Transient and steady-state analysis of a manufacturing system with setup changes

This paper deals with the optimal scheduling of a one-machine two-product manufacturing system with setup, operating in a continuous time dynamic environment. The machine is reliable. A known constant setup time is incurred when switching over from a part to the other. Each part has specified constant processing time and constant demand rate, as well as an infinite supply of raw material. The problem is formulated as a production flow control problem. The objective is to minimize the sum of the backlog and inventory costs incurred over a finite planning horizon. The global optimal solution, expressed as an optimal feedback control law, 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 (Limit Cycle) is determined. This is equivalent to solving a one-machine two-product Lot Scheduling Problem. To solve the transient case, the system's state space is partitioned into mutually exclusive regions such that with each region is associated an optimal control policy. A novel algorithm (Direction Sweeping Algorithm) is developed to obtain the optimal state trajectory (optimal policy that minimizes the sum of inventory and backlog costs) for this last case.     

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.     

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.     

Admission policies for the customized stochastic lot scheduling problem with strict due-dates

This papers considers admission control and scheduling of customer orders in a production system that produces different items on a single machine. Customer orders drive the production and belong to product families, and have family dependent due-date, size, and reward. When production changes from one family to another a setup time is incurred. Moreover, if an order cannot be accepted, it is considered lost upon arrival. The problem is to find a policy that accepts/rejects and schedules orders such that long run profit is maximized. This problem finds its motivation in batch industries in which suppliers have to realize high machine utilization while delivery times should be short and reliable and the production environment is subject to long setup times.We model the joint admission control/scheduling problem as a Markov decision process (MDP) to gain insight into the optimal control of the production system and use the MDP to benchmark the performance of a simple heuristic acceptance/scheduling policy. Numerical results show that the heuristic performs very well compared with the optimal policy for a wide range of parameter settings, including product family asymmetries in arrival rate, order size, and order reward.     

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.     

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.     

On the Effect of Product Variety in Production–Inventory Systems

In this paper, we examine the effect of product variety on inventory costs in a production–inventory system with finite capacity where products are made to stock and share the same manufacturing facility. The facility incurs a setup time whenever it switches from producing one product type to another. The production facility has a finite production rate and stochastic production times. In order to mitigate the effect of setups, products are produced in batches. In contrast to inventory systems with exogenous lead times, we show that inventory costs increase almost linearly in the number of products. More importantly, we show that the rate of increase is sensitive to system parameters including demand and process variability, demand and capacity levels, and setup times. The effect of these parameters can be counterintuitive. For example, we show that the relative increase in cost due to higher product variety is decreasing in demand and process variability. We also show that it is decreasing in expected production time. On the other hand, we find that the relative cost is increasing in expected setup time, setup time variability and aggregate demand rate. Furthermore, we show that the effect of product variety on optimal base stock levels is not monotonic. We use the model to draw several managerial insights regarding the value of variety-reducing strategies such as product consolidation and delayed differentiation.     

The Common Cycle Economic Lot Scheduling Problem with Backorders: Benefits of Controllable Production Rates

In this paper, we deal with the production scheduling ofseveral products that are produced periodically, in a fixed sequence, ona single machine. In the literature, this problem is usually referred to asthe Common Cycle Economic Lot Scheduling Problem. We extend thelatter to allow the production rates to be controllable at the beginningof as well as during each production run of a product. Also, we assumethat unsatisfied demand is completely backordered. The objective is todetermine the optimal schedule that satisfies the demand for all theproducts and that realizes the minimum average setup, inventoryholding and backlog cost per unit time. Comparison with previousresults (when production rates are fixed) reveals that averagecosts can be reduced up to 66% by allowing controllable productionrates.     

Shortening cycle times in multi-product,capacitated production environments through quality level improvements and setup reduction

This paper addresses the issue of investing in reduced setup times and defect rates for a manufacturer of several products operating in a JIT environment. Production cycle times can be shortened by investing in setup time and defect rate reductions, respectively. The objective is to determine optimal levels of setup time and defect rate reductions along with the corresponding optimal levels of investments respectively, and the optimal production cycle time for each product. The problem is constrained by demand requirements, process improvement budget limitations, and manufacturing and warehousing capacity constraints. We consider the cases of product-specific quality improvements and joint-product quality improvements. A general nonlinear optimization models of these problems are formulated. A convex geometric programming approximation of these models is developed respectively, in order to solve them. The approximation can be made to any desired degree of accuracy. Our empirical findings provide insights into a number of managerial issues surrounding investment decisions in product-specific quality improvements and setup reductions due to a product redesign as well as in joint-product improvements due to a process redesign.     

