Optimizing multi-stage production with constant lot size and varying numbers of batches

The model presented in this paper assumes that a uniform lot size is produced through a series of manufacturing stages, with a single set-up and without interruption at each stage. Transportation of partial lots, called batches, is allowed between stages after the whole batch is completed. The batch sizes must be equal at any particular stage, but the optimal number of equal-sized batches may differ across stages. Of course, the set-up costs, the inventory-holding costs and the transportation costs influence both the optimal batch-sizes at the various stages and the uniform lot size. An optimization method for this deterministic model is developed and is illustrated by an example.     

Cost models for lot streaming in a multistage flow shop

Lot streaming is a technique used to split a processing batch into several transfer batches. In this way, overlapping operations can be performed in different manufacturing stages, and production can be accelerated. This paper proposes two cost models for solving lot streaming problems in a multistage flow shop. The purpose is to determine the optimal processing batch size and the optimal number of transfer batches that minimize the total annual cost in each model. In the first model, a more complete and accurate method is developed to compute the costs of raw materials, work-in-process, and finished-product inventories. The total cost includes the setup cost, the transfer batch movement cost, the three-type inventory holding cost, and the finished-product shipment cost. The second model contains not only the four costs in the first model, but also the imputed cost associated with the makespan time. The total annual cost functions in both models are shown to be convex, and two solution approaches are suggested. An experiment consisting of three phases was conducted to explore the effect on the optimal solution when changing the value of one parameter at a time. The results indicate that three parameters have significant effects on the optimal solution.     

Setups and effective capacity: the impact of lot sizing techniques in an MRP environment

This study investigates how lot sizing techniques influence the profit performance, inventory level, and order lardiness of an assembly job shop controlled by MRP. Four single-level lot sizing techniques are compared by simulation analysis under two levels of master schedule instability and two levels of end item demand. A second analysis investigates the influence of a multilevel lot sizing technique, the generalized constrained-K (GCK) cost modification, on the four single-level techniques at low demand and low nervousness. The analyses reveal a previously unreported phenomenon. Given the same inventory costs, the single-level lot sizing techniques generate substantially different average batch sizes. The lot sizing techniques maintain the following order of increasing average batch size (and decreasing total setup time):

economic order quantity (EOQ)

period order quantity (POQ)

least total cost (LTC)

Silver-Meal heuristic (SML)

The causes for different average batch sizes among the lot sizing techniques are analysed and explained. Demand lumpiness, inherent in multilevel manufacturing systems controlled by MRP, is found to be a major factor. The number of setups each lot sizing technique generates is the primary determinant of profit performance, inventory level, and order tardiness. EOQ, a fixed order quantity technique, is less sensitive to nervousness than the discrete lot sizing techniques. EOQ_, however, generates the smallest average batch size, and, therefore, the most setups. Since setups consume capacity, EOQ, is more sensitive to higher demand. SML generates the largest average batch sizes, and is, therefore, less sensitive to increased demand. At low demand, the other lot sizing techniques perform better on all criteria. They generate smaller batches and, therefore, shorter actual lead times. The GCK cost modification increases the average batch size generated by each lot sizing technique. GCK improves the profit and customer service level of EOQ the lot sizing technique with the smallest batches. GCK causes the other lot sizing techniques to generate excessively large batches and, therefore, excessively long actual lead times.     

A joint economic lot size model for raw material ordering,manufacturing setup,and finished goods delivering

In this research we consider a single-manufacturer single-buyer supply chain problem where the manufacturer orders raw materials from its supplier, then using its manufacturing processes converts the raw materials to finished goods, and finally delivers the finished goods to its customer. The manufacturer produces the product in batches at a finite rate and periodically delivers the finished goods at a fixed lot size to its customer, who has a constant demand rate. An integrated inventory control model, making joint economic lot sizes of manufacturer's raw material ordering, production batch, and buyer's ordering, is developed to minimize the mean total cost per unit time of the raw materials ordering and holding, manufacturer's setup and finished goods holding, the buyer's ordering, and inventory holding. Numerical examples are also setup to illustrate that jointly considering the inventory costs above results in less mean total cost than that of considering them separately.     

