CAPACITATED LOT SIZING AND SCHEDULING PROBLEMS USING HYBRID GA/TS APPROACHES

The capacitated lot sizing and scheduling problem that involves indetermining the production amounts and release dates for several items over a given planning horizonare given to meet dynamic order demand without incurring backloggings. The problem consideringovertime capacity is studied. The mathematical model is presented, and a genetic algorithm (GA)approach is developed to solve the problem. The initial solutions are generated after usingheuristic method. Capacity balancing procedure is employed to stipulate the feasibility of thesolutions. In addition, a technique based on Tabu search (TS) is inserted into the genetic algorithmdeal with the scheduled overtime and help the convergence of algorithm. Computational simulation isconducted to test the efficiency of the proposed hybrid approach, which turns out to improve boththe solution quality and execution speed.

Collaborative supply chain planning under dynamic lot sizing costs with capacity decision

Studies show that supply chain cooperation improves supply chain performance. However, it remains a challenge to develop and implement the realistic supply chain cooperation scheme. We investigate a two-echelon supply chain planning problem with capacity acquisition decision under asymmetric cost and demand information. A simple negotiation-based coordination mechanism is developed to synchronize production/order strategies of a supplier and a buyer. The coordination scheme shows how the supplier and the buyer modify their production and order policy in order to find a joint economic lot sizing plan, which saves the overall supply chain cost. The allocation of the cooperation benefit is determined by negotiation. Due to the complexity of the multiple periods, multiple level supply chain lot sizing with capacity decision, a heuristic algorithm is developed to find coordination solutions. Finally, the results of the numerical study indicate the performance of supply chain coordination scheme.

Lot Sizing in Production Scheduling at a Personal Protection Equipment Company

This work presents an optimization model to support decisions during production planning and control in the personal protective equipment (PPE) industry (in particular, gloves). A case study was carried out at a Brazilian company with the aim of increasing productivity and improving customer service with respect to meeting deadlines. In this case study, the mixed integer linear programming model of Luche (2009) was revisited. A new model for single-stage lot sizing was applied to the production scheduling of gloves. Optimizing this scheduling was not a simple task because of the scale of the equipment setup time, the diversity of the products and the deadlines for the orders. The model was implemented in GAMS IDE and solved by CPLEX 12. The model and the associated heuristic produce better solutions than those currently used by the company.

MIP Formulations and Metaheuristics for Multi-Item Capacitated Lot-Sizing Problem with Non-Customer Specific Production Time Windows and Setup Times

Our research focuses on the development of two cooperative approaches for resolution of the multi-item capacitated lot-sizing problems with time windows and setup times (MICLSP-TW-ST). In this paper we combine variable neighborhood search and accurate mixed integer programming (VNS-MIP) to solve MICLSP-TW-ST. It concerns so a particularly important and difficult problem in production planning. This problem is NP-hard in the strong sense. Moreover, it is very difficult to solve with an exact method;it is for that reason we have made use of the approximate methods. We improved the variable neighborhood search (VNS) algorithm, which is efficient for solving hard combinatorial optimization problems. This problem can be viewed as an optimization problem with mixed variables (binary variables and real variables). The new VNS algorithm was tested against 540 benchmark problems. The performance of most of our approaches was satisfactory and performed better than the algorithms already proposed in the literature.

Optimal production lot size with process deterioration under an extended inspection policy

A mathematical model to determine the optimal production lot size for a deteriorating production system under an extended product inspection policy is developed. The last-K product inspection policy is considered so that the nonconforming items can be reduced, under which the last K products in a production lot are inspected and the nonconforming items from those inspected are reworked. Consider that the products produced towards the end of a production lot are more likely to be nonconforming, is proposed an extended product inspection policy for a deteriorating production system. That is, in a production lot, product inspections are performed among the middle K1 items and after inspections, all of the last K2 products are directly reworked without inspections. Our objective here is the joint optimization of the production lot size and the corresponding extended inspection policy such that the expected total cost per unit time is minimized. Since there is no closed form expression for our optimal policy, the existence for the optimal production inspection policy and an upper bound for the optimal lot size are obtained. Furthermore, an efficient solution procedure is provided to search for the optimal policy. Finally, numerical examples are given to illustrate the proposed model and indicate that the expected total cost per unit time of our product inspection model is less than that of the last-K inspection policy.

Effects of Imperfect Quality and Defective Items on Economic Production Lot Size

The classical EPQ model has been used for a long ti me and is widely accepted and implemented. Nevertheless, the analysis for finding an economic lot size has based on a number of unrealistic assumptions. A common unrealistic assumption in using EPQ is that all units produced are of good quali ty. The classical EPQ model shows that the optimal lot size will generate minimum ma nufacturing cost, thus producing minimum setup cost and inventory cost. However, this is only true if all products manufactured in the process are assumed to be of good quality (i.e. all products are within the specification limits). In rea lity this is not the case, therefore, it is necessary to consider the cost of im perfect quality items, because this cost can influence the economic lot size. Ma ny studies and recent papers have indicated that there is a significant relation ship between economic production lot size and process/product quality. However, their models included either the imperfect quality items (not necessarily de fective) which are to be sold at a discounted price or defective items which can be reworked or rejected. The aim of this paper is to provide a framework to integrate three different sit uations (discounted pricing/rework/reject) into a single model. 100% inspection is performed in order to distinguish the amount of good quality items, imper fect quality items and defective items in each lot. In this paper, a mathematica l model is developed, and a numerical example is presented to illustrate the sol ution procedures. It is found that the economic production lot size tends to inc rease as the average percentage of imperfect quality items and defectives (rejec ted items) increases.

