On the formulation of linear programming problems of the blending type

On the formulation of linear programming problems of the

blending type

Citation for published version (APA):

Benders, J. F. (1983). On the formulation of linear programming problems of the blending type. (Memorandum COSOR; Vol. 8310). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1983

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics and Computing Science

PROBABILITY THEORY, STATISTICS AND OPERATIONS RESEARCH GROUP

Memorandum COSOR 83-10 On the formulation of linear

programming problems of the blending type by

J.F. BENDERS

Eindhoven, April 1983 The Netherlands

ON THE FORMULATION OF LINEAR

PROGRAMMING PROBLEMS OF THE BLENDING TYPE by

J.F. Benders

Abstract

It is shown how a linear quality-blending problem can be expressed as a processing network problem.

In [I] a theoretical basis has been laid for the development of a linear programming solution system for solving processing network problems by fully exploiting their network structure. Here, processing network problems refer to network flow problems which do not consider only pure or generalized flows of materials through -,arcs of a network, but which do also allow processing of these materials in some of its nodes. The processes are either separation- or blending on recipe, i.e. any two com-ponE,'lnta:pr.oduced by separation-have a given mutual ration; also any two compo-nents used in the blending process; these ratio's constitute the recipe. Blending is an often occurring process in the oil and food industry. Blending on recipe however can be used only when one is interested in a single quality aspect of the blend. In general one is interested in a larger number of qualities. In such a case these qualities are known

2 -blem.

v· .

a

I

L

The blending problem is described then by the linear system

bound, or for which both a lower and an upper bound has been specified.

one for each of the above types. We also take only three components. considered: qualities for which either only a lower bound, only an upper linear programming problem can be written as a processing network pro-is already known from Koene [1, P.' 120] where it has been shown that any as a processing network problem. The possibility of such an expression

For simplicity we consider a blend with only three quality requirements, For the quality requirements of the blend three possibilities must be It will be shown now how such a quality blending problem can be expressed and if the available amount of component j is a., then the blend

satis-J

fies the linear system

required having these qualities between given limits.

upper qualities for the blend are specified by the vectors q and q, for the components (e.g. density and sulfur content) and the blend is

X 3

-\

network processing representation of a quality blending problem; one blend.

gl :;; q II xl + _qlZxZ + _ql3x3 qZlxl + _qzzxz + _qZ3x3 :s:; qz g3 :;; q3l x I + q3ZxZ + q33x3 :;; q3 xI + X z + x3 • X 0 :s:; xI :;; a_l 0 :;; _X z :;; aZ 0 :;; _x 3 :;; a3 Figure

lowing processing network problem:

fol-OSx Sa

3 3

4

-Figure 2. Network processing representation of a quality blending problem; two blends

5

-In general, a quality blending problem can be represented by a processing network problem involving a separation process, corresponding to each component, a blending process on recipe corresponding to the require-ments for the blend and a special separation process with two separation coefficients both equal to I. (a "duplicating" process) for each quality for which both a lower and an upper bound is specified for the blend.

(In Figure I,the arrows ~ andl~ represent slack activities~)

If two or more blends have to be produced from the same set of compo-nents, for each blend a similar processing network must be drawn for each blend, linked only by the common component sources. An example is given in Figure 2.

[IJ J. Koene. Minimal cost flow in processing networks; a primal approach. Amsterdam, Mathematical Centre, 1982.