Operation Research Midterm

Problem 4. (25 Points)
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