The stochastic economic lot sizing problem for non-stop multi-grade production with sequence-restricted setup changeovers

We study a variant of the stochastic economic lot scheduling problem (SELSP) encountered in process industries, in which a single production facility must produce several different grades of a family of products to meet random stationary demand for each grade from a common finished-goods (FG) inventory buffer that has limited storage capacity. When the facility is set up to produce a particular grade, the only allowable changeovers are from that grade to the next lower or higher grade. Raw material is always available, and the production facility produces continuously at a constant rate even during changeover transitions. All changeover times are constant and equal to each other, and demand that cannot be satisfied directly from inventory is lost. There is a changeover cost per changeover occasion, a spill-over cost per unit of product in excess whenever there is not enough space in the FG buffer to store the produced grade, and a lost-sales cost per unit short whenever there is not enough FG inventory to satisfy the demand. We model the SELSP as a discrete-time Markov decision process (MDP), where in each time period the decision is whether to initiate a changeover to a neighboring grade or keep the set up of the production facility unchanged, based on the current state of the system, which is defined by the current set up of the facility and the FG inventory levels of all the grades. The goal is to minimize the (long-run) expected average cost per period. For problems with more than three grades, we develop a heuristic solution procedure which is based on decomposing the original multi-grade problem into several 3-grade MDP sub-problems, numerically solving each sub-problem using value iteration, and constructing the final policy for the original problem by combining parts of the optimal policies of the sub-problems. We present numerical results for problem examples with 2–5 grades. For the 2- and 3-grade examples, we numerically solve the exact MDP problem using value iteration to obtain insights into the structure of the optimal changeover policy. For the 4- and 5-grade examples, we compare the performance of the decomposition-based heuristic (DBH) solution procedure against that obtained by numerically solving the exact problem. We also compare the performance of the DBH method against the performance of three simpler parameterized heuristics. Finally, we compare the performance of the DBH and the exact solution procedures for the case where the FG inventory storage consists of a number of separate general-purpose silos capable of storing any grade as long as it is not mixed with any other grade.     

Analysis of deteriorating inventory/production systems using a linear quadratic regulator

We consider an inventory-production system where items deteriorate at a constant rate. The objective is to develop an optimal production policy that minimizes the cost associated with inventory and production rate. The inventory problem is first modeled as a linear optimal control problem. Then linear quadratic regulator (LQR) technique is applied to the control problem in order to determine the optimal production policy. Examples are solved for three different demand functions. Sensitivity analysis is then conducted to study the effect of changing the cost parameters on the objective function.     

Controlling adverse effect on work in process inventory while reducing machine setup time

Earlier research has found that the presence of setup time variance can cause an adverse effect on waiting time and inventory as one reduces setup time for a product on a single machine that processes a number of products in a cyclic production system [Sarkar, D., Zangwill, W.I., 1991. Variance effects in cyclic production systems. Management Science 37 (4) 444–453; Zangwill, W.I., 1987. From EOQ towards ZI. Management Science 33 (10) 1209–1223]. This finding validates what other researchers had believed from a rather anecdotal perspective: “variability reduction” is extremely important for improving overall effectiveness of a pull or JIT system [Schonberger, Richard J., 1982. Japanese Manufacturing Techniques: Nine Hidden Lessons in Simplicity. The Free Press, New York]. In this paper, we offer explicit mathematical equations that characterize the variance levels in order to offer exact conditions under which WIP improves or worsens when one reduces setup time.     

On a production-inventory system of deteriorating items subject to random machine breakdowns with a fixed repair time

This paper considers the impact of random machine breakdowns on the classical Economic Production Quantity (EPQ) model for a product subject to exponential decay and under a no-resumption (NR) inventory control policy. A product is manufactured in batches on a machine that is subject to random breakdowns in order to meet a constant demand over an infinite planning horizon. The product is assumed to have a significant rate of deterioration and time to deterioration is described by an exponential distribution. Also, the time-to-breakdown is a random variable following an exponential distribution. Under the NR policy, when a breakdown occurs during a production run, the run is immediately aborted. A new run will not be started until all available inventories are depleted. Corrective maintenance of the production system is carried out immediately after a breakdown and it takes a fixed period of time to complete such an activity. The objective is to determine the optimal production uptime that minimizes the expected total cost per unit time consisting of setup, corrective maintenance, inventory carrying, deterioration, and lost sales costs. A near optimal production uptime is derived under conditions of continuous review, deterministic demand, and no shortages.     

