# Production Planning Models using Max-Plus Algebra

Arun N. Nambiar (California State University, USA), Aleksey Imaev (Ohio University, USA), Robert P. Judd (Ohio University, USA) and Hector J. Carlo (University of Puerto Rico - Mayaguez, Puerto Rico)
Copyright: © 2012 |Pages: 31
DOI: 10.4018/978-1-61350-047-7.ch011
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## Abstract

The chapter presents a novel building block approach to developing models of manufacturing systems. The approach is based on max-plus algebra. Within this algebra, manufacturing schedules are modeled as a set of coupled linear equations. These equations are solved to find performance metrics such as the make span. The chapter develops a generic modeling block with three inputs and three outputs. It is shown that this structure can model any manufacturing system. It is also shown that the structure is hierarchical, that is, a set of blocks can be reduced to a single block with the same three inputs and three output structure. Basic building blocks, like machining operations, assembly, and buffering are derived. Job shop, flow shop, and cellular system applications are given. Extensions of the theory to buffer allocation and stochastic systems are also outlined. Finally, several numerical examples are given throughout the development of the theory.
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## Max-Plus Algebra

This algebra (The notations used and the concepts given here have been adapted from (Heidergott, 2006)) has two main operators viz. the max operator (maximization) which is denoted by the symbol ⊕and the plus operator (addition) which is denoted by the symbol ⊗. The operators are defined as shown in Equations (1) and (2).

(1)
(2) where is the union of the set of real numbers and the zero element of max-plus algebra, ε=-∞, i.e. . For example

1⊕2=max(1,2)=21⊗2=1+2=3

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