# Linear programming formulation

Below is a linear programming formulation for a problem with two variables, a and b, and with three constraints, referred to as constraints #1, #2,and #3. Each of the three constraints is a ">=" (greater than or equal to) inequality; the objective is to minimize Z, where the objective function Z is shown in the formulation.

Minimize Z = a + 2b

such that a >= 5 (constraint #1)

b >= 5 (constraint #2)

2a + b >= 20 (constraint #3)

a, b >= 0 (non-negativity constraints)

In this linear programming problem, for the questions that follow: the `solution' refers to the values of the variables, a and b, and the `objective function value' refers to the value of Z at the solution. Please consider each question below as separate from and independent of all others; the changes described are not cumulative. For each change consider it a change to the original problem. Show any work that you can to support your answers.

a. Set up this problem in excel; find the optimal solution using excel solver, and the solution's sensitivity report. What are your solution and your objective function value? For each constraint: what is its shadow price, what is its slack (or, you may call it surplus since these are `>=' constraints), and is it tight (yes or no)?

b. Based on your sensitivity report in part a, what do you anticipate will happen to the solution and objective function value in part a if the right-hand side of constraint #1 goes up by 1? Please be specific.

c. Increase the right-hand side of constraint # 1 by 1 (from 5 to 6) and resolve the problem. What are your solution and your objective function value? Is this consistent with what you anticipated in part b?

d. Based on your sensitivity report in part a, what do you anticipate will happen to the solution and objective function value in part a if the right-hand side of constraint #2 goes up by 1? Please be specific.

e. Return the RHS (right hand side) of constraint 1 to its original value. Increase the right-hand side of constraint # 2 by 1 (from 5 to 6) and resolve the problem. What are your solution and your objective function value? Is this consistent with what you anticipated in part d?

f. Based on your sensitivity report in part a, what do you anticipate will happen to the solution and objective function value in part a if the right-hand side of constraint #3 goes up by 1? Please be specific.

g. Return the RHS (right hand side) of constraint 2 to its original value. Increase the right-hand side of constraint # 3 by 1 (from 20 to 21) and resolve the problem. What are your solution and your objective function value? Is this consistent with what you anticipated in part f?

h. Based on your sensitivity report in part a, what do you anticipate will happen to the solution and objective function value in part a if the right-hand sides of all three constraints go up by 1? Why? Please be specific and show work to support your answer.

i. Increase the right-hand sides of all three constraints by 1 and resolve the problem. What are your solution and your objective function value? Is this consistent with what you anticipated in part h?

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#### Solution Preview

Please see attached files.

Below is a linear programming formulation for a problem with two variables, a and b, and with three constraints, referred to as constraints #1, #2,and #3. Each of the three constraints is a ">" (greater than or equal to) inequality; the objective is to minimize Z, where the objective function Z is shown in the formulation.

Minimize Z = a + 2b

s.t. a >= 5 (constraint #1)

b >= 5 (constraint #2)

2a + b >= 20 (constraint #3)

a, b >= 0 (non-negativity constraints)

In this linear programming problem, for the questions that follow: the 'solution' refers to the values of the variables, a and b, and the 'objective function value' refers to the value of Z at the solution. Please consider each question below as separate from and independent of all others; the changes described are not cumulative. For each change consider it a change to the original problem. Show any work that you can to support your answers.

a. Set up this problem in excel; find the optimal solution using excel solver, and the solution's sensitivity report. What are your solution and your objective function value? For each constraint: what is its shadow price, what is its slack (or, you may call it surplus since these are '>=' constraints), and is it tight (yes or no)?

Please see attached excel file for problem setup and the solution details. The optimal ...

#### Solution Summary

Linear programming formulation is demonstrated in the solution.