Lot-sizing for a single-stage single-product production system with rework of perishable production defectives

We consider a single-stage single-product production system. Produced units may be non-defective, reworkable defective, or non-reworkable defective. The system switches between production and rework. After producing a fixed number (N) of units, all reworkable defective units are reworked. Reworkable defectives are perishable or can become technologically obsolete. We assume that the rework time and the rework cost increase linearly with the time that a unit is held in stock. Therefore, N should not be too large. On the other hand, N should not be too small either, since there are set-up times and costs associated with switching between production and rework. For a given N, we derive an explicit expression for the average profit (sales revenue minus costs). Using this expression, the optimal value for N can be determined numerically. Moreover, it is easy to perform a sensitivity analysis, as we illustrate. RID="*" ID="*"The research of Dr. Ruud H. Teunter has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences. The research presented in this paper is part of the research on re-use in the context of the EU sponsored TMR project REVersed LOGistics (ERB 4061 PL 97-5650) in which take part the Otto-von-Guericke Universitaet Magdeburg (D), the Erasmus University Rotterdam (NL), the Eindhoven University of Technology (NL), INSEAD (F), the Aristoteles University of Thessaloniki (GR), and the University of Piraeus (GR). We thank the anonymous referees for their many helpful comments. Correspondence to: R. H. Teunter     

Lot sizing with bounded inventory and lost sales

In production planning, there can be situations where the ability to meet customer demands is constrained by inventory capacity rather than production capacity. This situation often happens in petrochemical manufacturing, food processing, and glass manufacturing. Only a few studies can be found in the literature for this situation, and among these lost sales usually are not considered. In this paper, we consider the lot sizing problem with bounded inventory. We further consider that (1) lost sales are allowed; (2) production cost functions are non-increasing with respect to the time period; and (3) inventory capacity is non-decreasing with respect to the time period. With these considerations, we present a model as well as an algorithm which has a polynomial time complexity. An illustration is given to demonstrate both the application of our model and the algorithm.     

A genetic algorithm approach to the integrated inventory-distribution problem

We introduce a new genetic algorithm (GA) approach for the integrated inventory distribution problem (IIDP). We present the developed genetic representation and use a randomized version of a previously developed construction heuristic to generate the initial random population. We design suitable crossover and mutation operators for the GA improvement phase. The comparison of results shows the significance of the designed GA over the construction heuristic and demonstrates the capability of reaching solutions within 20% of the optimum on sets of randomly generated test problems.     

Heuristics for the N-product,M-stage,economic lot sizing and scheduling problem with dynamic demand

This paper presents a new composite heuristics approach for solving the N-product, M-stage lot sizing and scheduling problem with dynamic demands and limited production capacity. The first phase of these composite heuristics aims at finding a feasible solution. This solution is such that for each period and for each product, the lot size equals the net demand of the considered period plus the demand of a number of upcoming periods. If capacity does not satisfy all demands of a given period, we try to find earlier periods where we can produce the missing units. The second phase is an improvement procedure which recursively attempts to move back each lot, provided that it is both more economical to do so and capacity feasible. We also provide two variants of this heuristic to handle the case where production capacity can be increased by using overtime. Overtime is a usual practice in real life which, in many cases, allows a reduction of the overall cost. The first variant constructs the initial solution without recourse to overtime and introduces overtime only during the solution improvement phase. The second one considers overtime during both the first and second phases. The performance of the proposed heuristics is numerically assessed and the most efficient ones are identified.     

Analyzing the effects of inventory cost setting rules in a disassembly and recovery environment

In this study we consider a disassembly and recovery facility receiving end-of-life products and facing demand for a specific part that is disassembled from the product and then recovered. The disassembly and recovery operations can be either performed before hand, or upon customer arrival. In the latter case, a discount on the selling price is applied to compensate the customer for waiting for the completion of the disassembly and recovery operations. One of the difficulties faced in planning for such a system is the determination of the opportunity cost associated with carrying recovered parts inventory. The difficulty arises in seeking the value added to the part given the costs incurred for maintaining the product return, disassembly and recovery costs and revenue earned from the hulk, that is the remaining product after the disassembly of the part. The main objective of the study is to investigate the effect of different rules to determine this opportunity cost on the performance of the system. Six rules are considered in the study. The performance of the rules is assessed by a computational study under an approximate inventory control policy.     

Multi-item production routing problem with backordering: a MILP approach

The aim of this paper is to present mixed integer linear programming formulations for the production routing problem with backordering (PRP-B) and a new hybrid heuristic to solve the problem. The PRP-B is considered in the context of a supply chain consisting of a production facility with limited production and storage capacities and geographically dispersed points of sale with limited storage capacities. The PRP-B integrates multiple item lot sizing decisions and vehicle routing decisions to the points of sale, where backordering of end customer demands is allowed at a penalty. Two integrated mixed integer programming models are formulated and a solution procedure consisting of a relax-and-fix heuristic combined with a local search algorithm is proposed. The numerical results show that this hybrid heuristic outperforms a state-of-the-art MIP commercial solver, in terms of solution quality and CPU times.     

