Optimal lot-sizing problem with imperfect maintenance and imperfect production

In this paper, we develop an integrated model for the joint determination of both economic production quantity and level of preventive maintenance (PM) for an imperfect production process. This process has a general deterioration distribution with increasing hazard rate. The effect of PM activities on the deterioration pattern of the process is modelled using the imperfect maintenance concept. In this concept, it is assumed that after performing PM, the ageing of the system is reduced in proportion to the PM level. After a period of time in production, the process may shift to out-of-control states, either type I or type II. A minimal repair will remove the type I out-of-control state. If a type II out-of-control state occurs, the production process has to stop, and then restoration work is carried out. Examples of Weibull shock models are given to show that the use of PM reduces costs.     

Heuristics for the multi-item capacitated lot-sizing problem with lost sales

This paper deals with the multi-item capacitated lot-sizing problem with setup times and lost sales. Because of lost sales, demands can be partially or totally lost. To find a good lower bound, we use a Lagrangian relaxation of the capacity constraints, when single-item uncapacitated lot-sizing problems with lost sales have to be solved. Each subproblem is solved using an adaptation of the O(T²)$O(T^2)$ dynamic programming algorithm of Aksen et al. [5]. To find feasible solutions, we propose a non-myopic heuristic based on a probing strategy and a refining procedure. We also propose a metaheuristic based on the adaptive large neighborhood search principle to improve solutions. Some computational experiments showing the effectiveness and limitation of each approach are presented.     

Computationally efficient solution of the multi-item, capacitated lot-sizing problem

The problem of multi-item, single-level, dynamic lotsizing in the presence of a single capacitated resource is addressed. A model based on variable redefinition is developed leading to a solution strategy based on a branch-and-bound search with sharp low bounds. The multi-item low bound problems are solved by column generation with the capacity constraints as the linking constraints. The resulting subproblems separate into as many single-item, uncapacitated lotsizing problems as there are items. These subproblems are solved as shortest-path problems. Good upper bounds are also generated by solving an appropriate minimum-cost network flow problem at each node of the branch-and-bound tree. The resulting solution scheme is very efficient in terms of computation time. Its efficiency is demonstrated by computational testing on a ste of benchmark problem instances and is attributable to the sharpness of the lower bounds, the efficiency with which the low bound problems are solved and the frequent generation of good upper bounds; all of which leading to a high exclusion rate.     

Variable neighborhood descent heuristic for solving reverse logistics multi-item dynamic lot-sizing problems

The multi-product dynamic lot sizing problem with product returns and recovery is an important problem that appears in reverse logistics and is known to be NP-hard. In this paper we propose an efficient variable neighborhood descent heuristic algorithm for solving this problem. Furthermore, we present a new benchmark set with the largest instances in the literature. The computational results demonstrate that our approach outperforms the state-of-the-art Gurobi optimizer.     

Effective replenishment policies for the multi-item dynamic lot-sizing problem with storage capacities

We address the dynamic lot-sizing problem considering multiple items and storage capacity. Despite we can easily characterize a subset of optimal solutions just extending the properties of the single-item case, these results are not helpful to design an efficient algorithm. Accordingly, heuristics are appropriate approaches to obtain near-optimal solutions for this NP-hard problem. Thus, we propose a heuristic procedure based on the smoothing technique, which is tested on a large set of randomly generated instances. The computational results show that the method is able to build policies that are both easily implemented and very effective, since they are on average 5% above the best solution reported by CPLEX. Moreover, an additional computational experiment is carried out to show that the performance of this new heuristic is on average better and more robust than other methods previously proposed for this problem.     

The multi-item capacitated lot-sizing problem with safety stocks and demand shortage costs

We address a multi-item capacitated lot-sizing problem with setup times, safety stock and demand shortages. Demand cannot be backlogged, but can be totally or partially lost. Safety stock is an objective to reach rather than an industrial constraint to respect. The problem is np-hard. We propose a Lagrangian relaxation of the resource capacity constraints. We develop a dynamic programming algorithm to solve the induced sub-problems. An upper bound is also proposed using a Lagrangian heuristic with several smoothing algorithms. Some experimental results showing the effectiveness of the approach are reported.     

Optimal pricing, lot-sizing and marketing planning in a capacitated and imperfect production system

In the classical economic production quantity (EPQ) problem demand is considered to be known in advance. However, in the real-world, demand of a product is a function of factors such as product’s price, its quality, and marketing expenditures for promoting the product. Quality level of the product and specifications of the adopted manufacturing process also affect the unit product’s cost. Therefore, in this paper we consider a profit maximizing firm who wants to jointly determine the optimal lot-sizing, pricing, and marketing decisions along with manufacturing requirements in terms of flexibility and reliability of the process. Geometric programming (GP) technique is proposed to address the resulting nonlinear optimization problem. Using recent advances in optimization techniques we are able to optimally solve the developed, highly nonlinear, mathematical model. Finally, using numerical examples, we illustrate the solution approach and analyze the solution under different conditions.     

