1. The output for the below linear programming problem follows below:

Max: 25X_{1}+30X_{2}+15X_{3}

S.T.

1) 4X_{1}+5X_{2}+8X_{3}<1200

2) 9X_{1}+15X_{2}+3X_{3}<1500

Optimum Solution Output

Objective Function Value = 4,700.00

Variable | Value | Reduced Cost |

X_{1} |
140.000 | 0.000 |

X_{2} |
0.000 | 10.000 |

X_{3} |
80.000 | 0.000 |

Constraint | Slack/Surplus | Dual Price |

1 | 0.000 | 1.000 |

2 | 0.000 | 2.333 |

Objective Coefficient Ranges:

Variable | Lower Limit | Current Value | Upper Limit |

X_{1} |
19.286 | 25.000 | 45.000 |

X_{2} |
None | 30.000 | 40.000 |

X_{3} |
8.333 | 15.000 | 50.000 |

RHS Ranges:

Constraint | Lower Limit | Current Value | Upper Limit | ||

1 | 666.667 | 1200.000 | 4000.000 | ||

2 | 450.000 | 1500.000 | 2700.000 | ||

a. (10 Points): Give the complete optimal solution

b. (10 Points): Which constraints are binding?

c. (15 Points): What is the dual price for the second constraint? What does this mean?

d. (10 Points): Over what range can the objective function coefficient of x_{2} vary before a new solution point becomes optimal?

e. (10 Points): By how much can the amount of resource 2 decrease before the dual price will change?

f. (10 Points): What would happen if the first constraint’s right-hand side increased by 700 and the second’s decreased by 350?

2. Consider the following linear programming problem

Max 8X + 7Y

s.t. 15X + 5Y < 75

10X + 6Y < 60

X + Y < 8

X, Y ³ 0

a. (10 Points): Use a graph to show each constraint and the feasible region.

b. (15 Points): Identify the optimal solution point on your graph. What are the values of X and Y at the optimal solution?

c. (10 Points) What is the optimal value of the objective function?

## Expert Answer

Final value of variables are:

x1 = 140

x2 = 0

x3 = 80

b) Both constraints are binding as in table 2, surplus or slack of constraints are 0

c) Dual price for constraints 2 = 2.333. It means, with each unit increase of constraints 2, the optimal function will increase by 2.333

d) Coefficient range for x2 is: (None, 40). Thus, on lower side it can be anything but on upper limit it is 40. Beyond that, the optimal solution will change

e) The x2 has current value of 1500. It can change upto 450. Thus, 1050 can be changed without impacting dual price

f) In this case, optimal solution will change but dual price will remain same for both constraints as it is within range