Lot-sizing two-echelon assembly systems with random yields and rigid demand

We consider a two-echelon assembly system producing a single final product for which the demand is known. The first echelon consists of several parallel stages, whereas the second echelon consists of a single assembly stage. We assume that the yield at each stage is random and that demand needs to be satisfied in its entirety; thus, several production runs may be required. A production policy should specify, for each possible configuration of intermediate inventories, on which stage to produce next and the lot size to be processed. The objective is to minimize the expected total of setup and variable production costs.We prove that the expected cost of any production policy can be calculated by solving a finite set of linear equations whose solution is unique. The result is general in that it applies to any yield distribution. We also develop efficient algorithms leading to heuristic solutions with high precision and, as an example, provide numerical results for binomial yields.     

Determining Net Requirements for Material Requirements Planning

This paper presents a design guideline for netting systems that determine net requirements, for push-type production ordering. Two alternative netting systems (full and single) are formulated in an N stage production and inventory system. The performances of the two alternatives are discussed through numerically analysing production ordering variations and inventory level variations at each stage in the processes. The results obtained suggest that: (1) the full netting system is preferable at the production stages, where variance in the accumulated forecast consumption error by the immediately succeeding stage is smaller than the variance in the accumulated forecast market demand error over a production ordering interval and the lead time for the stage. (2) The single netting system may be preferable at the other stages, particularly in regard to production ordering variations.     

Multi-plant production planning in capacitated self-configuring two-stage serial systems

This research deals with a capacitated master production planning and capacity allocation problem for a multi-plant manufacturing system with two serial stages in each plant. We consider both cases in which a decoupling buffer is allowed or not between the two stages. Peculiar to the system considered is the capability of dynamically self-configuring its layout when buffering is disallowed, in the sense that, for each production run, different parallel machines in the second stage are grouped together and serially connected to a machine in the first stage. Although setup times and costs are considered negligible in our model, yet binary setup variables are introduced in order to account for minimum lot-sizes. The resulting mixed {0,1} linear programming model is solved by means of LP-based heuristic algorithms. The proposed modeling and solution procedure has been applied to problem instances originating from a real-world application, showing good results in practice.     

Numerical Approach of Multi-Objective Optimal Control Problem in Imprecise Environment

In this paper, realistic production-inventory models without shortages for deteriorating items with imprecise holding and production costs for optimal production have been formulated. Here, the rate of production is assumed to be a function of time and considered as a control variable. Also the demand is time dependent and known. The imprecise holding and production costs are assumed to be represented by fuzzy numbers which are transformed to corresponding interval numbers. Following interval mathematics, the objective function is changed to respective multi-objective functions and thus the single-objective problem is reduced to a multi-objective decision making(MODM) problem. The MODM problem is then again transformed to a single objective function with the help of weighted sum method and then solved using global criteria method, calculus method, the Kuhn–Tucker conditions and generalized reduced gradient(GRG) technique. The models have been illustrated by numerical data. The optimum results for different objectives are obtained for different types of production function. Numerical values of demand, production function and stock level are presented in both tabular and graphical forms     

Optimizing a Production System with a Fixed Delivery Schedule

This note considers a model in which a manufacturing company purchases a raw material, manufactures a product (at a finite rate) and ships a fixed quantity of the product to a single customer at fixed and regular intervals of time, as specified by the customer. In general there are several shipments made during each production run. The objective is to determine a purchasing and production schedule which minimises the total cost of purchasing, manufacturing and stockholding. It extends previously published work by considering the possibility that a single raw material purchase provides stock for several production runs or that several raw material purchases provide stock for a single production run.     

An Application of Travelling-Salesman Routines to Solve Pattern-Allocation Problems in the Glass Industry

This paper focusses on an often encountered constraint in real-life cutting-stock problems. The constraints require that pieces corresponding to the same order are not spread too much over the production run. This elimination of order spread is called pattern allocation or cutting sequencing. In this paper, a two-stage procedure to solve the two-dimensional pattern-allocation problem is suggested. The first stage consists of solving the cutting-stock problem without the sequencing constraint. In the second stage a sequencing problem is used for the ordering of the cutting patterns in an optimal or near-optimal way. The sequencing problem is formulated as a travelling-salesman model, and the model is solved by Lin's 3-optimal method. Computational experience is reported from a case study in the glass industry.     

Production Planning and Inventory Control on a Chemical Plant

In this paper formulae are derived to give the optimum run lengths on a plant producing two products alternately. The general case of n runs of one product to one run of the other is considered. The work is applied to a chemical plant and the difficulties of determining certain costs are discussed. It is estimated that appreciable savings can be made.An empirical method of calculating buffer stocks is then described, together with a method of planning production before an annual shut-down. This part of the work does not lead to any substantial savings in cost, but reduces the danger of running out of stock.     

