The Dynamic Lot-Sizing Problem for Multiple Items Under Limited Capacity

This paper addresses the problem of scheduling the timing and quantities of production of n different products over m periods for a single production facility with a prespecified capacity. We assume that the demand is deterministic and can vary from one period to another and from one product to another. The objective is to minimize the sum of production setup and inventory holding costs. For medium-size problems, optimal solution algorithms do not yet exist and therefore heuristic solution algorithms are of interest. Most of the existing heuristics make use of the “forward-pass” concept in one way or the other. Forward pass means we begin by determining the lot sizes for earlier periods before moving to study the later periods. In this paper we study the forward-pass approach as well as a different solution approach which we call the four-step algorithm. We develop the feasibility conditions for pure forward-pass algorithms. Finally, we perform a comparative evaluation study.     

One-machine, N-product-type, continuous-time scheduling with a common due date: a polynomially solvable case

The paper analyzes a manufacturing system made up of one-machine which produces TV-product-types with controllable production rates in response to product demands. The demands are characterized by different amounts of each product-type to be produced and a common due date. The objective is to minimize inventory and backlog costs which are incurred when meeting the due date results in inventory surpluses and shortages. With the aid of the maximum principle, the continuous-time scheduling problem is studied as an optimal control model and is reduced to a combinatorial one, polynomially solvable when the costs are either “agreeable” or when the number of the non-agreeable costs is limited.     

Combining cost-based and rule-based knowledge in complex resource allocation problems

A major challenge in the formulation of optimization models for large-scale, complex operational problems is that some data are impossible or uneconomical to collect, producing a cost model that suffers from incomplete information. As a result, even an optimal solution may be “wrong” in the sense that it is solving the wrong problem. In many operational settings, knowledgeable experts will already know, at least approximately, how a model should behave, and can express this knowledge in the form of low dimensional patterns: “high powered locomotives should pull intermodal trains” (because they need to move quickly) or “loaded C-141s should not be flown into Saudi Arabia” (for maintenance reasons). Unlike the literature on inverse optimization which uses observed actions to train the parameters of a cost model, we used exogenous patterns to guide the behavior of a model using a proximal point term that penalizes deviations from these patterns. Under the assumption that the patterns are derived from rational behaviors, we establish the conditions under which incorporating patterns will reduce actual costs rather than just the engineered costs. The effectiveness of the approach is demonstrated in a controlled, laboratory setting using data from a major railroad.     

A fast procedure for computing the total system cost of an appointment schedule for medical and kindred facilities

Scheduling outpatients and medical operation rooms has the following structure: Nusers are given appointment times to use a facility, the duration required by the facility to service each user is stochastic. The system incurs a “user idle cost” if a user arriving at the appointed time finds the facility still engaged by preceding users, while a “facility idle cost” is incurred if the facility becomes free before the next user arrives. We develop an accurate procedure to compute the expected total system costs for any given appointment schedule. Compared to earlier related procedures, ours is much faster and can handle larger problems as well as very general service-time distributions. We then show that this fast computation procedure enables one to determine easily the “lowest-cost appointment schedule” for any given “job” (i.e., “user”) sequence. This in turn will enable one to search for the optimal job sequence that has the best “lowest-cost appointment schedule”.     

Performance of infra-plant overtime policies

Production quotas smooth production in order to synchronize inter-plant production networks. To meet quotas, plants work overtime whenever they're behind. In addition to this 'mandatory' overtime required to satisfy the quota of finished product at the end of the line, many plants run selected stations within the plant overtime to fill critical buffers within the plant. We call this intra-plant overtime 'preventive' overtime because by spending a little overtime this shift on critical stations, the plant can avoid massive overtime next shift. We describe this real production control problem currently facing industry. The problem arises because of (often underestimated or ignored) production variability. We exhibit actual production data that shows the degree of variability often inherent in current manufacturing processes. We describe four policies and compare them using simulation.     

