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# Linear Programming

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Introduction

Linear Programming Part A Introduction "Linear programming was developed by George B. Dantzig in 1947 as a technique for planning the diversified activities of the U.S Air Force." Linear programming is a powerful mathematical technique that can be used to deal with the problem of allocating limited facilities and resources among many alternative uses in order to find out the optimal benefits. The main objective of the linear programming problem in management is to maximize profit or minimize cost. Linear programming has a wide variety of applications. It is used by oil companies to determine the best mixture of ingredients for blending gasoline. It is also plays an important part in making the optimal schedules for transportation, production, and construction. In addition, linear programming is a flexible problem-solving tool for portfolio selection in finance, budgeting advertising expenditures in marketing, assigning personnel in human resources management. Applications One of the most important applications of linear programming is the formulation of blends. Blending problems appear whenever a manager must decide how to blend tow or more recourse in order to produce one or more products. In these situations, the recourses often contains one or more essential components that must be mixed in a given pattern and the final product will contains specific percentage of the essential components. In most of these applications management then has to decide how much of each recourse to purchase in order to satisfy product specification and produce demand at minimum cost. ...read more.

Middle

Another case is where the solution might be unbounded. This means that the value of the solution to maximization or minimization problem is infinitely large or small respectively, without violating any of the constraints. This happens due to one or more constraints have been inadvertently omitted in the problem formulation. For the unbounded result, management has to change the objective function to find a realistic solution Finally, linear programming assumes that the production process is fully under control and the total profit is just simply the sum of all the objectives. However, in practice, the production process can be influenced by many factors, which means that there are other losses and costs occur, in this case the resources would not be allocated and used as what they were expected, and therefore leads to a miscalculating in the total profit. Part B Formulate this problem as a linear programming X= number of cows Y= number of pigs MAXIMISE: 40X +20Y (Objective Function) Constraints: (1)3X+Y ? 9 (2)X+Y ? 4 (3)X ? 4 (4)Y ? 6 X, Y ? 0 To solve the problem, we have to use the following equations; Line 1 3X+Y=9 Line 2 X+Y=4 Line 3 X=4 Line 4 Y=6 Use there four equations to draw the graph as below: (Hand-Drawing) The best Mix of purchase and the max-profit Suppose that the maximum profit is �80 (it could be any figure as the slope of iso profit line keeps not change, just easy for calculation) ...read more.

Conclusion

The difference between the two maximum profits is 173.33-160=�13.33, so the dual price for increasing/decreasing 1 unit of bushel is �13.33. Now we increase the number of pigs by 1 at the right-hand side of Y ? 6, then, Y=7, and keep 3X+Y = 9 as the same, solve them simultaneously, we get X=2/3, Y=7, plug them into objective function 40X+20Y we get a profit of �166.67. The difference between this profit and the original max-profit is 166.67-160=�6.67, which means the dual price for increasing/decreasing in purchasing 1 pig, is �6.67. Economic meaning for shadow price The economics meaning of shadow price is the improvement in the optimal value of the objective function per unit increase in the right-hand side of the constraint. In a profit maximization problem, the dual price is the same as the shadow price. Managers could get information from the performance of each constraint and therefore make decisions on any changes in a particular input factor or resource in order to increase profit. In this case, to get more profit, the farmer is recommended to increase the number of bushels rather than increase the amount of pigs. Reference: 'The Quantitative methods for business decision with cases', Lawrence L. Lapim, 6th Edition, Dryden, Chapter 9. 'An introduction to management science-quantitative approaches to decision making', David R.Anderson, Dennis J. Sweeney, Thomas A. Williams. . 6Th Edition, West, Chapter 4. ?? ?? ?? ?? ECON232 Linear Programming DEC 2004 Anting Andy Liu Page 1 ...read more.

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