Simultaneous determination of production lot size and process parameters under process deterioration and process breakdown

The purpose of this study is to combine production-inventory management with process-quality design for determining production lot size and process parameters under the possibility of process deterioration and breakdown. The total cost of such an integrated model includes: the combined setup cost (production setup and process resetting), the costs of quality loss, tolerance and mean costs for processes established, a penalty cost for process breakdown and carrying costs for cumulated inventory. The quadratic quality loss function is introduced to assess quality loss within the system. Decision variables include the initial setting (process mean) and process tolerance for process parameters determination, and production lot size for production-inventory management. The cycle time for production-inventory management is assumed to be the same as the resetting cycle for the new process-quality system. The contribution of this study lies in its development of an integrated model that enables process parameters, production lot size, and cycle time to be determined concurrently for quality and economic considerations, and at an earlier time in the process design and production management stage. An example is presented to demonstrate the proposed model.     

冷轧生产批量计划与调度问题模型及算法

对冷轧生产系统特点进行分析,将冷轧生产批量计划与调度问题抽象为多阶段、多品种带有中间库的批量计划与调度问题。针对该问题建立了数学模型,通过对库存成本和调整成本惩罚系数的控制可以协调库存水平和调整次数的关系。对所建立的模型,提出了基于二进制粒子群优化与局部搜索的混合求解算法。最后,通过对企业实际生产数据的计算和分析,验证了模型和算法的可行性和有效性。     

PRODUCTION SCHEDULING OF CONTINUOUS FLOW LINES: MULTIPLE PRODUCTS WITH SETUP TIMES AND COSTS

The problem of production planning and setup scheduling of multiple products on a single facility is studied in this paper. The facility can only produce one product at a time. A setup is required when the production switches from one type of product to another. Both setup times and setup costs are considered. The objective is to determine the setup schedule and production rate for each product that minimize the average total costs, which include the inventory, backlog, and setup costs. Under the assumption of a constant production rate, we obtain the optimal cyclic rotation schedule for the multiple products system. Besides the decision variables studied in the classical economic lot scheduling problem (ELSP), the production rate is also a decision variable in our model. We prove that our solutions improve the results of the classical ELSP.     

A COMPARISON OF DIRECT COST SAVINGS BETWEEN FLEXIBLE AUTOMATION AND LABOR WITH LEARNING

This paper compares flexible automation with labor‐intensive manufacturing processes in a batch production environment and considers learning, forgetting, inventory carrying costs, setup costs, production demand volume, previous production experience, and the proportion of material to labor cost. While flexible automation typically can reduce setup times, and therefore inventory carrying costs through smaller optimal batch sizes, the results of this research show that the effect of forgetting on relative cost savings is difficult to predict in some situations. When using optimal lot sizes in both the automated and labor environments, cost savings from flexible automation may be smaller than expected or may occur in different ways than anticipated.     

Coordinated scheduling of production and delivery stages with stage-dependent inventory holding costs

This paper considers a problem of integrated decision-making for job scheduling and delivery batching wherein different inventory holding costs between production and delivery stages are allowed. In the problem, jobs are processed on a facility at a production stage and then delivered at the subsequent delivery stage by a capacitated vehicle. The objective is to find the coordinated schedule of production and delivery that minimizes the total cost of the associated WIP inventory, finished product inventory and delivery, where both the inventory costs are characterized in terms of the weighted flow-time and the delivery cost is proportional to the required number of delivery batches. It is proved that the problem is NP-hard in the strong sense. Thereupon, three heuristic algorithms are derived. Some restricted cases are also characterized as being solvable in polynomial time. Numerical experiments are conducted to evaluate the performance of the derived heuristic algorithms.     