Optimal Production and Setup Scheduling: A One-Machine, Two-Product System

This paper studies the scheduling problem for two products on a single production facility. The objective is to specify a production and setup policy that minimizes the average inventory, backlog, and setup costs. Assuming that the production rate can be adjusted during the production runs, we provide a close form for an optimal production and setup schedule. Dynamic programming and Hamilton–Jacobi–Bellman equation is used to verify the optimality of the obtained policy.     

The single-server scheduling problem with convex costs

Being probably one of the oldest decision problems in queuing theory, the single-server scheduling problem continues to be a challenging one. The original formulations considered linear costs, and the resulting policy is puzzling in many ways. The main one is that, either for preemptive or nonpreemptive problems, it results in a priority ordering of the different classes of customers being served that is insensitive to the individual load each class imposes on the server and insensitive to the overall load the server experiences. This policy is known as the cμ-rule. We claim and show that for convex costs, the optimal policy depends on the individual loads. Therefore, there is a need for an alternative generalization of the cμ-rule. The main feature of our generalization consists on first-order differences of the single stage cost function, rather than on its derivatives. The resulting policy is able to reach near optimal performances and is a function of the individual loads.     

Multiple Lotsizing in Production to Order with Random Yields: Review of Recent Advances

This article provides a review of models, analytical results and insights pertaining to multiple lotsizing in production to order, with emphasis on recent work on multistage systems and inspection issues. The basic setting consists of an order of fixed size that has to be met in its entirety. Production is in lots and is imperfect, and inspection can take place only when lot has completed a stage. Processing each lot at each stage entails fixed and variable costs. The tradeoff is between running large lots, entailing potentially costly overproduction, and small lots, which may necessitate multiple production runs and their associated setups. Variations of this problem have been explored since the 1950's, but only recently has there been significant progress in solving exactly problems of realistic complexity. We first review single-stage models, including those which compute the variance of costs, those dealing with inspections, those with multiple grades and situations with uncertain demand. We then review multi-stage systems, starting with those with no bottlenecks, and progressing to one, and two bottlenecks. The implications of rework capability in the various systems are then explored.     

Coordination in decentralized assembly systems with uncertain component yields

The literature on assembly systems with random component yields has focused on centralized systems, where a single decision maker chooses all components’ production quantities and incurs all the costs. We consider a decentralized setting where the component suppliers choose their production quantities based solely on their own cost/reward structure, and the assembly firm makes ordering decisions based on its own cost/reward structure. When the suppliers control their inputs but the outputs exhibit random yields, coordination in such systems becomes quite complex. In such situations, incentive alignment control mechanisms are needed so that the suppliers will choose production quantities as in the centralized system case. One such mechanism is to penalize the supplier with the worse delivery performance. We analyze the conditions under which system coordination is achieved while respecting participation constraints. Further, we determine the optimal component ordering policy for the assembly firm and derive the optimal coordinating penalties.     

Optimal Ordering Policies for Inventory Systems with Emergency Ordering

This paper considers dynamic single- and multi-product inventory problems in which the demands in each period are independent and identically distributed random variables. The problems considered have the following common characteristics. At the beginning of each period two order quantities are determined for each product. A “normal order” quantity with a constant positive lead time of λ _n periods and an “emergency order” quantity with a lead time of λ _e periods, where λ _e = λ _n - 1. The ordering decisions are based on linear procurement costs for both methods of ordering and convex holding and penalty costs. The emergency ordering costs are assumed to be higher than the normal ordering costs. In addition, future costs are discounted.For the single-product problem the optimal ordering policy is shown to be the same for all periods with the exception of the last period in the N-period problem. For the multi-product problem the one- and N-period optimal ordering policy is characterized where it is assumed that there are resource constraints on the total amount that can be ordered or produced in each period.     

A fair method for resetting the target in interrupted one-day cricket matches

A method is described for setting revised target scores for the team batting second when a limited-overs cricket match has been forcibly shortened after it has commenced. It is designed so that neither team benefits or suffers from the shortening of the game and so is totally fair to both. It is easy to apply, requring nothing more than a single table of numbers and a pocket calculator, and is capable of dealing with any number of interruptions at any stage of either or both innings.The method is based on a simple model involving a two-factor relationship giving the number of runs which can be scored on average in the remainder of an innings as a function of the number of overs remaining and the number of wickets fallen. It is shown how the relationship enables the target score in an interrupted match to be recalculated to reflect the relative run scoring resources available to the two teams, that is overs and wickets in combination. The method was used in several international and domestic one-day competitions and tournaments in 1997.     