A RELATIVE BENEFIT ALGORITHM FOR BASIC ECONOMIC LOT SIZE PROBLEM

The paper develops an algorithm that solves economic lot size problem in O(n~2) time in the Wagner-Whitin case. The algorithm is based on the standard dynamic programming approach which requires the computation of the maximal relative benefit for some possible subplans of the production plan. In this algorithm the authors have studied the forward property and decomposition properties which can make computation easy. The proposed algorithm appears to perform quite reasonably for practical application.

POLYNOMIAL DYNAMIC PROGRAMMING ALGORITHMS FOR LOT SIZING MODELS WITH BOUNDED INVENTORY AND STOCKOUT AND/OR BACKLOGGING

This paper addresses a dynamic lot sizing problem with bounded inventory and stockout where both no backlogging and backlogging allowed cases are considered. The stockout option means that there is outsourcing in a period only when the inventory level at that period is non-positive. The production capacity is unlimited and production cost functions are linear but with fixed charges. The problem is that of satisfying all demands in the planning horizon at minimal total cost. We show that the no backlogging case can be solved in O（T^2） time with general concave inventory holding and outsourcing cost functions where T is the length of the planning horizon. The complexity can be reduced to O（T） when the inventory holding cost functions are also linear and have some realistic properties, even if the outsourcing cost functions remain general concave functions. When the inventory holding and outsourcing cost functions are linear, the backlogging case can be solved in O（ T^3 logT） time whether the outsourcing level at each period is bounded by the sum of the demand of that period and backlogging level from previous periods, or only by the demand of that period.

A LAGRANGIAN RELAXATION APPROACH FOR SUPPLY CHAIN PLANNING WITH ORDER/SETUP COSTS AND CAPACITY CONSTRAINTS

A heuristic approach is developed for supply chain planning modeled as multi-item multi-level capacitated lot sizing problems. The heuristic combines Lagrangian relaxation (LR) with local search. Different from existing LR approaches that relax capacity constraints and/or inventory balance constraints, our approach only relaxes the technical constraints that each 0-1 setup variable must take value 1 if its corresponding continuous variable is positive. The relaxed problem is approximately solved by using the simplex algorithm for linear programming, while Lagrange multipliers are updated by using a surrogate subgradient method that ensures the convergence of the dual problem in case of the approximate resolution of the relaxed problem. At each iteration, a feasible solution of the original problem is constructed from the solution of the relaxed problem. The feasible solution is further improved by a local search that changes the values of two setup variables at each time. By taking the advantages of a special structure of the lot-sizing problem, the local search can be implemented by using a modified simplex algorithm, which significantly reduces its computation time. Numerical experiments show that our approach can find very good solutions for problems of realistic sizes in a short computation time and is more effective than an existing commercial optimization code.

A REVIEW OF TACTICAL PLANNING MODELS

The aim of this paper is to give an overview on models and methods used to solve tactical planning problems. The modeling and the elaboration of the well-know tactical planning problems （master planning ＆ scheduling, material requirement planning and multi-site planning） are discussed. These problems are modeled from two ＂lot sizing＂ models called the Capacitated Lot Sizing Problem （CLSP） and Multi Level Capacitated Lot Sizing Problem （MLCLSP）. From both models, a lot of extensions has been proposed in the literature. The purpose of this paper is twofold： first, classifications of the CLSP and MLCLSP as well as their extensions are given. For each model, the major scientific contributions are mentioned. These classifications made from seventy papers give an overview of ＂lot sizing＂ models dedicated to the MPS, MRP and Multi-site and show the diversity of models. Second, from a classification, an analysis of methods used for each model is given. The instance size, best gap and reference for gap computation are given for each contribution, This work can be used to elaborate an optimization tool for tactical planning problematic such as Advanced Planning System.

Inventory and sales decisions in a vendor–buyer system with imperfect items and learning in inspection

In this study,we propose a joint economic lot-sizing model to include learning process along with errors in inspection and full backordering system.We aim to study pricing and inventory decisions in a two-level supply chain involving a single vendor and a single buyer in which the demand is sensitive to price and company’s advertisement efforts.The mathematical inventory model is developed analytically and solved using a proposed algorithm.The objective of the model is to maximise joint total profit by simultaneously determining optimal shipment size,number of deliveries,backorder quantity and product selling price.A numerical example is provided to show the application of the model and to investigate the impact of key parameter’s changes on model behaviour.By comparing the integrated/centralised model to the independent/decentralised model,we note that the integrated/centralised model provides better profit to the system,along with lower selling price and smaller amount of backorder.