Polling, production & priorities

The present monograph focuses on the so-called stochastic economic lot scheduling problem(SELSP), which deals with the make-to-stock production of multiple standardizedproducts on a single machine with limited capacity under random demands, possibly randomsetup times and possibly random production times. In the SELSP, a production policyis needed which describes for each possible state of the system whether to continue productionof the current product, whether to switch to another product or whether to idlethe machine. The objective of the present monograph is the development and the analysisof mathematical models that capture the behavior of the class of fixed-sequence base-stockpolicies. For given base-stock levels, it is shown that the analysis of a fixed-sequence basestockpolicy is tantamount to the analysis of the queue length distribution in a classicalqueueing model, the so-called polling system.The focus of the current research is mainly on the lot-sizing decision: what should thelength of the production run be? Within the context of this lot-sizing decision the presentmonograph is, in particular, concerned with the evaluation and comparison of the traditionalexhaustive and gated lot-sizing policies, on the one hand, and the more sophisticatedquantity-limited lot-sizing policy, on the other hand. The latter offers the possibility toprioritize among the different products for improving total system performance throughbounding the lengths of the production runs. Evaluation and optimization of these lotsizingdisciplines are achieved through state-of-the-art analysis of several polling systems.We study two research objectives as summarized below.Research objective 1. Development of a unifying exact framework for the analysis ofthe exhaustive and gated lot-sizing policies in terms of the average work-in-progress (WIP)levels under the assumption of Poisson demand processes. ¤In Chapter 3 an exact Mean Value Analysis (MVA) framework for the exhaustive andgated lot-sizing disciplines is presented, which computes the average WIP levels by exploitingdirect mean value arguments. Within this framework the individual WIP levels canbe efficiently obtained via the solution of a sparse set of linear equations, whereas for thetotal WIP level a closed-form expression is presented.The MVA framework gives rise to explicit closed-form expressions, allowing for back-ofthe-envelope calculations, for the individual WIP levels in the asymptotic regime of highutilization of capacity due to either customer demands or setup times. These expressionsexplicitly show the impact of all input parameters, yield insensitivity and monotonicityproperties and unearth the (dis)similarities between the two sources of high utilization.In particular, it is shown that the exhaustive and gated lot-sizing disciplines display undesirablebehavior if the utilization rate is high due to customer demand, which revealsitself, for example, in difficulties in the coordination between stages within the productionprocess.Motivated by the practical significance of the large setup times regime, we study thisregime in more detail for a general class of branching-type lot-sizing policies by usingmore advanced techniques. The most remarkable result of this analysis is the fact thatthe stochastic system converges to its deterministic counterpart in the limit of increasingsetup times implying that the exhaustive lot-sizing policy is optimal in terms of the WIPlevels and that, thus, production runs should not be bounded in systems with extremelylarge setup times. For general settings, the latter conclusion does not always hold whichwe analytically show in the analysis of the second research objective.Research objective 2. Development of an efficient and accurate approximate tool forthe analysis of the quantity-limited lot-sizing policy under the assumption of general demandprocesses. ¤In order to gain insights into the impact of bounding production runs and not to be divertedby other effects, Chapter 4 starts the analysis with a basic occurrence of the SELSPin an exact way. That is, we analyze a two-product system, in which a high-priority productis produced exhaustively and a low-priority product according to the quantity-limitedservice strategy. In this model, we observe significant cost reductions by application ofthe quantity-limited policy, compared to the standard exhaustive policies, indicating thepotential of the quantity-limited service discipline as lot-sizing rule in production environments.The results obtained in the two-product case provide us with theoretical evidence thatthe quantity-limited strategy may lead to considerable cost reductions compared to thewidely used (standard) exhaustive policy. Therefore, in Chapter 4 we develop an efficientand accurate approximate decomposition approach for the evaluation of quantity-limitedlot-sizing policies under the most general imaginable assumptions, i.e., general number ofproducts each with their own quantity limit in an environment with generally distributedarrival, service time and setup time distributions. The accuracy of the approximationscheme is verified by means of an extensive simulation study.The last part of Chapter 4 is devoted to a numerical simulation study assessing thequality of the quantity-limited lot-sizing policy as tool for prioritizing among products. Itis shown that the quantity-limited lot-sizing policy outperforms the standard exhaustivepolicy leading to improvements in system performance for a variety of environments.Finally, we would like to emphasize that the results of the present monograph are certainlynot limited to the described production setting, but may be used in the design andoptimization phase of many other fields of application such as communication, maintenance,manufacturing and transportation.