The constraint x$1+$ x$2+$ x$3+$ x$4<=2$ means that two out of the first four projects must be selected.

Question $1$ options:

a$)$ True

b$)$ False

Question $2(2\mathrm{.}5$ points$)$

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In an integer linear program,

Question $2$ options:

a$)$ all objective function coefficients must be integer.

b$)$ all right$-$hand side values must be integer.

c$)$ all variables must be integer.

d$)$ all objective function coefficients and right$-$hand side values must be integer.

Question $3(2\mathrm{.}5$ points$)$

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Shadow prices cannot be used for integer programming sensitivity analysis because they are designed for linear programs.

Question $3$ options:

a$)$ True

b$)$ False

Question $4(2\mathrm{.}5$ points$)$

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Binary variables are variables whose only possible values are $0$ or $1.$

Question $4$ options:

a$)$ True

b$)$ False

Question $5(2\mathrm{.}5$ points$)$

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How is an LP problem changed into an ILP problem?

Question $5$ options:

a$)$ by adding discontinuity constraints.

b$)$ by adding constraints that the decision variables be non$-$negative.

c$)$ by adding integrality conditions.

d$)$ by making the right hand side values integer.

Question $6(2\mathrm{.}5$ points$)$

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Which types of questions can be answered by Binary Integer Programming problems?

Question $6$ options:

a$)$ How much of a product should be produced?

b$)$ Should an investment be made?

c$)$ Should a plant be located at a particular location?

d$)$ b and c$.$

Question $7(2\mathrm{.}5$ points$)$

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BIP can be used in capital budgeting decisions to determine whether to invest a certain amount.

Question $7$ options:

a$)$ True

b$)$ False

Question $8(2\mathrm{.}5$ points$)$

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Let x$1,$ x$2,$ and x$3$ be Binary variables whose values indicate whether the projects are not done $(0)$ or are done $(1).$ Which answer below indicates that only one must be done?

Question $8$ options:

a$)$ x$1+$ x$2+$ x$3>=1$

b$)$ x$1+$ x$2+$ x$3=1$

c$)$ x$1+$ x$2+$ x$3<=1$

d$)$ x$1+$ x$2+$ x$3>=0$

Question $9(2\mathrm{.}5$ points$)$

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The constraint x$1+$ x$2+$ x$3<=3$ in a BIP represents mutually exclusive alternatives.

Question $9$ options:

a$)$ True

b$)$ False

Question $10(2\mathrm{.}5$ points$)$

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In an all$-$integer linear program,

Question $10$ options:

a$)$ all variables must be integer.

b$)$ all objective function coefficients and right$-$hand side values must be integer.

c$)$ all right$-$hand side values must be integer.

d$)$ all objective function coefficients must be integer.

Question $11(2\mathrm{.}5$ points$)$

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Linear programming must have integer solutions.

Question $11$ options:

a$)$ True

b$)$ False

Question $12(2\mathrm{.}5$ points$)$

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Problems where all the variables are binary variables is called a pure BIP problem.

Question $12$ options:

a$)$ True

b$)$ False

Question $13(2\mathrm{.}5$ points$)$

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Variables whose only possible values are $0$ and $1$ are called integer variables.

Question $13$ options:

a$)$ True

b$)$ False

Question $14(2\mathrm{.}5$ points$)$

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Rounding the solution of a Linear Programming model without an integer constraint on the variables to the nearest integer values provides

Question $14$ options:

a$)$ an infeasible solution.

b$)$ an integer solution that might be neither feasible nor optimal.

c$)$ a feasible but not necessarily optimal integer solution.

d$)$ an integer solution that is optimal.

Question $15(2\mathrm{.}5$ points$)$

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In a model, x$1>=0$ and integer, x$2>=0,$ and x$3=0,1.$ Which solution would not be feasible?

Question $15$ options:

a$)$ x$1=4,$ x$2=\mathrm{.}389,$ x$3=1$

b$)$ x$1=0,$ x$2=8,$ x$3=0$

c$)$ x$1=2,$ x$2=3,$ x$3=\mathrm{.}578$

d$)$ x$1=5,$ x$2=3,$ x$3=0$

Question $16(2\mathrm{.}5$ points$)$

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Binary variables can have the following values:

Question $16$ options:

a$)0.$

b$)1.$

c$)$ any integer value.

d$)$ a and b only.

e$)$ All of the above.

Question $17(2\mathrm{.}5$ points$)$

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Let x$1,$ x$2,$ and x$3$ be Binary variables whose values indicate whether the projects are not done $(0)$ or are done $(1).$ Which answer below indicates that at least two of the projects must be done?

Question $17$ options:

a$)$ x$1+$ x$2+$ x$3>=2$

b$)$ x$1+$ x$2+$ x$3=2$

c$)$ x$1+$ x$2+$ x$3<=2$

d$)$ x$1-$ x$2=0$

Question $18(2\mathrm{.}5$ points$)$

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Binary integer programming problems are those where all the decision variables restricted to integer values are further restricted to be binary variables.

Question $18$ options:

a$)$ True

b$)$ False

Question $19(2\mathrm{.}5$ points$)$

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In a BIP problem with $3$ mutually exclusive alternatives, x$1,$ x$2,$ and x$3,$ the following constraint needs to be added to the formulation:

Question $19$ options:

a$)$

x$1+$ x$2+$x$3<=1.$

b$)$

x$1+$ x$2+$x$3=1.$

c$)$

x$1-$ x$2-$x$3<=1.$

d$)$

x$1-$ x$2-$x$3=1.$

e$)$

None of the above.

Question $20(2\mathrm{.}5$ points$)$

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Rounded solutions to linear programs must be evaluated for

Question $20$ options:

a$)$ feasibility and optimality.

b$)$ sensitivity and duality.

c$)$ sensitivity and optimality.

d$)$ each of these choices are true.