A cross entropy-Lagrangean hybrid algorithm for the multi-item capacitated lot-sizing problem with setup times

The aim of this article is to introduce a hybrid algorithm for the multi-item multi-period capacitated lot-sizing problem with setups. In this problem, demands over a finite planning horizon must be met, where several items compete for space with limited resources in each period, and a portion of these resources is used by setups. The proposed scheme considers a Lagrangean relaxation of the problem and applies a cross entropy-based metaheuristic to the uncapacitated version of the original problem. A thorough experimental plan has been designed and implemented to test the effectiveness and the robustness of the algorithm: first, drawing inspiration from the response surface methodology, we calibrate the algorithm by identifying the optimal parameters value for any given instance size. Next, we carry out experiments on large scale instances, collecting information about solution quality and computational time, and comparing these results with those offered by a global optimizer.     

Optimal control laws for lot-sizing and timing of jobs on a single production facility

The optimal control of a single machine processing a certain number of jobs and modeled as a discrete-event dynamic system is considered. The number of jobs and their sequence are fixed, whereas their timing and sizes represent the control variables of the system. The objective function to be optimized is a weighted sum of the quadratic earliness and tardiness of each job, and of the quadratic deviations of job lot sizes and actual machine service speeds from those specified by the production demand and by the regular machine speeds. An optimization problem with quadratic cost function and nonlinear constraints is stated and formalized as a multistage optimal control problem. Necessary conditions to be satisfied by an optimal control sequence are derived. A simpler model is also considered in which the machine speeds are fixed; in this case, the control problem is solved by a procedure making use of dynamic programming techniques. The optimal control laws at each stage are thus obtained.     

A Lagrangean-based heuristic for multi-plant,multi-item,multi-period capacitated lot-sizing problems with inter-plant transfers

This paper addresses scheduling of lot sizes in a multi-plant, multi-item, multi-period, capacitated environment with inter-plant transfers. A real-world problem in a company manufacturing steel rolled products provided motivation to this research. A Lagrangean-based approach, embedded with a lot shifting–splitting–merging routine, has been used for solving the multi-plant, capacitated lot-sizing problem. A “good” solution procedure developed by Sambasivan (Ph.D. Dissertation, University of Alabama, Tuscaloosa, 1994) has been used for solving the relaxed problem. About 120 randomly generated instances of the problem have been solved and it has been found that Lagrangean-based approach works quite “efficiently” for this problem.     

Grouping in decomposition method for multi-item capacitated lot-sizing problem with immediate lost sales and joint and item-dependent setup cost

This article examines a dynamic and discrete multi-item capacitated lot-sizing problem in a completely deterministic production or procurement environment with limited production/procurement capacity where lost sales (the loss of customer demand) are permitted. There is no inventory space capacity and the production activity incurs a fixed charge linear cost function. Similarly, the inventory holding cost and the cost of lost demand are both associated with a linear no-fixed charge function. For the sake of simplicity, a unit of each item is assumed to consume one unit of production/procurement capacity. We analyse a different version of setup costs incurred by a production or procurement activity in a given period of the planning horizon. In this version, called the joint and item-dependent setup cost, an additional item-dependent setup cost is incurred separately for each produced or ordered item on top of the joint setup cost.     

Pricing for production and delivery flexibility in single-item lot-sizing

Adjusting prices to influence demand so as to increase revenue has become common practice. We investigate adjusting prices to influence demand so as to reduce cost. More specifically, we consider offering price discounts in return for production and delivery flexibility. We do so in the context of the single-item, single-level uncapacitated lot-sizing problem. We show that even though the resulting optimization problem has a nonlinear objective function it can still be solved in polynomial time. Furthermore, we report results of a computational study analyzing the benefits of offering price discounts in return for production and delivery flexibility in various settings.     

Group technology inventories with optimal production costs

A model of group technology (GT) production systems including the cost of raw material processing is developed. It is assumed that the throughput time in a given GT cell depends linearly on the lot size. The optimal GT lot–size formula is derived and its properties discussed. The variability with respect to changes in the annual demand rate, the average carrying cost rate, and the rate charged per unit of cell production time are examined and evaluated. A typical case study is worked out to illustrate the analytical results.     