Lot-sizing in a multi-stage flow line production system with energy consideration

In this paper, a single-item capacitated lot-sizing problem in a flow-shop system with energy consideration is studied. The planning horizon is defined by a set of periods where each one is characterised by a length, an allowed maximal power, an electricity price, a power price and a demand. The objective is to determine the quantities to be produced by each machine at each period while minimising the production cost in terms of electrical, inventory, set-up and power required costs. For medium- and large-scale problems, lot-sizing problems are hard to solve. Therefore, in this study, two heuristics are developed to solve this problem in a reasonable time. To evaluate the performances of these heuristics, computational experiments are presented and numerical results are discussed and analysed.     

Modeling industrial lot sizing problems: a review

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.     

考虑运输能力约束的VMI补货发货动态批量研究

考虑由一个供应商和多个零售商构成的VMI供应链,在供应商运输能力有限、可采用外包策略的情况下,研究了动态需求环境下供应链补货及发货批量策略问题.分析得出供应商最优补货及发货期结构满足"零库存补货"性质,进而利用动态规划方法提出一个多项式算法优化补货及发货策略,其计算复杂度为O(T3),且通过平衡各种成本,可得每个发货周...     

Production-inventory policy for a deteriorating item with a single vendor-buyer system

In this paper, production-inventory models for a deteriorating item in a single vendor-buyer system has been developed with constant production and demand rate. Shortages at the buyer (when it is allowed) depends on time. The models have been formulated as cost minimization problem via both integrated and non-integrated approaches and solved using genetic algorithms developed to solve the single and multiobjective production inventory problems. Numerical illustrations of the models have been presented and the sensitivity analysis with respect to rates of production, demand and deterioration are performed.     

On the equivalence of small batch assembly line balancing and lot streaming in a flow shop

In this article we compare two apparently dissimilar scheduling problems. The Small Batch Assembly Line Balancing problem is the process of dividing operations over multiple stations so as to produce a series of parts whose processing times are characterized by learning. The Lot Streaming in a Flow Shop problem is the process of splitting a given lot or job to allow overlap of successive operations in multi-stage production systems, thereby reducing the makespan of the corresponding schedule. We show that the two problems are formally equivalent. By exploring the mathematical equivalence of the two problems, a number of unexpected new results have been obtained.     

Capacitated lot sizing with alternative routings and overtime decisions

Material requirements planning (MRP) is a basic tool for performing detailed material planning function in the manufacture of component parts and their assembly into finished items. MRP's managerial objective is to provide ‘the right part at the right time’ to meet the schedules for completed products. However satisfying end customer demands faster with lower inventories implies smarter scheduling which must simultaneously reflect actual capacity conditions. Therefore, the need is to schedule both capacity and materials simultaneously. Since MRP does not consider the availability of capacity resources to schedule production, consequently the schedules so developed are usually capacity infeasible. This paper proposes a three-step procedure to develop capacity feasible material and production schedules in a finite capacity environment. In the first step, an LP model produces capacity feasible but lot size relaxed planned order releases for all end products and assembly components which are then fed into a MRP processor, where a bill of material (BOM) explosion process generates material plans. Finally, these material plans are introduced to another LP model which assures that capacity feasibility is again restored. The mathematical models developed consider restrictions on lot sizes as well as alternative production routings and overtime decisions. A numerical example also is provided and some future research directions are outlined.     

Resorption system with simultaneous heat and cold production

A resorption system with simultaneous cold and heat production was studied. The heat produced could be used for sanitary or process purposes, or to drive another heat-powered machine. The resorption reactors had MnCl₂ and NH₄Cl as reactant (which are impregnated in expanded graphite) and NH₃ as refrigerant. The combined coefficient of performance and amplification (COPA) of this system reached 1.3 when the cooling effect was produced at 0 °C and heating effect at 75 °C with the regeneration temperature of 140 °C. Its COP was 0.35 with a specific cooling power (SCP) of 1.12 MJ kg⁻¹ day⁻¹, and the heat sink in this case remained below the cooling temperature for more than 5 h. Because of the heat production at certain temperature level (from 70 °C to 80 °C) in this study, the released heat could be used to power a silica gel-water adsorption chiller and the overall COP of the combined system would increase dramatically.     

Pricing and lot-sizing decisions in a two-echelon system with transportation costs

We consider a single-buyer single-supplier system. The market demand is sensitive to the selling price set by the buyer. Both the buyer and the supplier operate with unit product costs, inventory holding costs, and order placement costs. In addition, the buyer is responsible for the freight cost. We formulate a model for determining the optimal lot-sizing and pricing decisions. Existing models for the problem do not consider the transportation costs with price sensitive market demand, and determine the optimal decisions through an exhaustive search. We propose an approximate solution procedure, and report the computational results on the effectiveness of the proposed procedure.     

