c h a p t e r2. l p t h e g r a p h i c a l m e t h o d( lecture1)（c h a p t e r - 2 . l p t h e g r a p h i c a l t h o m e d(lecture1)）.doc

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c h a p t e r2. l p t h e g r a p h i c a l m e t h o d( lecture1)（c h a p t e r - 2 . l p t h e g r a p h i c a l t h o m e d(lecture1)）.doc

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c h a p t e r2. l p t h e g r a p h i c a l m e t h o d( lecture1)（c h a p t e r - 2 . l p t h e g r a p h i c a l t h o m e d(lecture1)）

LP: THE GRAPHICAL METHOD Today’s Topic: A Simple LP Example Graphical solution of LP Special Cases Graphical Sensitivity Analysis Change of an objective function coefficient Change of a right-hand-side Simultaneous changes of several coefficients 1. A Simple LP Example RMC, Inc. is a firm that produces chemical-based products. Three raw materials are used to produce two products. The material requirements per ton are shown below. Product Material 1 Material 2 Material 3 Fuel additive 2/5 0 3/5 Solvent base 1/2 1/5 3/10 For the current production period, RMC has available the following quantities of each raw material. Because of spoilage, any materials not used for current production must be discarded. Material Material 1 Material 2 Material 3 Available 20 5 21 If the contribution to profit is $40 for each ton of fuel additive and $30 for each ton of solvent base, how many tons of each product should be produced in order to maximize the total contribution to profit? -- LP formulation Decision variables: Objective function: Constraints: -- Interpretation feasible vs. infeasible feasible solution vs. optimal solution slack vs. surplus slack = rhs - lhs ≤ constraints surplus = lhs - rhs ≥ constraints 0 all resources used requirement just satisfied + extra resource over performed - not enough resource (infeasible) not satisfied (infeasible) 2. Graphical Solution of LP -- for LP with only 2 variables Step 1. Draw the solution space (describe all feasible solutions) -- For each constraint 1. draw the equation line 2. decide the direction Step 2. Draw the objective function line 1. Set obj. fun. Z = convenient value 2. Draw the obj. equation line 3. Determine the direction of improvement for obj. fun. Z Step 3. Shift the obj. line along the improvement direction and identify an optimal solution. Step 4. Calculate the optimal solution and the corresponding objective value -- Observations from graphical solution: