A polyhedral study of the single-item lot-sizing problem with continuous start-up costs

Coordinated deterministic dynamic demand lot-sizing problem: A review of models and algorithms

In this study, we investigated the time-varying capacitated lot-sizing problem under a fast-changing production environment, where production factors such as the setup costs, inventory-holding costs, production capacities, or even material prices may be subject to continuous changes during the entire planning horizon. Traditional lot-sizing theorems and algorithms, which often assume a constant production environment, are no longer fit for this situation. We analyzed the time-varying environment of today’s agile enterprises and modeled the time-varying setup costs and the time-varying production capacities. Based on these, we presented two mixed-integer linear programming models for the time-varying capacitated single-level lot-sizing problem and the time-varying capacitated multi-level lot-sizing problem, respectively, with considerations on the impact of time-varying environments and dynamic capacity constraints. New properties of these models were analyzed on the solution’s feasibility and optimality. The solution quality was evaluated in terms of the entropy which indicated that the optimized production system had a lower value than that of the unoptimized one. A number of computational experiments were conducted on well-known benchmark problem instances using the AMPL/CPLEX to verify the proposed models and to test the computational effectiveness and efficiency, which showed that the new models are applicable to the time-varying environment. Two of the benchmark problems were updated with new best-known solutions in the experiments.

Hybrid proactive approach for solving maintenance and planning problems in the scenario of Industry 4.0

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 research aims to detect national income variable influence, credit rate of interest level and domestic investment previous period towards domestic investment with detects which variable that has influence dominantest towards domestic investment. Method which is used in this researcd is double linear regression PArtisl Adjustement Model used for perceiving short-range and long-range responsivenees from dependent variable to one changed unit of independent variable value. After conducted by t-test known that national income and previous period of the domestic investment individually having influence significantly to the level domestic investment. But credit rate of interest level individually not having influence significantly to the level domestic investment. The influence variable to the level of domestic investment is the level of previous period domestic investment model because analysis model that used is adjustment of partial model. Outside independent variable, in the reality variabel that influencing to the level of domestic investment is national income.

We address the problem of coordinated replenishment of products when the products can be produced only in fixed proportion to each other. Such problems commonly arise in the manufacture of sheet/plate metal parts or die-cast parts. The problem is a variant of the well-known Joint Replenishment Problem. We call this problem the Strong Interaction Problem (SIP). After giving a mathematical formulation of the problem, we show that the general problem is NP-hard. An important variant of the problem, in which products are unique to a family, is shown to be polynomially solvable. We present several lower bounds, an exact algorithm and a heuristic for the problem. Computational testing on randomly generated problems suggests that our exact algorithm performs very well when compared with a commercially available integer programming solver. The heuristic method also gives good solutions.

Superdirectivity can yield large directivity using a small array aperture, but it suffers from high sensitivity to random errors. In a previously proposed model, robust Nth-order superdirectivity was obtained by combining low-order superdirectivity sub-solutions with orders from 0 to N. However, the orders N were exclusively positive integers, which may lead to an over-improved robustness with an excessively degraded directivity. This paper presents an arbitrary-order superdirectivity model of circular sensor arrays, which can achieve a more flexible compromise between directivity and robustness than the previous model. A coefficient η is weighted at the (N + 1)th-order entry of the optimal weighting vector and the new weighting vector will consist of the 0th- to Nth-order and the weighted (N + 1)th-order entries. The beampattern, the directivity factor, and the error sensitivity function can then be expressed as functions of the coefficient η. Moreover, broadband superdirective beampatterns with a desired directivity factor or error sensitivity function can be readily synthesized using the closed-form solutions derived from the arbitrary-order superdirectivity model. Simulations and experimental results show that the proposed model can easily synthesize superdirective beampatterns with arbitrary orders, and it is more flexible than the previous integer-order model. It is also demonstrated that the performance of the proposed method is similar to that of the optimization-based method, but it uses a simple closed-form weighting vector rather than a numerical algorithm.

The Football Pool Polytope

We consider a multiple item Economic Lot Sizing problem where the demands for items depend on their stock quantities. The objective is to find a production plan such that the resulting stock levels (and hence demands) maximize total profit over a finite planning horizon. The single item version of this problem has been studied in the literature, and a polynomial time algorithm has been proposed when there are no bounds on production. It has also been proven that the single item version is NP-hard even when there are constant (i.e., time-invariant) finite capacities on production. We extend this single item model by considering multiple items and production capacities. We propose a Lagrangian relaxation method to find an initial solution to the problem. This solution is a hybrid solution obtained by combining two distinct solutions generated in the process of solving the Lagrangian dual problem. Starting with this initial solution, we then implement a Tabu Search algorithm to find better solutions. The performance of the proposed solution method is compared with the performance of a standard commercial software that works on a mixed integer programming formulation of the problem. We show that our solution approach finds better solutions within a predetermined time limit in general.

Production planning is a process to determine number of production in a particular period to fulfill consumers orders according to the available production capacity. Production planning by combining Integer Linear Programming (ILP) and simulation is done in iterative way. ILP has an objective function to minimize cost, which has a number of methods, such as reguler production capacity, overtime production capacity and inventory constrain, for each type of tyres for monthly during constraints in three month (January – March 2004). Simulation model representes stochastic events on the production floor, so that actual production capacity is obtained. On each iteration, number of production was compared with the available capacity from the result of simulation model. If the simulated number of production is more than or equal to the available capacity, the iteration need an adjustment capacity. And then, if the simulated number of production is less than the available capacity, the iteration is stopped, so that the number of production proper with the actual capacity is obtained. Keywords : production planning, Integer Linear Programming, Simulation, capacity

A hybrid heuristic for capacitated three-level lot-sizing and replenishment problems with a distribution structure

We present structural and computational investigations of a new class of weak forecast horizons—minimal forecast horizons under the assumption that future demands are integer multiples of a given positive real number—for a specific class of dynamic lot-size (DLS) problems. Apart from being appropriate in most practical instances, the discreteness assumption offers a significant reduction in the length of a minimal forecast horizon over the one using the classical notion of continuous future demands. We provide several conditions under which a discrete-demand forecast horizon is also a continuous-demand forecast horizon. We also show that the increase in the cost resulting from using a discrete minimal forecast horizon instead of the classical minimal forecast horizon is modest. The discreteness assumption allows us to characterize forecast horizons as feasibility/optimality questions in 0-1 mixed-integer programs. On an extensive test bed, we demonstrate the computational tractability of the integer programming approach. Owing to its prevalence in practice, our computational experiments emphasize the special case of integer future demands.

: This project was concerned with the application of mathematical programming models and techniques to logistics planning problems. Basic research was performed on a new approach, called inverse optimization, to the parametric analysis of mixed integer programming models. The approach was implemented and tested for the capacitated plant location problem. Basic research was also performed on three other logistics planning models with cyclic structures; namely, lot-size problems when demand and costs are cyclic, vehicle routing and cyclic staffing. A final research effort, partially suggested by the contract, was the construction and optimization, using decomposition methods, of a model of the U.S. coal supply and demand markets.

Production Lot Sizes on a Single Imperfect Machine: FPTAS vs ILP models

A goal programming model for joint decision making of inventory lot-size, supplier selection and carrier selection