Tooling choices and product performance

One of the dilemmas that manufacturers face involves the tradeoff between the cost of maintaining a variety of production processes, and the cost of not having the ideal process for every product that they produce. This issue is continuing to become more of a problem as manufacturers are forced by market conditions to offer a wider selection of products. We study an instance of this problem in the manufacture of sheet metal parts. We model the problem of selecting and/or designing tools to punch holes in these parts. The cost of not having an “ideal process” is the cost of not having a tool that precisely matches a hole's design diameter. We consider both general “process deviation” costs as well as the Taguchi loss function. Solution procedures are provided for several versions of the problem.     

A Study of Lot Sizing Algorithms

The many MRP lot-sizing algorithms proposed imply four underlying time-phasing requirements: minimizing inventory, maintaining uniform batch sizes, maintaining regular order cycles, and attaining minimum costs. This study examines the cost behavior of four exact and two classical algorithms addressing these structures. Several parameters, describing the demand profile, planning horizon and component costs enable effective evaluation and prediction of cost behaviour. The study shows that: (1) imposing regularity constraints on an MRP schedule leads to costs 14% above the minimum for fixed batch sizing, and 2% for regular order cycling; (2) fixed order cycle schedules show costs close to the minimum attainable, thus supporting the use of this approach in practice; (3) the classical POQ schedules also show good cost performance (11% above minimum); and (4) the cost behaviour of a wide range of MRP demand profiles and cost components can be fairly well predicted (to 7% on the average) using simplified “ideal” relationships.     

Proposed Short Runs Multivariate Control Charts for the Process Mean

In recent years, there is a trend in manufacturing industries to produce smaller lot sizes or low volume production. This trend is due to increased emphasis on just-in-time techniques (JIT), synchronous manufacturing, job-shop settings, the reduction of in-process inventory, and costs. The term used to describe such low volume production is “Short Runs Production” or more commonly “Short Runs.” In such an environment, the run of a process is short, usually fewer than 50. Therefore, a principal practical problem is the need to chart a large number of different processes and the consequent large number of charts required. This article discusses proposed multivariate control charts for short runs based on individual measurements and subgrouped data.     

A Future Value Approach to Determining Project Duration

Optimal project duration for situations where project duration can be shortened by “crashing” activities is analyzed. The cost components considered are; regular direct costs, crashing costs and overhead costs. The regular direct costs are those associated with carrying out the various activities of the project. The crashing costs are those associated with shortening the project duration and represent the time-cost tradeoff. The overhead costs are assumed to be constant throughout the project's life. To facilitate the analysis, the costs are approximated by mathematical expressions. Calculating the cumulative future value of all costs leads to determining the optimal project duration.     

Choosing the appropriate capacity for a make-to-forecast production environment using a Markov analysis approach

In the “make-to-forecast” production environment, competitive market dynamics require customer delivery times substantially shorter than the fixed manufacturing lead times, which require the release of units into production without prior knowledge of customers' desires. As a result, there is the possibility of either getting more orders than can be accommodated causing the rejection of some, or getting too few orders leading to a finished unit without a buyer, which we term an “orphan”. The physical size and dollar value of the units make storing of the orphans, if not completely impossible in some situations, at least extremely undesirable. The likelihood of these two undesirable events depends on the managerially predetermined production capacity relative to the exogenous average order arrival rate. If the capacity is excessive, too many units will be orphaned, whereas insufficient capacity will result in too many rejected orders. We present a Markov model to analyze the behavior of the system in regard to orphan and order rejection levels. The analysis provides management and researchers with highly generalizable insights into managing this common dilemma under various demand and policy scenarios to make more informed capacity level decisions.     

The capacitated lot sizing problem with overtime decisions and setup times

The Capacitated Lot Sizing-Problem (CLSP) consists of planning the lot sizes of multiple items over a planning horizon with the objective of minimizing setup and inventory holding costs. In each period that an item is produced a setup cost is incurred. Capacity is limited and homogeneous. Here, the CLSP is extended to include overtime decisions and capacity consuming setups. The objective function consists of minimizing inventory holding and overtime costs. Setups incur costs implicitly via overtime costs, that is, they lead to additional overtime costs when setup times contribute to the use of overtime capacity in a certain period. The resulting problem becomes more complicated than the standard CLSP and requires methods different from the ones proposed for the latter. Consequently, new heuristic approaches are developed to deal with this problem. Among the heuristic approaches are the classical HPP approach and its modifications, an iterative approach omitting binary variables in the model, a Genetic Algorithm approach based on the transportation-like formulation of the single item production planning model with dynamic demand and a Simulated Annealing approach based on shifting family lot sizes among consecutive periods. Computational results demonstrate that the Simulated Annealing approach produces high quality schedules and is computationally most efficient.     

