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# Graphical Solutions To Linear Programming Problems

#### Increasingly new tools and techniques are coming into use by managers as they seek to make intelligent decisions and control operations. At the heart of management’s responsibility is the best or optimum use of limited resources including money, personnel, materials, facilities, and time.

Solutions To Linear Programming Problems : What is Linear programming? Linear programming is a mathematical technique that permits determination of the best use which can be made of available resources. It is a valuable aid to management because it provides a systematic and efficient procedure which can be used as a guide in decision making.

For an example, imagine the simple problem of a small machine shop that manufactures two models, standard and deluxe. Each standard model requires two hours of grinding and four hours of polishing; each deluxe model requires five hours of grinding and two hours of polishing. The manufacturer has three grinders and two polishers ; therefore, in a 40-hour week there are 120 hours of grinding capacity and 80 hours of polishing capacity. There is a contribution margin of \$3 on each standard model and \$4 on each deluxe model and a ready market for both models. The management must decide on (1) the allocation of the available production capacity to standard and deluxe models and (2) the number of units of each model in order to maximize the total contribution margin.

### Techniques available to solve Linear Programming Problems

To solve linear programming problems, the symbol X is assigned to the number of standard models and Y to the number of deluxe models. The contribution margin from making X standard models and Y deluxe models then is 3X + 4Y dollars. The contribution margin per unit is the selling price per unit less the unit variable cost. Total contribution margin is the per unit contribution multiplied by the number of units.

The restrictions on machine capacity are expressed in this manner : To manufacture one standard unit requires two hours of grinding time, so that making X standard models uses 2X hours. Similarly, the production of Y deluxe models uses 5Y hours of grinding time. With 120 hours of grinding time available, the grinding capacity is written: 2X + 5YÂ â‰¤ 120 hours of grinding capacity per week. The limitation on polishing capacity is expressed : 4X + 2Y â‰¤ 80 hours per week. In summary, the relevant information is
:
GrindingÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  PolishingÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  Contribution
TimeÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  TimeÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  Margin
Standard modelÂ Â Â Â Â Â Â Â Â Â Â  2 hoursÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  4 hoursÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  \$3
Deluxe modelÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  5 hoursÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  2 hoursÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  4
Plant capacityÂ Â Â Â Â Â Â Â Â Â Â Â  120 hoursÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  80 hours

There are two basic Linear programming problems solving techniques such as the graphic method and the Linear programming simplex method are used to solve Linear Programming Problems. There areÂ  some linear programming software also available to solve these Linear Programming Problems.

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