Single-Item Dynamic Lot-Sizing Models With Bounded Inventory and Outsourcing

Dynamic lot sizing problem with continuous-time Markovian production cost

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.

A production inventory model with Just-In-Time (JIT) set-up cost has been developed in which inflation and time value of money are considered under an imperfect production process. The demand rate is considered to be a function of advertisement cost and selling price. Unit production cost is considered incorporating several features like energy and labour cost, raw material cost and development cost of the manufacturing system. Development cost is assumed to be a function of reliability parameter. Considering these phenomena, an analytic expression is obtained for the total profit of the model. The model provides an analytical solution to maximize the total profit function.A numerical example is presented to illustrate the model along with graphical analysis. Sensitivity analysis has been carried out to identify the most sensitive parameters of the model.

Integrated lot sizing and energy-efficient job shop scheduling problem in manufacturing/remanufacturing systems

In this paper we give an overview of recent developments in the field of modeling deterministic single-level dynamic lot sizing problems. The focus of this paper is on the modeling of various industrial extensions and not on the solution approaches. The timeliness of such a review stems from the growing industry need to solve more realistic and comprehensive production planning problems. First, several different basic lot sizing problems are defined. Many extensions of these problems have been proposed and the research basically expands in two opposite directions. The first line of research focuses on modeling the operational aspects in more detail. The discussion is organized around five aspects: the set ups, the characteristics of the production process, the inventory, demand side and rolling horizon. The second direction is towards more tactical and strategic models in which the lot sizing problem is a core substructure, such as integrated production–distribution planning or supplier selection. Recent advances in both directions are discussed. Finally, we give some concluding remarks and point out interesting areas for future research.

The aim of this work is to propose a solution approach for a capacitated lot sizing and scheduling real problem with parallel machines and shared buffers, arising in a packaging company producing yoghurt. The problem has been formulated as a hybrid Continuous Set-up and Capacitated Lot Sizing Problem (CSLP–CLSP). A new effective two stage optimisation heuristic based on the decomposition of the problem into a lot sizing problem and a scheduling problem has been developed. An assignment of mixture to buffers is made in the first stage, and therefore the corresponding orders are scheduled on the production lines by performing a local search. Computational tests have been performed on the real data provided by the company. The heuristic exhibits near-optimal solutions, all obtained in a very short computational time.

This paper considers a single item, imperfect economic production lot-size (EPL) model with stock-dependent demand and partial rework. In real life EPL models, defective production commences from the out-of-control state, after the passage of some time from production commencement. Its occurrence is random after the lapse of certain time and imposed here through a chance constraint. The set-up cost is partly production dependent. Unit production cost is also production dependent and a part of it is taken as environment protection cost. Defective rate is also assumed to be random and production dependent. The model is formulated as an average cost minimisation problem subject to a chance constraint and solved using a nonlinear optimisation technique - generalised reduced gradient (GRG) method through LINGO 11.0. Several special cases are derived and more specifically, the present investigation the works of Sana (2010) and Khouja and Mehrez (1994). Numerical experiments are performed to illustrate the general and particular models. Some sensitivity analyses are presented against few model parameters.

A parallel algorithm for the dynamic lot-sizing problem

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

In our work, a multi-stage stochastic programming model was considered to develop the multi-drug dynamic lot-size and delivery scheduling problem (MDLSDSP) under local emergencies. In the real world, the demand and delivery capacity are uncertain but largely conform to a discrete distribution, estimated based on data from Internet of Things (IoT). To address the numerous scenarios, a modified sample average approximation (SAA) method and a progressive hedging algorithm (PHA) are explored for solving large-scale models. Practical experiments regarding the replenishment and delivery of medical materials under local public health emergencies are designed to evaluate the designed algorithms. Experimental results show that the gaps between the two proposed algorithms and the optimal results are less than 0.7% and 0.9% on average, while the efficiency is much faster for large-scale experiments. The top three costs of the MDLSDSP are the product cost, delivery cost, and setup cost, with a total of over 95%. The total cost from external suppliers accounts for over 15% when the capacities are tight, which indicates that local emergencies have a significant impact on the normal operation of the supply chain system. Moreover, although the productions from external suppliers are far less than the productions from the system, product cost from external suppliers accounts for 10% on average, which indicates that the total cost will be significantly reduced if the demand is more fulfilled by suppliers within the system. So, it is crucial to accurately predict the duration and scope of the impact of local emergencies, which is precisely the advantage of digital information technology, including big data, IoT, artificial intelligence.

The aim of this paper is to present a branch and bound approach to solve a k-stage hybrid flowshop scheduling problem in order to minimize the total completion time. This scheduling problem encloses three subproblems: lot sizing, scheduling and assignment problems. In this paper only scheduling and assignment are taken into account because we suppose that the lot sizing problem has already been solved. The method we propose is an optimal method, based on a special architecture of search tree which allows us to solve the two subproblems considered. Some lower bounds are proposed. Then computational results are given and proved that results are acceptable. Finally, two theorems demonstrate some dominance relations.

The determination of lot sizes and production schedules of multiple products in multiple periods is difficult when sequence-dependent set-up costs are present. By using an initial set-up cost estimator (ISCE) the joint problem can be divided and solved sequentially. A search of the literature reveals that, while there is considerable research on the individual problems of lot sizing and scheduling, there is a dearth of study on either the joint problem or on the effect of ISCE selection on the total cost solution. Using a variety of data sets, this paper evaluates the sensitivity of ISCE selection on the solution to the joint problem. Results of the study demonstrate that the choice of an ISCE can have a significant effect on the final lot sizes and schedules. Additionally, the research shows that ‘common sense’ a priori assumptions about ISCE performance may lead to the selection of an inappropriate ISCE.

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 paper develops a fuzzy stochastic multi-objective linear programming (FSMOLP) model for a multi-level, multi-item capacitated lot sizing problem (CLSP) in a multi assembly shop. The proposed model attempts to minimize the total cost consisting of total production variation cost, inventory cost, backlog cost and total setup cost while maximizing the resource utilization simultaneously. To cope with the uncertainty associated with the most of the input data, e.g., the demand and process-related parameters they are treated as fuzzy stochastic parameters with identical stochastic membership function during the planning horizon. To show the usefulness of the proposed solution method, a numerical example is first solved. Then, the usefulness of the proposed model is validated over a set of randomly generated test problems.

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.