Solve the following problem graphically (Please be neat). Draw the polytope on the x-y coordinate system (can be done either by hand or computer). Show all intersection of the polytope and identify the point (x,y coordinate) where the objective function is maximized and provide that value.

Maximize Z = 3x1 + 2x2

Subject to:

1x1 + 1x2 ≤ 10

8x1 + 1x2 ≤ 24

and

x1, x2 ≥ 0

Solution :

Point (a) is the origin (0,0) where Z(a) = 3*0 + 2*0 = 0

Point (b) is the intersection of line 2 and X-axis (3,0) where Z(b) = 3*3 + 2*0 = 9 Point (c) is the intersection of line 1 and line 2 (2,8) where Z(c) = 3*2 + 2*8=22 . (Optimum Solution) Point (d) is the intersection of line 1 and Y-axis (0,10) where Z(d) = 3*0 + 2*10 = 20 Y

X

d

a

b

c

I

II

Problem 5. (30 Points)

Work through the simplex method (in algebraic form) step by step to solve the following problem. Show all work and provide the solutions for each variable at every iteration of the simplex.

Maximize z = 4x1 + 3x2 + 4x3

Subject to:

2x1 + 2x2 + 1x3 ≤ 20

2x1 + 1x2 + 2x3 ≤ 14

1x1 + 1x2 + 3x3 ≤ 15

and

x1, x2, x3 ≥ 0

Solution :

Problem 6. (30 Points)

The Weigelt Corporation has three branch plants with excess production capacity. Fortunately, the corporation has a new product ready to begin production, and all three plants have this capability, so some of the excess capacity can be used in this way. This product can be made in three sizes--large, medium, and small--that yield a net unit profit of $420, $360, and $300, respectively. Plants 1, 2, and 3 have the excess capacity to produce 750, 900, and 450 units per day of this product, respectively, regardless of the size or combination of sizes involved.

The amount of available in-process storage space also imposes a limitation on the production rates of the new product. Plants 1, 2, and 3 have 13,000, 12,000, and 5,000 square feet, respectively, of in-process storage space available for a day's production of this product. Each unit of the large, medium, and small sizes produced per day requires 20, 15, and 12 square feet, respectively.

Sales forecasts indicate that if available, 900, 1,200, and 750 units of the large, medium, and small sizes, respectively, would be sold per day.

At each plant, some employees will need to be laid off unless most of the plant’s excess production capacity can be used to produce the new product. To avoid layoffs if possible, management has decided that the plants should use the same percentage of their excess capacity to produce the new product.

Management wishes to know how much of each of the sizes should be produced by each of the plants to maximize profit.

1. Formulate a linear programming model for this problem by: A. Listing and labeling all of the decision variables. B. Creating an objective function for the model. C. List all of the constraints for the model.

I want a complete model, not just an Excel sheet.

2. Solve the model using Excel® solver or Open Office Solver. Give the value for each decision variable and the objective function.

Solution :

2.

Product 1 Plant 1| X11| 517| | |

Product 1 Plant 2| X12| 0| | |

Product 1 Plant 3| X13| 0| | |

Product 2 Plant 1| X21| 178| | |

Product 2 Plant 2| X22| 667| | |

Product 2 Plant 3| X23| 0| | |

Product 3 Plant 1| X31| 0| | |

Product 3 Plant 2| X32| 167| | |

Product 3 Plant 3| X33| 417| | |

| | | | |

Objective Function| | | | |

MaxZ=420(X11+X12+X13)+360(X21+X22+X23)+300(X31+X32+X33)| 696000.0044| | | | | |

Constraints | | | | Surplus|...