Inventory reservation and real-time order promising in a Make-to-Stock system

In this paper we consider a Make-to-Stock order fulfillment system facing random demand with random due date preferences from two classes of customers. We develop an integrated approach for reserving inventory in anticipation of future order arrivals from high priority customers and for order promising in real-time. Our research exhibits three distinct features: (1) we explicitly model uncertain due date preferences of the customers; (2) we consider multiple receipts in the planning horizon that can be utilized to fulfill customer orders; and (3) we choose to utilize a service level measure for reserving inventory rather than estimating short- and long-term implications of order promising with a penalty cost function. We propose an algorithm that exploits the time structure in order arrivals and time-phased material receipts to determine inventory reservations for high priority orders. Numerical experiments are conducted to investigate the performance and the benefits of the inventory reservation and order promising approach under varying system parameters.     

Optimal control of a continuum model for single-product re-entrant manufacturing systems

A semiconductor manufacturing system that involves a large number of items and many steps can be modelled through conservation laws for a continuous density variable on a production process. In this paper, the basic hyperbolic partial differential equation (PDE) models for multiple re-entrant manufacturing systems are proposed. However, through numerical examples, the basic continuum models do not perform well for small-scale multiple re-entrant systems, so a new state equation taking into account the re-entrant degree of the product is introduced to improve the basic continuum models. The applicability of the modified continuum model is illustrated through a numerical example. Based on the modified continuous model, this paper studies the optimal control problems for multiple re-entrant manufacturing systems. The gradient of the cost function with respect to the influx is solved by the adjoint approach, and then the optimal influx is computed by the steepest descent method. Finally, numerical examples on optimal influx profiles for steps in demand rate, linear demand rate and periodically varying demand rate are given. The relationships among influx, outflux and demand are also discussed in detail.     

Optimal stocking strategies for inventory mechanism with a stochastic short-term price discount and partial backordering

Price discount is an important research topic in the field of inventory management. The existing research on this topic mainly considers fixed price discount, but ignores the situation in which stochastic short-term price discount may be involved. In this paper, we study an inventory problem considering stochastic short-term price discount and partial backordering. To address this problem, we propose an optimal replenishment and stocking model to maximise the retailers' profit. After that, a cost–benefit analysis-based heuristic method for solving the developed model is presented by considering two scenarios depending on whether a replenishment point belongs to a discount period or not. Furthermore, an algorithm is provided to elicit an optimal ordering policy from multiple solutions derived from the given heuristic solution method. Finally, a real case is offered to demonstrate the application of the proposed model, followed by a sensitivity analysis. The results indicate that a retailer can identify the optimal replenishment policy with the aim of achieving maximal profit in situations where stochastic short-term price discount and partial backordering are considered for certain inventory problems at hand. In addition, sensitivity analysis illustrates a fact that different values of the introduced parameters may influence the optimal replenishment policy.     

Optimal ordering and pricing policy for an inventory system with order cancellations

This paper investigates the problem of ordering and pricing over a finite time planning horizon for an inventory system with advance sales and spot sales. It is assumed that the planning horizon is divided into several sales cycles each of which is divided into an advance sales period and a spot sales period. During the advance sales period, all customers are required to make reservations for their orders and will receive them at the arrival time of the replenishment orders. In the case of the spot sales periods, all customers receive their orders at the time of the purchase. In actual practice, since customers with reservations may cancel their orders before receiving them, this paper considers this phenomenon and develops a continuous time inventory model to deal with the proposed problem. This paper maximizes the total profit over a finite time planning horizon by determining the optimal advance sales price, spot sales price, order size, and replenishment frequency. Analysis of results shows that a simple algorithm can be developed to arrive at an optimal decision.     

The value of Electronic Marketplace in a perishable product inventory system with auto-correlated demand

This paper studies a periodic review inventory system for perishable products with correlated demand on a finite horizon. In this system, a fixed quantity of products from the supplier is received in each period. This quantity must be determined before the first period and cannot be changed thereafter. The inventory level can be adjusted through purchasing and selling products in an electronic marketplace at the beginning of each period. The available supply and demand quantities in the electronic marketplace depend on the prices offered by the retailer. The retailer’s optimal purchasing and selling quantities, and respective prices in the electronic marketplace are computed, and the expected total cost is shown to be convex with respect to the order quantity from the supplier, which enables an efficient algorithm in obtaining the optimal order quantity. Numerical experiments show that greater cost savings from electronic marketplace are obtained when demands in different periods are strongly correlated and greatly differ from each other.     

Preventive maintenance and optimal buffer inventory for products sold with warranty in an imperfect production system

This paper deals with an imperfect production system with allowable shortages due to regular preventive maintenance for products sold with free minimal repair warranty. Preventive maintenance is an essential factor of the just-in-time structure that results in a shutdown of the production process for a certain period of time. During such an interruption, a buffer stock is needed to adjust the market demand. The study includes the possibility of imperfect production and determines the optimum buffer level and production run time by trading off the holding cost, shortage cost, rework cost, repair cost for warranty, labour/energy costs, material cost and cost for maintenance so that the cost per unit product is minimised.