An Experimental Analysis of Capacity-Sensitive Setup Parameters for MRP Lot Sizing

Most material requirements planning (MRP) systems apply standard costing (absorption costing) approaches to define setup costs that are used as fixed (time invariant) setup parameters in single-level lot-sizing methods. This paper presents a computationally simple approach for estimating more appropriate setup parameters based on estimates of work-center shadow prices. These setup parameters then are used in traditional single-level MRP lot-sizing procedures. The shadow price of capacity at each work center is calculated as the increase in the overall inventory carrying cost for each additional hour of capacity lost to setups. The opportunity cost of a setup for an order subsequently is determined based on the routing information for each order and is used by traditional MRP lot-sizing procedures to calculate lot sizes. A simulation experiment compares the performance period order quantity lot sizing with capacity-sensitive setup parameters with the fixed accounting-based setup parameters. The simulation replicates the planning and control functions of a typical MRP system. The results of the experiment show that capacity-sensitive setup parameters can make significant reductions in both carrying cost and lateness and can achieve many of the benefits of optimized production technology in the context of an MRP system.     

Joint Lot Sizing and Scheduling of Multiple Items with Sequence-Dependent Setup Costs

The objective of this research is to investigate the effects of setup-cost estimating methods on the lot sizing and scheduling of multiple products in multiple periods. These initial setup cost estimators (ISCEs) are used to estimate sequence-independent initial setup costs from sequence-dependent setup costs. A search of the literature reveals that, although sequence-dependent setup costs are frequently found in practice and ISCEs are frequently used, there is a dearth of research concerning the effect of using ISCEs. After a review of the literature, a mixed integer formulation of the joint problem of lot sizing and scheduling is presented, followed by a discussion of the difficulty in solving the formulation. Next, the six ISCEs evaluated are presented. These ISCEs range from simple (select the minimum setup cost) to complex (use the branch-and-bound solution to a traveling salesman-type problem). Each ISCE is evaluated using a full factorial design with five independent variables: demand distribution (three levels), demand trend (three levels), setup to inventory level (six levels), setup distribution (three levels), and setup variability (two levels). Two hypotheses are researched. Do the more computationally complex ISCEs produce lower overall costs than do the simpler ISCEs? Does the reduction in total cost justify the additional computation cost? The results of this study demonstrate that it may be incorrect to use “conventional wisdom'’when selecting an ISCE.     

Solving the Economic Lot-Scheduling Problem Using the Method of Prime Subperiods

In recent years the reported successes of Japanese production systems, particularly the just-in-time approach to inventory control, has caused managers to focus more of their attention on efficient decision-making procedures for determining production schedules that minimize inventory costs. One such potential area of attention is the economic lot-scheduling problem (ELSP), which occurs in a variety of manufacturing environments where machining operations are prevalent. The economic lot-scheduling problem addresses the determination of lot sizes for N products with constant demand (and cycled through one machine with a given production rate) to minimize setup and inventory costs. The most successful solution approaches to the ELSP have been based on the concept of a basic period that is of sufficient length for the production of all items, even though each item might not be produced during each repetition of the basic period. This paper proposes a heuristic approach to the solution of the ELSP (referred to as the method of prime subperiods), which is an extension of the basic period approaches. The procedure is described and demonstrated via an example and then tested using a set of six example problems previously employed in the literature related to the ELSP. The results indicate as good or superior performance by the proposed method of prime subperiods.     

Optimal two-stage production-inventory policy with whole lot and split lots

This paper deals with the problem of determining the optimal lot sizes for a two-stage production-inventory system. The first stage has a large lot to be processed in a single batch and is called the ‘whole lot’; this is split in the second stage into smaller lots of equal size called ‘split lots’. The lot in stage 1 is inspected for its conformity to standards before it is passed on to stage 2. We use a single sampling attribute plan for inspection and examine the consequences of rejecting or accepting the lot. The rejected lots are 100 % inspected and defectives are eliminated..The accepted lots contain defectives in the uninspected part which are identified during stage 2 processing. We have combined these aspects of inspection with the usual inventory costs and developed a model to determine the optimal sizes of the whole lot and split lots     

