introduction to mathematical optimization , operation research
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OPERATIONS RESEARCH HOMEWORK
Acme food processing plant which manufactures hamburger buns and burger patties. They grind their own flour for the hamburger buns at a maximum rate of 200 kilograms per week. Each hamburger bun requires 0.1 kgs of flour. They currently have a contract with Seawood, Inc. which specifies that a delivery of 800 kgs of meat product is delivered every Monday. Each hamburger patty requires ¼ kgs of minced meat. All the other ingredients in the hamburger patties and hamburger buns are in plentiful supply. Finally, the labor force at Acme consists of 5 employees working full time (40 hours per week each). Each hamburger patty requires 3 minutes of labor, and each hamburger bun requires 2 minutes of labor. Each hamburger patty yields a profit of $0.20, and each hamburger ban yields a profit of $0.10.
Acme would like to know how many hamburger patties and how many hamburger buns they should produce each week so as
to achieve the highest possible profit.
(a) Formulate a linear programming model for this problem.
(b) Use the graphical method to solve this model.
The Metsan co. desires to blend a new alloy of 40 percent copper, 35 percent brass, and 25 percent silver from several available alloys having the following properties:
Alloy
1 2 3 4 5
Percentage of copper 60 25 45 20 50
Percentage of brass 10 15 45 50 40
Percentage of silver 30 60 10 30 10
Cost ($/kg) 110 100 125 120 135
The objective is to determine the proportions of these alloys that should be blended to produce the new alloy at a minimum cost.
(a) Formulate a linear programming model for this problem.
(b) Solve this model by the simplex method.
The feasible region for an LP model is given below:
The objective is to maximize the total profit from the two activities. The unit profit for activity 1 is $1,000 and the unit profit for activity 2 is $2,000.
(a) Calculate the Z value for each cornerpoint basic feasible solution. Use this information to find an optimal solution.
(b) Identify the sequence of basic feasible solutions that would be obtained by the simplex method to reach an optimal solution.
Consider the following problem.
Maximize Z = 2x1 + 5x2 + 3x3,
subject to
x1 - 2x2 + x3 ≥ 20
2x1 + 4x2 + x3 = 50
and
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.
(a) Using the Big M method, construct the complete first simplex tableau for the simplex method and identify the corresponding initial (artificial) basic feasible solution. Also identify the initial entering basic variable and the leaving basic variable.
(b) Work through the simplex method step-by-step to solve the problem.
(c) Using the two-phase method, construct the complete first simplex tableau for phase 1 and identify the corresponding initial (artificial) basic feasible solution. Also identify the initial entering
basic variable and the leaving basic variable.
(d) Work through phase 1 step-by-step.
(e) Construct the complete first simplex tableau for phase 2.
(f ) Work through phase 2 step-by-step to solve the problem.
(g) Compare the sequence of BF solutions obtained in part (b) with that in parts (d) and ( f ). Which of these solutions are feasible only for the artificial problem obtained by introducing artificial
variables and which are actually feasible for the real problem?
Project ID: #28908105
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