Use of path algebra tools for a unified description of a large class of pull control policies

Abstract

Many pull policies can be found in the literature for controlling multi-stage production/inventory systems. In this paper we present a framework that enables us to describe the dynamics of a large class of pull control policies, using the same set of canonical functions. The class of policies we consider includes well-known pull policies such as kanban, CONWIP, basestock, generalised kanban, and extended kanban, and also many other hybrid policies, and their extensions to systems producing batches. Each of these policies is characterised by certain parameter values. These parameter values are calculated using a computational algorithm that relies on the use of path algebra tools, especially (min, +) algebra tools. This canonical formulation allows us to identify under which values of the control parameters, two different policies will exhibit the same dynamic behaviour. It also enables us to derive methods for evaluating and comparing the performance of various pull control policies.

Keywords:

• JIT performance measurement
• kanban
• Markov modelling
• performance analysis
• performance measures
• queueing models
• stochastic methods
• stochastic models

Acknowledgement

We are grateful to the anonymous referees for their careful reading and constructive comments that improved the presentation of the paper.

Notes

1. Note that f _i can be a function of the time (then we have a scalar value inside the parentheses) or a function of the state (then we have a vector inside the parentheses).

2. Note that, as all the weights on the arcs have positive values, we can use Dijkstra's algorithm to compute the weight of the shortest path of the graph.

3. We denote by ⌊αt⌋ the largest integer smaller than or equal to α, and by ⌊α⌋_β = β⌊α/β⌋ the largest multiple of β smaller than or equal to α.

4. The set ℝ_min has to be completed using (−∞), and it is then denoted by . It is assumed that (−∞) + (+∞) = +∞.