a detailed comparison of the three formulations and evaluates the impact of key problem parameters. We conclude that the proposed extended reformulations outperform the standard formulation for a majority of the instances. We also provide insights on the impact of the problem parameters on the computational performance.     

An algorithm for single-item economic lot-sizing problem with general inventory cost, non-decreasing capacity, and non-increasing setup and production cost

This paper considers an economic lot-sizing model with non-decreasing capacity constraint, non-increasing setup cost and production cost, and a general inventory cost. We prove that when periodic starting inventory is not less than a certain critical value, it is optimal to produce nothing; this critical value can be computed easily which results in a new effective algorithm.     

Two-stage stochastic lot-sizing problem under cost uncertainty

For production planning problems, cost parameters can be uncertain due to marketing activities and interest rate fluctuation. In this paper, we consider a single-item two-stage stochastic lot-sizing problem under cost parameter uncertainty. Assuming cost parameters will increase or decrease after time period p each with certain probability, we minimize the total expected cost for a finite horizon problem. We develop an extended linear programming formulation in a higher dimensional space that can provide integral solutions by showing that its constraint matrix is totally unimodular. We also project this extended formulation to a lower dimensional space and obtain a corresponding extended formulation in the lower dimensional space. Final computational experiments demonstrate that the extended formulation is more efficient and performs more stable than the two-stage stochastic mixed-integer programming formulation.     

A primal–dual algorithm for the economic lot-sizing problem with multi-mode replenishment

The classical economic lot-sizing problem assumes that a single supplier and a single transportation mode are used to replenish the inventory. This paper studies an extension of this problem where several suppliers and transportation modes are available. The decision-making process in this case involves identifying (i) the timing for an order; (ii) the choice of shipment modes; and (iii) the order size for each mode. The problem is defined as a network flow problem with multiple setups cost function and additional side constraints. This study provides an MIP formulation for the problem. We also provide an additional formulation of the problem by redefining its decision variables and show that the dual of the corresponding LP-relaxation has a special structure. We take advantage of the structure of the dual problem to develop a primal–dual algorithm that generates tight lower and upper bounds. Computational results demonstrate the effectiveness of the algorithm.     

Dynamic lot-sizing problem with demand time windows and container-based transportation cost

In this paper, we study the dynamic lot-sizing problem with demand time windows and container-based transportation cost. For each particular demand, there are corresponding earliest and latest times, and the duration between such earliest and latest times is the demand time window. If a demand is satisfied by a delivery within demand time window, then there is no holding or backlogging cost incurred. Our purpose is to satisfy demand at a minimum total cost, including setup cost, procurement cost, container cost, and inventory holding cost. This research is supported in part by Hong Kong RGC grant HKUST 6010/02E and NUS ARF grant R-266-000-019-112.     

A single-item economic lot-sizing problem with a non-uniform resource: Approximation

We study a generalization of the classical single-item capacitated economic lot-sizing problem to the case of a non-uniform resource usage for production. The general problem and several special cases are shown to be non-approximable with any polynomially computable relative error in polynomial time. An optimal dynamic programming algorithm and its approximate modification are presented for the general problem. Fully polynomial time approximation schemes are developed for two NP-hard special cases: (1) cost functions of total production are separable and holding and backlogging cost functions are linear with polynomially related slopes, and (2) all holding costs are equal to zero.     

Dynamic lot-sizing with setup cost reduction

One of the fundamental tenets of the Just-in-Time (JIT) manufacturing philosophy is that reduction or even elimination of inventory conserves valuable resources and reduces wasteful spending. In many cases, to achieve inventory reductions requires investment in reduction of setup costs. For this reason, certain proposals for incorporating means for reducing setup costs into classical production-inventory models have been offered in recent years. This article considers a dynamic lot-sizing model M where the values of the setup costs can be reduced by various amounts depending upon the level of funds R committed to this reduction. We show that for each fixed value of R, the model can be represented as a shortest path problem. By minimizing the optimal value function V(R) of the shortest path problem over R, model M can, in theory, be solved. In practice, the viability of this approach depends crucially upon the properties of the function V. Since these properties depend upon the nature of the setup cost function K used in model M, we analyze how V varies as K varies. This allows us to propose two exact, finite algorithms for solving model M, one for the case when K is a concave function, the other for the case when K is convex. Computational results for the convex case are presented. The problems solved demonstrate that, in practice, setup cost reductions chosen according to model M have the potential to significantly reduce both inventory levels and total costs.     