An Exploratory Study on Stopping a Paced Line When Incompletions Occur

This is an exploratory study of “hybrid” lines, i.e., “paced” lines whose cycle time can be extended (something like an “unpaced” line) when some task incompletions occur. Such a line should have its tasks classified into two categories; only the tasks in the first category have sufficiently high incompletion costs to justify extending the cycle time to ensure their completion. A heuristic procedure is developed to perform this classification, and we found that such hybrid lines may have a lower total cost (i.e. sum of labor and incompletion costs) than standard paced lines.     

Scheduling with multiple-job-on-one-processor pattern

Most scheduling literature considers a “one-job-on-one-processor” pattern, which assumes that a processor processes exactly one job at a time. In this paper we consider a new scheduling problem with a “multiple-job-on-one-processor” pattern, where several jobs can be processed by a single processor simultaneously, provided that the total size of the jobs being processed does not exceed the capacity of the processor at any point in time. This problem is motivated by the operation of berth allocation, which is to allocate vessels (jobs) to a berth (processor), where the vessels, if small in dimension, may share the berth with some other vessels for loading/unloading the goods. We consider the problem to minimize the makespan of the schedule. The well-known First-Fit Decreasing heuristic is generalized and applied to several variations of the problem, and the worst-case behavior of the generalized heuristics is studied. Worst-case error bounds are obtained for those models. Computational experiments are conducted to test the heuristics. The results suggest that the heuristics are effective in producing near-optimal solutions.     

Supply chain capacity and outsourcing decisions: the dynamic interplay of demand and supply uncertainty

We study the interplay of demand and supply uncertainty in capacity and outsourcing decisions in multi-stage supply chains. We consider a firm's investment in two stages of a supply chain (Stage 1 models the “core” activities of the firm, while Stage 2 are the “non-core” activities). The firm invests in these two stages in order to maximize the multi-period, discounted profit. We consider how non-stationary stochastic demand affects the outsourcing decisions. We also consider how investment levels are affected by non-stationary stochastic supply when the market responds to the firm's investments. We characterize the optimal capacity investment decisions Tor the single- and multi-period versions of our model and focus on how changes in supply and demand uncertainly affect the extent of outsourcing. We find that as the responsiveness of the market to investments made by the firm increases, the reliance on outsourcing generally increases. While greater supply and greater demand have the expected effect on investments, decreases in variability are not as straightforward. Greater supply uncertainty increases the need for vertical integration while greater demand uncertainty increases the reliance on outsourcing. In the multi-period model, we find that the nature of adjustments in capacity based on changes in demand or supply follows from the comparative statics of the single-period model, although whether outsourcing increases or decreases depends on the costs of adjusting capacity.     

Configuring a manufacturing firm's supply network with multiple suppliers

Consider a supply network consisting of a manufacturer and its suppliers. The manufacturer produces different types of products, using a common set of inputs (e.g., raw materials and/or component parts) from the suppliers: but, each product needs a different mix of these inputs. The manufacturer sells its finished products to the market at the end of the current decision horizon, facing an uncertain market demand. In situations where a manufacturer has outsourced its parts production to contract manufacturers, the contract manufacturer's capacity available for the given manufacturer in a particular time period may be limited by “capacity reservation” agreements made in advance. Thus, in making production mix decisions for the current planning horizon, the manufacturer has to take into account both its own and the component suppliers' capacity restrictions. We develop a mathematical model and an iterative algorithm that helps the manufacturer solve its supply configuration problem, that is, how much of each raw material and/or component part to order from which supplier, given capacity limits of suppliers as well as the manufacturer. The model takes into account such factors as market demand uncertainty, costs and product characteristics. We present an numerical example to illustrate the interacting effects among critical parameters in the model, and apply the model to a real-world case of a computer manufacturer.     

