Simplex Linear Programming - Essay Example

Writer Name] [Supervisor Name] [Subject] [Date] PROBABILITIES 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. Simplex Linear Programming The simplex method is an iterative procedure which gradually explores feasible solutions to a problem until the optimal solution is reached. The basis on which it works is matrix algebra, requiring a stepwise process of substitutions or transformations. In cases where there are many products and many constraints the calculations are tedious and, for most people, far from easy. Fortunately, there are a number of computer software packages (like excel solver) available which will handle the complexities of the algebra with ease, providing a perfect example of a situation in which the computer becomes a valuable (and perhaps indispensable) tool for the management accountant. The use of a computer changes the nature of the task from one requiring laborious calculation to one in which attention need be focused only on the formulation of the problem and the interpretation of the results (Miller, pp.55-60). Formulate this problem as a linear programming In any decision situation we need to choose between alternatives in order to meet a desired objective. In the make-or-buy situation our objective is to maximize advantage-in-making: Maximize z = 5(x1) + 7(x2) (Objective Function) In trying to achieve this objective we must take into account the various constraints which exist (Luenberger, pp. 120) Constraints are sets by limits to production since production must therefore less then or equal to the maximum production units for each product. This gives a limit against each product: X1 and X2 ≥ 10 hours a day (available time for each bulb usage) In production of smith’s light bulbs, each resources shell, fitting and testing time have limits which cannot be exceeded. We therefore need to show that time used up by units produced must be equal to or less than, the total time available: Constraints: Fitting time: 20 X1 + 30 X2 ≤ 1500/day Testing time: 10 X1 + 30 X2 ≤ 1500/day X1 ≥10 X2 ≥ 10 Graph with Feasible Solution After a number of iterations the following optimal position would be reached: (Please refer excel solver tutorial below) To solve the problem, we have to use the following equations; Line 1   20 X1 + 30 X2 = 1500 Line 2   10 X1 + 30 X2 = 1500  Line 3   X1 = 10 Line 4   X2 = 10 Use these four equations to draw the graph as below: Feasible Solution z = 5(10) + 7(10) = 120 z = 5(10) + 7(43) = 351 z = 5(60) + 7(10) = 370 Solutions are (10, 10), (10, 43) and (60, 10) but the feasible solution is (60, 10) which gives the maximum profitability i.e. 370. So by graph we can observe that yes, the feasible is the set of all points that satisfies all the LPs constrains and sign restrictions. MS Excel Solver Question: Maximizes the profit (z) Input data X1 X2 Total Limits objective 5 7 370 Fitting 20 30 1500 1500 Testing 10 30 900 1500 >=10 >=10 Output results X1 X2 Z 60 10 370 Microsoft Excel 11.0 Limits Report Target   Name Value objective total 370 Adjustable   Lower Target Upper Target Name Value Limit Result Limit Result X1 60 10 120 60 370 X2 10 10 370 10 370 Microsoft Excel 11.0 Sensitivity Report Adjustable Cells   Final Reduced Objective Allowable Allowable Name Value Cost Coefficient Increase Decrease X1 60 0 5 1E+30 0.333333333 X2 10 -0.5 7 0.5 1E+30 Constraints   Final Shadow Constraint Allowable Allowable Name Value Price R.H. Side Increase Decrease Fitting total 1500 0.25 1500 1200 1000 Testing total 900 0 1500 1E+30 600 So through shadow price we can improve the optimal objective value because of a unit transform in the right-hand side of a constraint Microsoft Excel 11.0 Answer Report Target Cell (Max) Name Original Value Final Value objective total 0 370 Adjustable Cells Name Original Value Final Value X1 0 60 X2 0 10 Constraints Name Cell Value Formula Status Slack Fitting total 1500 =10 Binding 0 Conclusion Hence the optimal solution has cost 370. As Smith’s produce two types of eco-friendly industrial light bulbs: one with very low energy usage and one with higher energy usage, and two machines are used for assembly: one for fitting and one for testing, both are available for 10 hours a day. According to given conditions and applying solver calculation, we observe that by taking 60 units of low energy usage and only 10 units of high energy usage, Smith’s got the maximum profit. It can be tricky by Excel solver, when one can handle and managing profit maximizing data by constructing linear programming models (Rodin, pp.60-90). Linear programming is very useful tool that can be used to deal with problems in manufacturing industry, such as the product-mix problem. In this situation, the objective of the manager is to determine the production levels that will allow the company to meet the product demand requirements, given limitations on labour capacity, machine hour’s capacity and so on, at the same time, to make the cost of production to minimum. The solutions that we calculated by using linear programming would satisfy the given limitations and to enable all resources used in manufacturing-raw materials, labours, facilities are employed most efficiently.  The application of linear programming for production scheduling is beneficial because those problems will recur in the next production period; so managers just have to resolve the problem according to different limitations. Therefore ant unnecessary costs would be reduced (George, pp.65). Smith should identify that these assurances are not free. If we want to recognize the impact of the uncertainty in production data, we have to build up models that exclusively incorporate uncertainty (Frederick, pp.27). References Charnes A., Cooper W., Lewin A., and L. Seiford, Data Envelopment Analysis: Theory, Methodology and Applications, Kluwer Academic Publications, 1994, pp. 110-125. Dempe S., Foundations of Bilevel Programming, Kluwer, 2002, pp. 12-17. Diwekar U., 2003. Introduction to Applied Optimization, Kluwer Academic Publishers. Covers almost all the above techniques. pp. 198-263. Frederick Hillier & Gerald Lieberman (1995) "Introduction to Operations Research", sixth edition, McGraw-Hill George B. Dantzig (1963) "Linear Programming and Extensions", Princeton University Press, Princeton, N.J, pp.65-76. Liu B., and A. Esogbue, Decision Criteria and Optimal Inventory Processes, Kluwer, 1999, pp. 521. Luenberger, (2003). Linear and Nonlinear Programming, Kluwer Academic Publishers. pp. 120-170. MigdalasA., Pardalos p., and P. Varbrand, Multilevel Optimization: Algorithms and Applications, Kluwer, (1998). Miller R., Optimization: Foundations and Applications, Wiley, (1999). pp. 55-60. Rodin R., Optimization in Operations Research, Prentice Hall, New Jersey, 2000, pp. 60-90. The Cobb Group (1994) "Running Excel 5 for Windows", fourth edition, Microsoft Press Read More