Consider the following all-integer linear program.1x₁ + 1x₂5X₁ + 6x₂ ≤ 311x₁ + 5x₂ ≤ 202X₁ + 1x₂ ≤ 11X₁, X₂ ≥ 0 and integer(a) Graph the constraints for this problem. Use dots to indicate all feasible integer solutions.X286X28Maxs.t.62244=6(b) Solve the LP Relaxation of this problem.at (x₁, x₂)(c) Find the optimal integer solution.at (x₁, x₂) =688X1X2842X28622244668CO8X1ड्र@

Linear programming is a mathematical technique that is also used in operations management…

Answered: Consider the following all-integer…

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter2: Introduction To Spreadsheet Modeling

Section: Chapter Questions

Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...

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Maximize Z = x₁ + x₂, subject to X₁ X₂ ≤-3 2x₁ + 7x₂ ≤ 4 and x₁ ≥ 0, x₂ ≥ 0.
A farmer has 300 acres of available land and $60,000 to spend. He wants to plant the combination of crops which maximizes his profit. Complete parts (a) through (c). Profits and Constraints for Crops Potatoes Corn x2 $160 $60 Number of Acres Cost (per acre) Profit (per acre) (a) Give the dual problem. subject to: X₁ $400 $180 with W = y₁ + Y₁ + 400y2 Y₁ + 160y₂ Y2 180 60 90 Planting acres of potatoes, (Simplify your answers.) Cabbage X3 $280 $90 VI VI II Y₁ + 280y₂ Y₁ ≥ 0, y₂ ≥0 > (b) The solution to the dual problem can be interpreted as shadow profits. Use shadow profits to estimate the farmer's profit if land is cut to 270 acres but capital increases to $63,000. The farmer's estimated profit is $ (Simplify your answer.) acres of corn, and = Total 300 $60,000 Z (c) Suppose the farmer has 330 acres of land but only $57,000. Find the optimum profit and the planting strategy that will produce this profit. acres of cabbage will produce an optimal profit of $¯.
a. Identify the feasible solution. b. Obtain the optimal solution. Note: Assumer or determine the objective function from available data.
example 5a
Estimate a range for the optimal objective value for the following LPs: *(a) Minimize z = 5x₁ + 2x₂ subject to (b) Maximize z subject to x₁ + 5x₂ + 3x3 X₁ X₂ ≥ 3 2x₁ + 3x₂ = 5 X1, X₂ ≥ 0 x₁ + 2x₂ + x3 = 30 2x₁ - x₂ = 40 X1, X2, X3 ≥ 0
b) Maximize Z = −40X1 −100X2s.t 10X1 + 5X2 ≤ 2502X1 + 5X2 ≤ 1002X1 + 3X2 ≤ 90X1, X2 ≥ 0Solve by simplex method, what are the solutions? Show that this problem hasmultiple solutions and find the solutions?
Consider the apportionment of 27 projectors for a school district with four campus locations labeled A, B, C, and D. The following table shows the apportionment of the projectors using the Hamilton method. A B 836 1936 C 312 Campus Enrollment Apportionment of 27 projectors 4 9 1 D 2736 13 (a) If the number of projectors to be apportioned increases from 27 to 28, what will be the apportionment if the Hamilton method is used? Campus A Campus B Campus C Campus D Did the Alabama paradox occur? O Yes O No (b) If the number of projectors to be apportioned using the Hamilton method increases from 28 to 29, will the Alabama paradox occur? O Yes O No
For each of the following constraints, draw a separate graph to show the nonnegative solutions that satisfy this constraint.
The manager of Jokitian Iron Works Company received an order to produce anumber of pipes which require the use of material A costing $3 and material Bcosting $8 per unit. For each pipe no more than 12 units of Material A and at least16 units of material B must be used. While each unit of A weighs 4 pounds andeach unit of B weighs 6 pounds, the final product must weigh exactly 120 pounds.a. Present the objective function.b. Present the constraint functions.c. Show and label the graph of the feasible solution.d. How many units of each raw material should be used in order to producethe ordered pipes most economically?e. What is the cost of each pipe?
b) The figure below presents an unbounded feasible region of a five constrained LP problem. 10- |(0, 6) 6. (1, 4) 4 (3, 2) 2+ (9, 0) 8 10 i. Write down in terms of x and y the inequalities that represent these constraints. ii. If the objective function is Z = 0.12x + 0.15y, determine the optimal solution to this problem (show the movement of the isocost line). ii. Identify the binding and non-binding constraints.
4a) Convert the following problem to Standard Form. max z := -2x1 + x2 s.t. 2x1 + x2 1 x1 > 0, x2 free. 4b) Write the augmented form of the following standard form problem. max z := 4.x1 – 25x2 x1,x2,x3,x4 s.t. x1 – x3 = 5 X2 – x3 + x4 = 0 x > 0

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Consider the following all-integer linear program. 1x₁ + 1x₂ Max s.t. 5X1 + 6x₂ 31 1x₁ + 5x₂ ≤ 20 2x₁ + 1x₂ ≤ 11 X₁, X₂20 and integer (a) Graph the constraints for this problem. Use dots to indicate all feasible integer solutions. X2 X2