Quantifying the advantages of production quota flexibility in the automotive industry

Manufacturers use production quotas to smooth and synchronize production. We use a Markov chain to quantify the effect of quota flexibility on overtime and inventory costs. This quantification allows us to trade off these costs to find the optimal amount of quota flexibility. We define quota flexibility as the permitted amount and duration of deviations from the cumulative quota. We specifically analyse the cost savings as a function of the maximum allowed deviation amount from the cumulative quota for four policies. These example policies are the four permutations of two factors: whether or not backlogging is permitted; and whether or not actual production and the quota must be reconciled weekly. We apply the model to three automotive assembly plants and conclude that permitting modest flexibility in production quotas (of ? 1 h of production) significantly reduces total cost.     

Production Scheduling with Sequence Dependent Setup Costs

The problem of scheduling products, with constant demand rates, on a single facility is difficult. This difficulty is compounded if the setup costs are not constant for each product but depend on the sequence in which the products are made. A heuristic that iterates between solving the scheduling problem with constant setup costs and solving a “traveling salesman” formulation with sequence dependent setup costs is presented. The heuristic works well in practice and always provides a feasible solution if one exists. Some computational experience is also given.     

Process-oriented tolerancing for multi-station assembly systems

In multi-station manufacturing systems, the quality of final products is significantly affected by both product design as well as process variables. Historically, however, tolerance research has primarily focused on allocating tolerances based on the product design characteristics of each component. Currently, there are no analytical approaches to optimally allocate tolerances to integrate product and process variables in multi-station manufacturing processes at minimum costs. The concept of process-oriented tolerancing expands the current tolerancing practices, which bound errors related to product variables, to explicitly include process variables. The resulting methodology extends the concept of “part interchangeability” into “process interchangeability,” which is critical due to increasing requirements related to the selection of suppliers and benchmarking. The proposed methodology is based on the development and integration of three models: (i) the tolerance-variation relation; (ii) variation propagation; and (iii) process degradation. The tolerance-variation model is based on a pin-hole fixture mechanism in multi-station assembly processes. The variation propagation model utilizes a state space representation but uses a station index instead of a time index. Dynamic process effects such as tool wear are also incorporated into the framework of process-oriented tolerancing, which provides the capability to design tolerances for the whole life-cycle of a production system. The tolerances of process variables are optimally allocated through solving a nonlinear constrained optimization problem. An industry case study is used to illustrate the proposed approach.     

An integrated approach to inventory and flexible capacity management subject to fixed costs and non-stationary stochastic demand

In a manufacturing system with flexible capacity, inventory management can be coupled with capacity management in order to handle fluctuations in demand more effectively. Typical examples include the effective use of temporary workforce and overtime production. In this paper, we discuss an integrated model for inventory and flexible capacity management under non-stationary stochastic demand with the possibility of positive fixed costs, both for initiating production and for using contingent capacity. We analyze the characteristics of the optimal policies for the integrated problem. We also evaluate the value of utilizing flexible capacity under different settings, which enable us to develop managerial insights.     

Understanding and controlling rotations in factor analytic models

Positive Matrix Factorization (PMF) is a least-squares approach for solving the factor analysis problem. It has been implemented in several forms. Initially, a program called PMF2 was used. Subsequently, a new, more flexible modeling tool, the Multilinear Engine, was developed. These programs can utilize different approaches to handle the problem of rotational indeterminacy. Although both utilize non-negativity constraints to reduce rotational freedom, such constraints are generally insufficient to wholly eliminate the rotational problem. Additional approaches to control rotations are discussed in this paper: (1) global imposition of additions among “scores” and subtractions among the corresponding “loadings” (or vice versa), (2) constraining individual factor elements, either scores and/or loadings, toward zero values, (3) prescribing values for ratios of certain key factor elements, or (4) specifying certain columns of the loadings matrix as known fixed values. It is emphasized that application of these techniques must be based on some external information about acceptable or desirable shapes of factors. If no such a priori information exists, then the full range of possible rotations can be explored, but there is no basis for choosing one of these rotations as the “best” result. Methods for estimating the rotational ambiguity in any specific result are discussed.