Real time disruption management for a two-stage batch production–inventory system with reliability considerations

In this research, a two-stage batch production–inventory system is introduced. In this system, the production may be disrupted, for a given period of time, either at one or both stages. In this paper, firstly, a mathematical model has been developed to suggest a recovery plan for a single occurrence of disruption at either stage. Secondly, multiple disruptions have been considered, for which a new disruption may or may not affect the recovery plan of earlier disruptions. We propose a new approach that deals with a series of disruptions over a period of time, which can be implemented for disruption recovery on a real time basis. In this approach, the model formulated for single disruption has been integrated to generate initial solutions for individual disruptions and the solutions have been revised for multiple dependent disruptions with changed parameters. With the proposed approach, an optimal recovery plan can be obtained in real time, whenever the production system experiences either a sudden disruption or a series of disruptions, at different points in time. Some numerical examples and a real-world case study are presented to explain the benefits of our proposed approach.     

To run or not?: Some dynamic programming models in cricket

In cricket, particularly near the end of an innings, batsmen of different abilities need to manage the rate at which they score runs. Either batsman can choose to bat aggressively or defensively, which alters their chances of scoring runs or being dismissed. Since they change ends when they score a run and at the end of an over, by scoring an odd or even number of runs the two batsmen also determine which of them will face the next ball. It may be worthwhile to refuse a run to keep the slower or lower scoring batsman from the strike. Some dynamic programming models are developed which could be used to maximise the total number of runs scored.     

Inventory control of service parts in the final phase

We consider an appliance manufacturer's problem of controlling the inventory of a service part in its final phase. That phase begins when the production of the appliance containing that part is discontinued (time 0), and ends when the last service contract on that appliance expires. Thus, the planning horizon is deterministic and known. There is no setup cost for ordering. However, if a part is not ordered at time 0, its price will be higher. The objective is to minimize the total expected undiscounted costs of replenishment, inventory holding, backorder, and disposal (of unused parts at the end of the planning horizon). We propose an ordering policy consisting of an initial order-up-to level at time 0, and a subsequent series of decreasing order-up-to levels for various intervals of the planning horizon. We present a method of calculating the optimal policy, along with a numerical example and sensitivity analysis.     

Optimizing Multi-stage Production for an Assembly-Type Supply Chain with Unequal Sized Batch Shipments

This paper studies the multi-stage logistics and inventory problem in an?assembly-type supply chain where a uniform lot size is produced uninterruptedly with a single setup at each stage. Partial lots, or sub-batches, can be transported to next stage upon completion. Unequal sub-batch sizes at each stage follow geometric series and the numbers of sub-batches across stages are allowed to be different. Since the mainline and each branch line of an assembly-type supply chain are series-type supply chains, a model of the series-type supply chain is first established and a model of the assembly-type supply chain is subsequently developed. Optimization algorithms that determine the economic lot sizes, the optimal sub-batch sizes and the number of sub-batches for each stage are developed. The polynomial-time algorithms incorporate the optimality properties derived in the paper to find the lower and upper bounds of the solutions by constructing the solution ranges and then the optimal solutions accordingly.     

Order Quantities with Temporary Price Reductions

The temporary price-change problem is studied, in which the objective is to minimize discounted cash flows. As pointed out by Goyal in an earlier paper, only the cash transactions at purchase times (i.e. the payments for the goods and the ordering costs) were considered. The cash flows associated with `inventory maintenance' costs which occur more or less continuously over time were neglected, which changes the structure of the model. Examples of these costs include storage, insurance, record-keeping, deterioration and obsolescence costs. In this paper, these continuously generated cash flows are included in the analysis, thereby making the new model more applicable to practical situations. This model is of interest because order-quantity decisions often must be made under conditions of both temporary price reductions and/or imminent price increases. These changes occur frequently in practice.     

A production planning model for an unreliable production facility: Case of finite horizon and single demand

We study a two-level inventory system that is subject to failures and repairs. The objective is to minimize the expected total cost so as to determine the production plan for a single quantity demand. The expected total cost consists of the inventory carrying costs for finished and unfinished items, the backlog cost for not meeting the demand due-date, and the planning costs associated with the ordering schedule of unfinished items. The production plan consists of the optimal number of lot sizes, the optimal size for each lot, the optimal ordering schedule for unfinished items, and the optimal due-date to be assigned to the demand. To gain insight, we solve special cases and use their results to device an efficient solution approach for the main model. The models are solved to optimality and the solution is either obtained in closed form or through very efficient algorithms.     

Power generator scheduling by dynamic programming

Dynamic programming is applied to the problem of determining an optimal short-run schedule for a series of power generating units to meet a time varying system load. Typical scheduling procedures include a priority ordering constraint which requires that units be committed to or removed from power production in a fixed order. A dynamic programming model which relaxes this constraints yields a large combinatorial problem whose state space depends on the number of feasible combinations of generating units. Reduction of storage requirements is achieved through the implementation of data structure techniques. An example of the dynamic programming procedure with data from an actual system suggests significant coast reductions. A comparison with a case in the literature also results in significant savings. Perhaps more important, however, are the substantial energy savings, in light of the current emphasis on energy conservation.