Optimal manufacturer's pricing and lot-sizing policies under trade credit financing

In this paper, we extend Goyal's economic order quantity (EOQ) model to allow for the following four important facts: (1) the manufacturer's selling price per unit is necessarily higher than its unit cost, (2) the interest rate charged by a bank is not necessarily higher than the manufacturer's investment return rate, (3) the demand rate is a downward‐sloping function of the price, and (4) an economic production quantity (EPQ) model is a generalized EOQ model. We then establish an appropriate EPQ model accordingly, in which the manufacturer receives the supplier trade credit and provides the customer trade credit simultaneously. As a result, the proposed model is in a general framework that includes numerous previous models as special cases. Furthermore, we provide an easy‐to‐use closed‐form optimal solution to the problem for any given price. Finally, we develop an algorithm for the manufacturer to determine its optimal price and lot size simultaneously.     

Impact of forecasting error on the performance of capacitated multi-item production systems

A computer model is built to simulate master production scheduling activities in a capacitated multi-item production system under demand uncertainty and a rolling time horizon. The output from the simulation is analyzed through statistical software. The results of the study show that forecasting errors have significant impacts on total cost, schedule instability and system service level, and the performance of forecasting errors is significantly influenced by some operational factors, such as capacity tightness and cost structure. Furthermore, the selection of the master production schedule freezing parameters is also significantly influenced by forecasting errors. The findings from this study can help managers optimize their production plans by selecting more reasonable forecasting methods and scheduling parameters, thus improving the performance of production systems.     

Optimal pricing and lot-sizing for perishable inventory with price and time dependent ramp-type demand

Product perishability is an important aspect of inventory control. To minimise the effect of deterioration, retailers in supermarkets, departmental store managers, etc. always want higher inventory depletion rate. In this article, we propose a dynamic pre- and post-deterioration cumulative discount policy to enhance inventory depletion rate resulting low volume of deterioration cost, holding cost and hence higher profit. It is assumed that demand is a price and time dependent ramp-type function and the product starts to deteriorate after certain amount of time. Unlike the conventional inventory models with pricing strategies, which are restricted to a fixed number of price changes and to a fixed cycle length, we allow the number of price changes before as well as after the start of deterioration and the replenishment cycle length to be the decision variables. Before start of deterioration, discounts on unit selling price are provided cumulatively in successive pricing cycles. After the start of deterioration, discounts on reduced unit selling price are also provided in a cumulative way. A mathematical model is developed and the existence of the optimal solution is verified. A numerical example is presented, which indicates that under the cumulative effect of price discounting, dynamic pricing policy outperforms static pricing strategy. Sensitivity analysis of the model is carried out.     

Multiple lot-sizing decisions with an interrupted geometric yield and variable production time

This study examines a multiple lot-sizing problem for a single-stage production system with an interrupted geometric distribution, which is distinguished in involving variable production lead-time. In a finite number of setups, this study determined the optimal lot-size for each period that minimizes total expected cost. The following cost items are considered in optimum lot-sizing decisions: setup cost, variable production cost, inventory holding cost, and shortage cost. A dynamic programming model is formulated in which the duration between current time and due date is a stage variable, and remaining demand and work-in-process status are state variables. This study then presents an algorithm for solving the dynamic programming problem. Additionally, this study examines how total expected costs of optimal lot-sizing decisions vary when parameters are changed. Numerical results show that the optimum lot-size as a function of demand is not always monotonic.     

Optimal lot-sizing policy for a manufacturer with defective items in a supply chain with up-stream and down-stream trade credits

In this paper, we establish an economic production quantity model for a manufacturer (or wholesaler) with defective items when its supplier offers an up-stream trade credit M while it in turn provides its buyers (or retailers) a down-stream trade credit N. The proposed model is in a general framework that includes numerous previous models as special cases. In contrast to the traditional differential calculus approach, we use a simple-to-understand and easy-to-apply arithmetic–geometric inequality method to find the optimal solution. Furthermore, we provide some theoretical results to characterize the optimal solution. Finally, several numerical examples are presented to illustrate the proposed model and the optimal solution.     

A volume flexible economic production lot-sizing problem with imperfect quality and random machine failure in fuzzy-stochastic environment

An “economic production lot size” (EPLS) model for an item with imperfect quality is developed by considering random machine failure. Breakdown of the manufacturing machines is taken into account by considering its failure rate to be random (continuous). The production rate is treated as a decision variable. It is assumed that some defective units are produced during the production process. Machine breakdown resulting in idle time of the respective machine which leads to additional cost for loss of manpower is taken into account. It is assumed that the production of the imperfect quality units is a random variable and all these units are treated as scrap items that are completely wasted. The models have been formulated as profit maximization problems in stochastic and fuzzy-stochastic environments by considering some inventory parameters as imprecise in nature. In a fuzzy-stochastic environment, using interval arithmetic technique, the interval objective function has been transformed into an equivalent deterministic multi-objective problem. Finally, multi-objective problem is solved by Global Criteria Method (GCM). Stochastic and fuzzy-stochastic problems and their significant features are illustrated by numerical examples. Using the result of the stochastic model, sensitivity of the nearer optimal solution due to changes of some key parameters are analysed.     

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