Optimal production control of a dynamic two-product manufacturing system with setup costs and setup times

This paper deals with the optimal control of a one-machine two-product manufacturing system with setup changes, operating in a continuous time dynamic environment. The system is deterministic. When production is switched from one product to the other, a known constant setup time and a setup cost are incurred. Each product has specified constant processing time and constant demand rate, as well as an infinite supply of raw material. The problem is formulated as a feedback control problem. The objective is to minimize the total backlog, inventory and setup costs incurred over a finite horizon. The optimal solution provides the optimal production rate and setup switching epochs as a function of the state of the system (backlog and inventory levels). For the steady state, the optimal cyclic schedule is determined. To solve the transient case, the system's state space is partitioned into mutually exclusive regions such that with each region, the optimal control policy is determined analytically.     

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.     

Average Cost Optimality for an Unreliable Two-Machine Flowshop with Limited Internal Buffer

We consider a production planning problem in a two-machine flowshop subject to breakdown and repair of machines and subject to nonnegativity and upper bound constraints on work-in-process. The objective is to choose machine production rates over time to minimize the long-run average inventory/backlog and production costs. For sufficiently large upper bound on the work-in-process, the problem is formulated as a stochastic dynamic program. We then establish a verification theorem and a partial characterization of the optimal control policy if it exists.     

Two-mode control of Brownian Motion with quadratic loss and switching costs

We consider a simple problem in the optimal control of Brownian Motion. There are two modes of control available, each with its own drift and diffusion coefficients, and switching costs are incurred whenever the control mode is changed. Finally, holding costs are incurred according to a quadratic function of the state of the system, and all costs are continuously discounted. It is shown that there exists an optimal policy involving just two critical numbers, and formulas are given for computation of the critical numbers.     

Optimal production planning in pull flow lines with multiple products

The paper is concerned with the problem of optimal production planning in deterministic pull flow lines with multiple products. The objective is to specify the production policy that minimizes the total inventory and backlog costs overtime. Assuming constant product demands and non-decreasing unit holding costs along the flow, an algorithm which obtains the optimal production policy is developed. This algorithm works for the discounted-cost function as well. The HJB equation is used to verify the optimality of the policy, and the computational complexity of the algorithm is discussed. Some illustrative examples are also included.     

Two-product inventory management with fixed costs and supply uncertainty

This paper determines the optimal ordering policy for a two-product, periodic-review inventory problem in which the probability of supply availability is unknown. Moreover, there are two different fixed costs assigned to each product. Demand rates are random variables with known probability density functions, and the supply availability for each product is updated at the beginning of each time period. We prove the optimality of (s,S) policy with a monotone switching curve that indicates the priority of production, where the order-up-to levels and the reorder points are functions of supply availability information. A simple computation is proposed to calculate the two threshold levels. Bayesian updating helps to manage the optimal ordering policy by updating supply disruption information. Numerical results show that improving the accuracy of the forecast leads to making a better ordering decision and eliminating the negative effect of supply disruption on the total cost.     

An impulse control of a geometric Brownian motion with quadratic costs

We examine an optimal impulse control problem of a stochastic system whose state follows a geometric Brownian motion. We suppose that, when an agent intervenes in the system, it requires costs consisting of a quadratic form of the system state. Besides the intervention costs, running costs are continuously incurred to the system, and they are also of a quadratic form. Our objective is to find an optimal impulse control of minimizing the expected total discounted sum of the intervention costs and running costs incurred over the infinite time horizon. In order to solve this problem, we formulate it as a stochastic impulse control problem, which is approached via quasi-variational inequalities (QVI). Under a suitable set of sufficient conditions on the given problem parameters, we prove the existence of an optimal impulse control such that, whenever the system state reaches a certain level, the agent intervenes in the system. Consequently it instantaneously reduces to another level.     

Variance minimization for constrained discounted continuous-time MDPs with exponentially distributed stopping times

This paper deals with minimization of the variances of the total discounted costs for constrained Continuous-Time Markov Decision Processes (CTMDPs). The costs consist of cumulative costs incurred between jumps and instant costs incurred at jump epochs. We interpret discounting as an exponentially distributed stopping time. According to existing theory, for the expected total discounted costs optimal policies exist in the forms of randomized stationary and switching stationary policies. While the former is typically unique, the latter forms a finite set whose number of elements grows exponentially with the number of constraints. This paper investigates the problem when the process stops immediately after the first jump. For costs up to the first jump we provide an index for selection of actions by switching stationary policies and show that the indexed switching policy achieves a smaller variance than the randomized stationary policy. For problems without instant costs, the indexed switching policy achieves the minimum variance of costs up to the first jump among all the equivalent switching policies.     

