In a linear model with m constraints there will never

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Unformatted text preview: rge a change can be made? –  What if more than one change is made? Shadow Prices •  If we added 1 hour of grinding time, we would increase profit by 6.25. We could add as much as 96 (=384-288) hours, and gain 96 x 6.25 = 600. •  If we added 1 hour of manpower time, we would increase profit by 23.75. We could add as much as 22.1(=406.1-384) hours, and gain 22.1 x 23.75=524.875 •  In either situation, the amount produced will change. Change in Manpower Adjacent Solutions •  •  •  Notice the extreme solutions, at most two products are made. Is it ever optimal to make three different products? More than three? –  It is possible for three products to be in the optimal solution. –  However, there will never be more than three non-zero products because there are only three constraints. In a linear model with m constraints, there will never be more than m non-zero variables in an optimal basic (extreme point) solution. In a nonlinear model, the optimal solution is not always an extreme point solution (may be interior). Reduced Costs and Obj Ranging SensiFvity Analysis: Types of LP Changes Six types of changes in an LP’s parameters change the optimal solution: 1.  Changing the objective function coefficient of a nonbasic variable. 2.  Changing the objective function coefficient of a basic variable. 3.  Changing the right-hand side of a constraint. 4.  Changing the column of a nonbasic variable. 5.  Adding a new variable or activity. 6.  Adding a new constraint. Multiple Changes: Parametric Programming Multiple Changes: Parametric Programming Multiple Changes: The 100% Rule 100% Rule for Changing Objective Function Coefficients Depending on whether the objective function coefficient of any variable with a zero reduced cost in the optimal tableau is changed, there are two cases to consider: Case 1 – All variables whose objective function coefficients are changed have nonzero reduced costs in the optimal row 0. •  the current basis remains optimal if and only if the objective function coefficient for each variable remains within the allowable range. •  If the current basis remains optimal, both the values of the decision variables and objective function remain unchanged. •  If the objective coefficient for any variable is outside the allowable range, the current basis is no longer optimal. Case 2 – at least one variable whose objective function coefficient is changed has a reduced cost of zero. •  Apple the 100% rule. Diet Problem Define: BR = # of Brownies IC = # of Scoops of Ice Cream COLA = # of Cans of Cola PC = # of Pieces of Pineapple MIN 50 BR + 20 IC + 30 COLA + 80 PC SUBJECT TO 400 BR + 200 IC + 150 COLA + 500 PC >= 500 (Calorie constraint) 3 BR + 2 IC >= 6 (Chocolate constraint) 2 BR + 2 IC + 4 COLA + 4 PC >= 10 (Sugar constraint) 2 BR + 4 IC + COLA + 5 PC >= 8 (Fat constraint) END OBJECTIVE FUNCTION VALUE 1) 90.00000 VARIABLE VALUE REDUCED COST BR 0.000000 27.500000...
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