Convexity Properties and Comparative Statics for M/M/S Queues with Balking and Reneging

We use sample path arguments to derive convexity properties of an M/M/S queue with impatient customers that balk and renege. First, assuming that the balking probability and reneging rate are increasing and concave in the total number of customers in the system (head-count), we prove that the expected head-count is convex decreasing in the capacity (service rate). Second, with linear reneging and balking, we show that the expected lost sales rate is convex decreasing in the capacity. Finally, we employ a sample-path sub-modularity approach to comparative statics. That is, we employ sample path arguments to show how the optimal capacity changes as we vary the parameters of customer demand and impatience. We find that the optimal capacity increases in the demand rate and decreases with the balking probability, but is not monotone in the reneging rate. This means, surprisingly, that failure to account for customersâ reneging may result in over-investment in capacity. Finally, we show that a seemingly minor change in system structure, customer commitment during service, produces qualitatively different convexity properties and comparative statics.Operations Management Working Papers Serie

Convexity Properties and Comparative Statics for M/M/S Queues with Balking and Reneging

We use sample path arguments to derive convexity properties of an M/M/S queue with impatient customers that balk and renege. First, assuming that the balking probability and reneging rate are increasing and concave in the total number of customers in the system (head-count), we prove that the expected head-count is convex decreasing in the capacity (service rate). Second, with linear reneging and balking, we show that the expected lost sales rate is convex decreasing in the capacity. Finally, we employ a sample-path sub-modularity approach to comparative statics. That is, we employ sample path arguments to show how the optimal capacity changes as we vary the parameters of customer demand and impatience. We find that the optimal capacity increases in the demand rate and decreases with the balking probability, but is not monotone in the reneging rate. This means, surprisingly, that failure to account for customersâ reneging may result in over-investment in capacity. Finally, we show that a seemingly minor change in system structure, customer commitment during service, produces qualitatively different convexity properties and comparative statics.Operations Management Working Papers Serie

Optimal and Heuristic Lead-Time Quotation For an Integrated Steel Mill With a Minimum Batch Size

This paper presents a model of lead-time policies for a production system, such as an integrated steel mill, in which the bottleneck process requires a minimum batch size. An accurate understanding of internal lead-time quotations is necessary for making good customer delivery-date promises, which must take into account processing time, queueing time and time for arrival of the requisite volume of orders to complete the minimum batch size requirement. The problem is modeled as a stochastic dynamic program with a large state space. A computational study demonstrates that lead time for an arriving order should generally be a decreasing function of the amount of that product already on order (and waiting for minimum batch size to accumulate), which leads to a very fast and accurate heuristic. The computational study also provides insights into the relationship between lead-time quotation, arrival rate, and the sensitivity of customers to the length of delivery promises

Lead-Time Quotation When Customers are Sensitive to Reputation

Firms consider a variety of factors when making lead-time promises, including current shop status and the size of the incoming order. The profit-maximising model presented in this paper is the first to include reputation effects explicitly in a lend-time optimisation model. Reputation is considered to be the lasting effect on the market of a firm\u27s delivery performance over time, and so it affects the future as well as the current profits. The model is complicated, and a counter-example demonstrates that qualitative monotonicity results are not obtainable. A computational study explores the relationships between shop status, order size, reputation, market characteristics and the lead-time decision. Regression analysis sheds light on these relationships and suggests three heuristics, which provide near-optimal solutions with relatively short running times

Spare parts planning and control for maintenance operations

This paper presents a framework for planning and control of the spare parts supply chain inorganizations that use and maintain high-value capital assets. Decisions in the framework aredecomposed hierarchically and interfaces are described. We provide relevant literature to aiddecision making and identify open research topics. The framework can be used to increasethe e¿ciency, consistency and sustainability of decisions on how to plan and control a spareparts supply chain. This point is illustrated by applying it in a case-study. Applicability of theframework in di¿erent environments is also investigated

Maintenance Centered Service Parts Inventory Control

High-tech capital goods enable the production of many services and articles that have become a part of our daily lives. Examples include the refineries that produce the gasoline we put in our cars, the photolithography systems that enable the production of the chips in our cell phones and laptops, the trains and railway infrastructure that facilitate public transport and the aircraft that permit us to travel long distances. To prevent costly production disruptions of such systems when failures occur, it is crucial that service parts are readily available to replace any failed parts. However, service parts represent significant investments and failures are unpredictable, so it is unclear which parts should be stocked and in what quantity. In this thesis, analytical models and solution methods are developed to aid companies in making this decision. Amongst other things, we analyze systems in which multiple parts need replacement after a failure, a situation that is frequently encountered in practice. This affects the ability to complete repairs in a timely fashion. We develop new modeling techniques in order to successfully apply scalable deterministic approaches, such as column generation techniques and sample average approximation methods, to this stochastic problem. This leads to solution techniques that, unlike traditional methods, can ensure that all parts needed to complete maintenance are readily available. The approach is capable of meeting the challenging requirements of a real-life repair shop