Capacity and Production Managment in a Single Product Manufacturing System

In planning and managing production systems, manufacturers have two main strategies for responding to uncertainty: they build inventory to hedge against periods in which the production capacity is not sufficient to satisfy demand, or they temporarily increase the production capacity by “purchasing” extra capacity. We consider the problem of minimizing the long-run average cost of holding inventory and/or purchasing extra capacity for a single facility producing a single part-type and assume that the driving uncertainty is demand fluctuation. We show that the optimal production policy is of a hedging point policy type where two hedging levels are associated with each discrete state of the system: a positive hedging level (inventory target) and a negative one (backlog level below which extra capacity should be purchased). We establish some ordering of the hedging levels, derive equations satisfied by the steady-state probability distribution of the inventory/backlog, and give a more detailed analysis of the optimal control policy in a two state (high and low demand rate) model.     

Flexible servers in tandem lines with setup costs

We study the dynamic assignment of flexible servers to stations in the presence of setup costs that are incurred when servers move between stations. The goal is to maximize the long-run average profit. We provide a general problem formulation and some structural results, and then concentrate on tandem lines with two stations, two servers, and a finite buffer between the stations. We investigate how the optimal server assignment policy for such systems depends on the magnitude of the setup costs, as well as on the homogeneity of servers and tasks. More specifically, for systems with either homogeneous servers or homogeneous tasks, small buffer sizes, and constant setup cost, we prove the optimality of “multiple threshold” policies (where servers’ movement between stations depends on both the number of jobs in the system and the locations of the servers) and determine the values of the thresholds. For systems with heterogeneous servers and tasks, small buffers, and constant setup cost, we provide results that partially characterize the optimal server assignment policy. Finally, for systems with larger buffer sizes and various service rate and setup cost configurations, we present structural results for the optimal policy and provide numerical results that strongly support the optimality of multiple threshold policies.     

A Hybrid Genetic Algorithm for the Single Machine Scheduling Problem

A hybrid genetic algorithm (HGA) is proposed for the single machine, single stage, scheduling problem in a sequence dependent setup time environment within a fixed planning horizon (SSSDP). It incorporates the elitist ranking method, genetic operators, and a hill-climbing technique in each searching area. To improve the performance and efficiency, hill climbing is performed by uniting the Wagner-Whitin Algorithm with the problem-specific knowledge. The objective of the HGA is to minimize the sum of setup cost, inventory cost, and backlog cost. The HGA is able to obtain a superior solution, if it is not optimal, in a reasonable time. The computational results of this algorithm on real life SSSDP problems are promising. In our test cases, the HGA performed up to 50% better than the Just-In-Time heuristics and 30% better than the complete batching heuristics.     

Optimal policy for brownian inventory models with general convex inventory cost

We study an inventory system in which products are ordered from outside to meet demands, and the cumulative demand is governed by a Brownian motion. Excessive demand is backlogged. We suppose that the shortage and holding costs associated with the inventory are given by a general convex function. The product ordering from outside incurs a linear ordering cost and a setup fee. There is a constant leadtime when placing an order. The optimal policy is established so as to minimize the discounted cost including the inventory cost and ordering cost.     

A two-level hedging point policy for controlling a manufacturing system with time-delay,demand uncertainty and extra capacity

This paper focuses on the production control of a manufacturing system with time-delay, demand uncertainty and extra capacity. Time-delay is a typical feature of networked manufacturing systems (NMS), because an NMS is composed of many manufacturing systems with transportation channels among them and the transportation of materials needs time. Besides this, for a manufacturing system in an NMS, the uncertainty of the demand from its downstream manufacturing system is considered; and it is assumed that there exist two-levels of demand rates, i.e., the normal one and the higher one, and that the time between the switching of demand rates are exponentially distributed. To avoid the backlog of demands, it is also assumed that extra production capacity can be used when the work-in-process (WIP) cannot buffer the high-level demands rate. For such a manufacturing system with time-delay, demand uncertainty and extra capacity, the mathematical model for its production control problem is established, with the objective of minimizing the mean costs for WIP inventory and occupation of extra production capacity. To solve the problem, a two-level hedging point policy is proposed. By analyzing the probability distribution of system states, optimal values of the two hedging levels are obtained. Finally, numerical experiments are done to verify the effectiveness of the control policy and the optimality of the hedging levels.     

