Clearing function in the context of the invariant manifold method

Клиринговая функция в контексте метода инвариантного многообразия
Mustafin A.T. Kantarbayeva A.K.
2023 Saint Petersburg State University

Vestnik Sankt-Peterburgskogo Universiteta, Prikladnaya Matematika, Informatika, Protsessy Upravleniya
2023 #19 Issue 2 185 - 198 pp.

Clearing functions (CFs), which express a mathematical relationship between the expected throughput of a production facility in a planning period and its workload (or work-in- progress, WIP) in that period have shown considerable promise for modeling WIP-dependent cycle times in production planning. While steady-state queueing models are commonly used to derive analytic expressions for CFs, the finite length of planning periods calls their validity into question. We apply a different approach to propose a mechanistic model for one-resource, one-product factory shop based on the analogy between the operation of machine and enzyme molecule. The model is reduced to a singularly perturbed system of two differential equations for slow (WIP) and fast (busy machines) variables, respectively. The analysis of this slow-fast system finds that CF is nothing but a result of the asymptotic expansion of the slow invariant manifold. The validity of C F is ultimately determined by how small is the parameter multiplying the derivative of the fast variable. It is shown that sufficiently small characteristic ratio working machines : WIP guarantees the applicability of CF approximation in unsteady-state operation.

enzyme catalysis , production model , quasi-steady-state approximation , singular perturbation , work in progress

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Satbayev University, 22, Satpayeva ul., Almaty, 050013, Kazakhstan
Farabi Kazakh National University, 71, al-Farabi pr., Almaty, 050040, Kazakhstan

Satbayev University
Farabi Kazakh National University

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