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Quantitative Methods - Integer programming

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1. In using rounding of a linear programming model to obtain an integer solution, the solution is:
a. always feasible.
b. always optimal.
c. sometimes optimal and feasible.
d. always optimal and feasible.
e. never optimal and feasible.

2. The linear programming relaxation contains the objective function and the original constraints of the integer-programming problem but drops all ________.
a. decision variables
b. different variables
c. slack values
d. integer restrictions
e. nonnegativity constraints

3. If we are solving a 0-1 integer programming problem, the constraint x1 <= x2 is a ________ constraint.
a. Corequisite
b. mutually exclusive
c. conditional
d. multiple-choice
e. none of the above

4. Assume that we are using 0-1 integer programming model to solve a capital budgeting problem and xj = 1 if project j is selected and xj = 0 otherwise.
The constraint (x1 + x2 + x3 + x4 <= 2) means that ________ out of the 4 projects must be selected.
a. exactly 2, 4
b. at least 2, 4
c. exactly 1, 4
d. at most 2, 4

5. The branch and bound method of solving linear integer programming problems is ________.
a. a graphical solution
b. an enumeration method
c. an integer method
d. a relaxation method

6. The Wiethoff Company has a contract to produce 10000 garden hoses for a customer. Wiethoff has 4 different machines that can produce this kind of hose. Because these machines are from different manufacturers and use differing technologies, their specifications are not the same.

Fixed cost to set
Machine up production run Variable cost per hose Capacity
1 750 1.25 6000
2 500 1.50 7500
3 1000 1.00 4000
4 300 2.00 5000
The company wants to minimize total cost. Give the objective function.

a. Min 750y1+500y2+1000y3+300y4
b. 1.25x1+1.5x2+x3+2x4
c. Min 750y1+500y2+1000y3+300y4+1.25x1+1.5x2+x3+2x4
d. none of the above

7. If a maximization linear programming problem consists of all less than or equal to constraints with all positive coefficients and the objective function consists of all positive objective function coefficients, then rounding down the linear programming optimal solution values of the decision variables will ________ result in a feasible solution to the integer linear programming problem.

a. Sometimes
b. Never
c. Always

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Please see explanation and answer for each question as given below. Also see word document. Hope this helps you.

1. In using rounding of a linear programming model to obtain an integer solution, the solution is:
a. always feasible.
b. always optimal.
c. sometimes optimal and feasible.
d. always optimal and feasible.
e. never optimal and feasible.
à Explanation: Rounding of fractional values may result in infeasible solution also. In case of maximization problem rounding up may give infeasible solution, but rounding down may give feasible and optimal solution. In case of minimization problem rounding down may give infeasible solution, but rounding up will feasible and if fortunate optimal. Therefore, for maximization problem rounding down may give feasible and optimal solution and for minimization problem rounding up may result in feasible and optimal solution. Answer is 'c'.

2. The linear programming relaxation contains the objective function and the original constraints of the integer-programming problem but drops all ________.
a. decision variables
b. different variables
c. slack values
d. integer restrictions
e. nonnegativity constraints
à Explanation: Relaxation problem is the problem with same objective function and constraints set, but without integer restriction. Relaxations are used to solve original problem, ...

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