Optimal production control of a failure-prone machine

We consider a problem of optimal production control of a single unreliable machine. The objective is to minimize a discounted convex inventory/backlog cost over an infinite horizon. Using the variational analysis methodology, we develop the necessary conditions of optimality in terms of the co-state dynamics. We show that an inventory-threshold control policy is optimal when the work and repair times are exponentially distributed, and demonstrate how to find the value of the threshold in this case. We consider also a class of distributions concentrated on finite intervals and prove properties of the optimal trajectories, as well as properties of an optimal inventory threshold that is time dependent in this case.     

Using aggregate fill rate for dynamic scheduling of multi-class systems

For dynamic scheduling of multi-class systems where backorder cost is incurred per unit backordered regardless of the time needed to satisfy backordered demand, the following models are considered: the cost model to minimize the sum of expected average inventory holding and backorder costs and the service model to minimize expected average inventory holding cost under an aggregate fill rate constraint. Use of aggregate fill rate constraint in the service model instead of an individual fill rate constraint for each class is justified by deriving equivalence relations between the considered cost and service models. Based on the numerical investigation that the optimal policy for the cost model is a base-stock policy with switching curves and fixed base-stock levels, an alternative service model is considered over the class of base-stock controlled dynamic scheduling policies to minimize the total inventory (base-stock) investment under an aggregate fill rate constraint. The policy that solves this alternative model is proposed as an approximation of the optimal policy of the original cost and the equivalent service models. Very accurate heuristics are devised to approximate the proposed policy for given base-stock levels. Comparison with base-stock controlled First Come First Served (FCFS) and Longest Queue (LQ) policies and an extension of LQ policy (Δ policy) shows that the proposed policy performs much better to solve the service models under consideration, especially when the traffic intensity is high.     

Analysis of the Best Double Frequency Policy in the Single Link Problem with Discrete Shipping Times

We study an inventory–transportation problem where one product has to be shipped from an origin to a destination by vehicles of given capacity over an infinite time horizon. The product is made available at the origin and consumed at the destination at the same constant rate. The intershipment time must be not lower than a given minimum value. The problem is to decide when to make the shipments and how to load the vehicles to minimize the sum of the transportation and the inventory costs at the origin and at the destination per time unit. We study the case in which the intershipment time is a multiple of the minimum value, i.e., the problem with discrete shipping times. We show that, in this case, the best double frequency policy has a tight performance bound of about 1.1603 with respect to the optimal periodic policy and of about 1.1538 with respect to the best frequency-based policy. Moreover, we show that, from the worst-case point of view, the best double frequency policy is the optimal frequency-based policy.     

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.     

Coordinated control of distribution supply chains in the presence of fuzzy customer demand

This paper considers a single product inventory control in a Distribution Supply Chain (DSC). The DSC operates in the presence of uncertainty in customer demands. The demands are described by imprecise linguistic expressions that are modelled by discrete fuzzy sets. Inventories at each facility within the DSC are replenished by applying periodic review policies with optimal order up-to-quantities. Fuzzy customer demands imply fuzziness in inventory positions at the end of review intervals and in incurred relevant costs per unit time interval. The determination of the minimum of defuzzified total cost of the DSC is a complex problem which is solved by applying decomposition; the original problem is decomposed into a number of simpler independent optimisation subproblems, where each retailer and the warehouse determine their optimum periodic reviews and order up-to-quantities. An iterative coordination mechanism is proposed for changing the review periods and order up-to-quantities for each retailer and the warehouse in such a way that all parties within the DSC are satisfied with respect to total incurred costs per unit time interval. Coordination is performed by introducing fuzzy constraints on review periods and fuzzy tolerances on retailers and warehouse costs in local optimisation subproblems.     

Analysis of a newsvendor which has errors in inventory data records

Although supply chain scholars very often assume the availability of error free data pertaining to the flow of goods that come in and go out of an inventory system as well as the on hand inventory level, some recent investigations show that this may not be true even in facilities where advanced item identification and data capture technologies such as the barcode system are used. This paper proposes a single period model where the inventory data capture process using the barcode system is prone to errors that lead to inaccuracies. In the first part of our work, we derive analytically the optimal policy in presence of errors when both demand and errors are uniformly distributed. In the second part, we examine quantitatively the impact of record inaccuracies on the performance of an inventory system, in terms of additional overage and shortage costs incurred.