Linear Programming in Industry: Theory and Applications. An by Prof. Dr. Sven Danø (auth.)

By Prof. Dr. Sven Danø (auth.)

A. making plans corporation Operations: the overall challenge At kind of typical periods, the administration of an commercial input­ prise is faced with the matter of making plans operations for a coming interval. inside this class of administration difficulties falls not just the final making plans of the company's combination creation yet difficulties of a extra constrained nature reminiscent of, for instance, figuring the least-cost combina­ tion of uncooked fabrics for given output or the optimum transportation agenda. this type of challenge of creation making plans is so much rationally solved in levels: (i) the 1st level is to figure out the possible possible choices. for instance, what replacement construction schedules are in any respect suitable with the given means obstacles? What combos of uncooked fabrics fulfill the given caliber requirements for the goods? and so forth. the knowledge required for fixing this a part of the matter are mostly of a technological nature. (ii) the second one is to choose between between those possible choices one that is economically optimum: for instance, the mixture construction programme in order to result in greatest revenue, or the least-cost blend of uncooked fabrics. this can be the place the economist is available in; certainly, any monetary challenge is anxious with creating a selection be.tween possible choices, utilizing a few criterion of optimum usage of resources.

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4 Exercise: What price and cost data are needed to compute the coefficients of the profit function? Should the costs of the cracking and electrolysis processes be taken into account? 5 Exercise: In the gasoline blending problem above, the introduction of sales restrictions implies the additional side conditions Xl ~ Xl and X2 ~ xa. Show geometrically how this mayor may not affect the optimal solution, depending on the values of Xl and X2' 36 Industrial Applications prices; in the present case, however, the market is such that the prices are fixed by a larger firm which dominates the trade and are taken as data by all other companies, including the one we are considering (price leadership).

We shall first illustrate the problem by a simple hypothetical example. An oil company produces two grades of gasoline by blending three crude products in the proportions 2: 1: 2 and 1: 1 : 3, these coefficients indicating the amounts of the respective crude gasolines present in a unit of each of the finished products. , per day), because of capacity limitations in the company's refinery departments. Gross profit per unit of each finished product being $ 3 and $ 2 respectively, the problem is to plan production such as to maximize total profit per period, subject to the restrictions that total consumption of each crude I This case is formally analogous to the occurrence of setup costs in problems of planning production subject to machine capacity limitations.

2X31 =0. (3a) Similarly, for product no. 1 X32 =0. (3b) Now let profit per unit of product be 4and 3 ¢ respectively. Then the optimal allocation of the raw materials can be found by maximizing the profit function J =4lX ll +X21 +X31)+3 (X 12 +x 22 +X 32 ) subject to the restrictions (2) - (3)3. The coefficients of this linear programming problem are given in the following table. Output no. 2 Input no. ~15 2 3 Quality specification for output no. 1 =0 =0 3 3 =j=max. 3 It is characteristic of this type of blending problem that the blending proportions are not technologically fixed - there is more than one solution to (2) - (3) - but are unknowns to be determined by the optimization procedure.

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