Mathematical Programming in Use pp 30-42 | Cite as

# A blending problem using integer programming on-line

## Abstract

The UK photographic company, Ilford Ltd., found that the manual construction of product blends was proving wasteful and inefficient. This paper describes and integer programming model developed to generate optimum blends and its integration into an on-line production system.

The problem arises because of the variability of qualities between the product batches used to construct the blends. Additionally, it is desirable to use batches in chronological order as far as possible. Several times a day a blend of the product is required for a manufacturing process and its qualities must lie within specified ranges. A number of operational factors influence the composition of the optimum blend and these give rise to several integer constraints. These can be represented in ordinary mixed integer terms, but in practive were conveniently incorporated into Special Ordered Sets of Type 2. The objective function is largely heuristic as it must balance the chief aim of selecting a blend having the specified qualities together with the desirability of using older batches.

A shortcoming of the initial model was that it was frequently infeasible and so did not provide useful solution output. Accordingly, the formulation was amended so as to always produce a meaningful report. In order to meet peak requirements, the system had to be able to handle up to nine separate products simultaneously. The operational requirement was that results should be available on the shop floor within an hour or two of initiating a run. Consequently the entire application was designed to run on a teletype operated by factory staff situated in an office adjacent to the blending apparatus.

Statistics from a number of runs are included.

## Key words

Chemical Variability Operational Requirements On-line Factory Floor System Complex Objective Integer Program Alternative Formulations Special Ordered Sets Flexible Software Infeasibility Suppression Robust System Run Statistics Savings## Preview

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## References

- [1]H.P. Williams, “Experiments in the formulation of integer programming problems”,
*Mathematical Programming Study*2 (1974) 180–197.Google Scholar - [2]E.M.L. Beale and J.J.H. Forrest, “Global Optimization using S2 Sets”,
*Mathematical Programming*10 (1976) 52–69.MATHCrossRefMathSciNetGoogle Scholar - [3]A. Brearley, “An investigation into the effects of different algorithmic heuristics on different formulations of the Paint Blending Problem”, internal paper of the University of Warwick (1975).Google Scholar
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*Proceedings of the 5th International conference on Operations Research*(Tavistock Publications, London 1969).Google Scholar