A stochastic production planning model with a dynamic chance constraint

This paper deals with the optimal production planning for a single product over a finite horizon. The holding and production costs are assumed quadratic as in Holt, Modigliani, Muth and Simon (HMMS) [7] model. The cumulative demand is compound Poisson and a chance constraint is included to guarantee that the inventory level is positive with a probability of at least α at each time point. The resulting stochastic optimization problem is transformed into a deterministic optimal control problem with control variable and of the optimal solution is presented. The form of state variable inequality constraints. A discussion the optimal control (production rate) is obtained as follows: if there exists a time t₁ such that t₁?[O, T]where T is the end of the planning period, then (i) produce nothing until t₁ and (ii) produce at a rate equal to the expected demand plus a ‘correction factor’ between t₁ and T. If t₁ is found to be greater than T, then the optimal decision is to produce nothing and always meet the demand from the inventory.     

Hierarchical decision making in production and repair/replacement planning with imperfect repairs under uncertainties

In this paper, we analyse an optimal production, repair and replacement problem for a manufacturing system subject to random machine breakdowns. The system produces parts, and upon machine breakdown, either an imperfect repair is undertaken or the machine is replaced with a new identical one. The decision variables of the system are the production rate and the repair/replacement policy. The objective of the control problem is to find decision variables that minimize total incurred costs over an infinite planning horizon. Firstly, a hierarchical decision making approach, based on a semi-Markov decision model (SMDM), is used to determine the optimal repair and replacement policy. Secondly, the production rate is determined, given the obtained repair and replacement policy. Optimality conditions are given and numerical methods are used to solve them and to determine the control policy. We show that the number of parts to hold in inventory in order to hedge against breakdowns must be readjusted to a higher level as the number of breakdowns increases or as the machine ages. We go from the traditional policy with only one high threshold level to a policy with several threshold levels, which depend on the number of breakdowns. Numerical examples and sensitivity analyses are presented to illustrate the usefulness of the proposed approach.     

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.     

Setup cost reduction in (<Emphasis Type="Italic">Q,r</Emphasis>) policy with lot size,setup time and lead-time interactions

It is often assumed in most deterministic and stochastic inventory models that lead-time is a given parameter and the optimal operating policy is determined on the basis of this unrealistic assumption. However, the manufacturing lead-time is made up of several components (moving time, waiting time, setup time, lot size, and rework time) most of which should be treated as controllable variables. In this paper the effect of setup cost reduction is addressed in a stochastic continuous review inventory system with lead-time depending on lot size and setup time. An efficient iterative procedure is developed to determine the near optimal lot size, reorder point and setup time. Furthermore, a sensitivity analysis is carried out to assess the cost savings that can be realised by investing in setup.     

基于随机动态规划的有限库存ATO系统优化控制

本文研究n维组件单一产品,有限库存的ATO系统。通过建立马尔可夫决策过程模型(MDP),构造优化算法,研究组件生产与库存的最优控制策略。最优策路可以表示为状态依赖型库存阈值,系统内任一组件的控制策略受其它组件库存状态的影响。利用最优控制理论动态规划方法和数值计算方法对最优控制策略的存在性、最优值的数值计算进行研究,建立更符合实际生产的ATO系统决策模型,进行相应的理论和实验验证,研究系统参数对最优策略的影响。     

Lot sizing for a single product recovery system with variable setup numbers

Inventory systems for joint remanufacturing and manufacturing have recently received considerable attention. In such systems, used products are collected from customers and are kept at the recoverable inventory warehouse for future remanufacturing. In this paper a production–remanufacturing inventory system is considered, where the demand can be satisfied by production and remanufacturing. The cost structure consists of the EOQ-type setup costs, holding costs and shortage costs. The model with no shortage case in serviceable inventory is first studied. The serviceable inventory shortage case is discussed next. Both models are considered for the case of variable setup numbers of equal sized batches for production and remanufacturing processes. For these two models sufficient conditions for the optimal type of policy, referring to the parameters of the models, are proposed.