Optimal production plans and shipment schedules in a supply-chain system with multiple suppliers and multiple buyers

This research addresses an optimal policy for production and procurement in a supply-chain system with multiple non-competing suppliers, a manufacturer and multiple non-identical buyers. The manufacturer procures raw materials from suppliers, converts them to finished products and ships the products to each buyer at a fixed-interval of time over a finite planning horizon. The demand of finished product is given by buyers and the shipment size to each buyer is fixed. The problem is to determine the production start time, the initial and ending inventory, the cycle beginning and ending time, the number of orders of raw materials in each cycle, and the number of cycles for a finite planning horizon so as to minimize the system cost. A surrogate network representation of the problem developed to obtain an efficient, optimal solution to determine the production cycle and cycle costs with predetermined shipment schedules in the planning horizon. This research prescribes optimal policies for a multi-stage production and procurements for all shipments scheduled over the planning horizon. Numerical examples are also provided to illustrate the system.     

Optimal solutions for the double row layout problem

The double row layout problem is how to allocate a given set of n machines on both sides of a straight line corridor so that the total cost of transporting materials between machines is minimized. This is a very difficult combinatorial optimization problem with important applications in industry. We formulate the problem as a mixed-integer program. Computational tests show that the proposed formulation presents a far superior performance than that of a previously published model.     

A particle swarm optimization for solving lot-sizing problem with fluctuating demand and preservation technology cost under trade credit

In this paper, we consider the effect of preservation technology cost investing on preservation equipment for reducing deterioration rate under two-level trade credit. The preservation technology cost is allowed for periodical upward or downward adjustments due to the time varying demand and the strategy of trade credit within the planning horizon. We establish a deterministic economic order quantity model for a retailer to determine his/her optimal preservation technology cost per replenishment cycle, the trade credit policies, the replenishment number and replenishment schedule that will maximize the present value of total profit. A particle swarm optimization with constriction factor is coded and used to solve the mixed-integer nonlinear programming problem by employing the properties derived from this paper. Some numerical examples are used to illustrate the features of the proposed model.     

Optimal cost and policy for a Markovian replacement problem

We consider the computation of the optimal cost and policy associated with a two-dimensional Markov replacement problem with partial observations, for two special cases of observation quality. Relying on structural results available for the optimal policy associated with these two particular models, we show that, in both cases, the infinitehorizon, optimal discounted cost function is piecewise linear, and provide formulas for computing the cost and the policy. Several examples illustrate the usefulness of the results.This research was supported by the Air Force Office of Scientific Research Grant AFOSR-86-0029, by the National Science Foundation Grant ECS-86-17860, by the Advanced Technology Program of the State of Texas, and by the Air Force Office of Scientific Research (AFSC) Contract F49620-89-C-0044.     

Optimal solutions for a dock assignment problem with trailer transportation

This paper presents a model for a dock assignment problem, where trailers need to be assigned to gates for a given period of time for loading or unloading activities. The parking lot is used as a buffer zone. Transportation between the parking lot and the gates is performed by additional resources called terminal tractors. The problem is modeled as a three-stage flexible flow shop, where the first and the third stage share the same identical parallel machines and the second stage consists of a different set of identical parallel machines. We examine multiple integer-programming formulations for the parallel-machine model in stage two and for the three-stage flow shop and we provide extensive computational results. Our goal is to explore the limits of the instance sizes that can be solved to guaranteed optimality within acceptable running times using integer programming.     

Polynomial cases of the economic lot sizing problem with cost discounts

In this paper we study the economic lot sizing problem with cost discounts. In the economic lot sizing problem a facility faces known demands over a discrete finite horizon. At each period, the ordering cost function and the holding cost function are given and they can be different from period to period. There are no constraints on the quantity ordered in each period and backlogging is not allowed. The objective is to decide when and how much to order so as to minimize the total ordering and holding costs over the finite horizon without any shortages. We study two different cost discount functions. The modified all-unit discount cost function alternates increasing and flat sections, starting with a flat section that indicates a minimum charge for small quantities. While in general the economic lot sizing problem with modified all-unit discount cost function is known to be NP-hard, we assume that the cost functions do not vary from period to period and identify a polynomial case. Then we study the incremental discount cost function which is an increasing piecewise linear function with no flat sections. The efficiency of the solution algorithms follows from properties of the optimal solution. We computationally test the polynomial algorithms against the use of CPLEX.     

An elliptic resonance problem with multiple solutions

In this paper we study the existence of multiple nontrivial solutions for a semilinear elliptic equation at resonance by the minimax methods and Morse theory.