SETUP REDUCTION AND MACHINE AVAILABILITY

This paper is an extension of Billington, who used the framework of the economic production quantity (EPQ) to model setup cost reduction. In the present paper, we use the EPQ model as a starting point to investigate the nature of setup costs and the effect of setup time reduction on the increase in available capacity. Reducing setup is vital to a company's success because a lengthy changeover of machinery is expensive: it demands long production runs to justify its cost, and these, in turn, lead to excessive inventory and to a slow response to customer needs. As in Billington, setup reduction is modeled as a function of an annual amortized investment. The paper examines the behavior of the setup time, the inventory cost, the lot size, and the freeing up of machine time in the face of a capacity constraint. A solution algorithm is provided to find setup times that minimize the sum of setup and holding cost, subject to a constraint on machine availability. The analysis sheds light on the true nature of setup cost and on the opportunity cost of not reducing setups. In the constrained optimization, the Lagrangian multiplier gives an estimate of the marginal value of adding one time unit of machine capacity, or, alternatively, of reducing one unit of setup time.     

A flexibly structured lot sizing heuristic for a static remanufacturing system

An effective planning of lot sizes is a key strategy to efficiently manage a combined manufacturing/remanufacturing system in the presence of substantial setup costs. Due to its complex interdependencies, optimal policies and solutions have not been identified so far, but several heuristic approaches have been analyzed in recent contributions. The main heuristic shortcuts are forcing equally sized lot sizes over the planning horizon as well as imposing a specific cycle structure, i.e., a sequence of manufacturing batches is followed by a sequence of remanufacturing batches. We are instead proposing a flexibly structured heuristic that allows for differently sized remanufacturing batches. We show in a comprehensive numerical study that our approach outperforms other existing approaches in more than half of all instances by up to 17%.     

EXISTENCE AND DERIVATION OF FORECAST HORIZONS IN A DYNAMIC LOT SIZE MODEL WITH NONDECREASING HOLDING COSTS

We are concerned with a discrete-time undiscounted dynamic lot size model in which demand and the production setup cost are constant for an initial few periods and the holding cost of inventory is an arbitrary nondecreasing function assumed to be stationary (i.e., explicitly independent of time) in the same initial few periods. We show that there exists a finite forecast horizon in our model and obtain an explicit formula for it. In addition, we obtain fairly general conditions under which the existence of a solution horizon in the model implies the existence of a forecast horizon. We also derive an explicit formula for the minimal solution horizon. These results extend the earlier ones obtained for the dynamic lot size model with linearly increasing holding costs.     

Setup cost reduction a multi-stage order quantity model

In this paper, a single item, multi-stage serial order quantity (MSOQ) model with constant demand is discussed. The objective of the model is to minimize the total cost which includes the setup cost and the inventory holding cost. This paper examines and analyses the investment in a one-time cost to reduce the (current) setup level and adds a per unit item amortization of this cost to the other costs associated with the MSOQ model. We consider the setup cost to be decreased on each stage with the same rate and the cost of the joint setup cost reduction is a logarithmic cost function.     

Replanning the Master Production Schedule for a Capacity-Constrained Job Shop

This research examines the use of both frozen and replanning intervals for planning the master production schedule (MPS) for a capacity-constrained job shop. The results show that forecast error, demand lumpiness, setup time, planned lead time, and order size have a greater impact on the mean total backlog, total inventory, and number of setups than the frozen and replanning intervals. The study also shows that a repetitive lot dispatching rule reduces the importance of lot sizing, and a combination of repetitive lot dispatching rule and single-period order size consistently produces the lowest mean total backlog and total inventory. The results also indicate that rescheduling the open orders every period produces a lower mean total backlog and total inventory when the forecast errors are large relative to the order sizes. This result suggests that the due date of an open order should be updated only when a significant portion of the order is actually needed on the new due date.     

Synchronization of material flow to aid production planning in a job shop

This paper discusses how synchronizing finish times for correspondingly numbered process operations of mating component parts (batches) can help a job shop take advantage of some aspects of the just-in-time philosophy.This paper proposes that the synchronization of process operations of mating parts (batches) can be an effective production planning tool that may actually decrease total throughput time where manufacturers are attempting to decrease in-process inventory. An example problem is included to demonstrate the situation.A production planner must keep in perspective all production orders when trying to decrease in-process inventory. The throughput times of all orders may be increased, while too much effort is concentrated on reducing in-process inventory costs of individual orders.