OPRS 3111
Dr. Franz Rothe
October 14, 2010
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Name:
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Homework
10 Problem 1.1. Solve the problem graphically. How many optimal solutions do
exist? How do you see that?
max 20x1 + 15x2
subject to x1
≤ 100
x2
≤ 100
50x1 + 35x2 ≤ 6000
20x1 + 15x2 ≥ 2000
Answer. The constraints yield the sides BC, DA, AB and CD of the blue quadrilateral
which is the feasible region. Shifting an equiprofit line as much as possible to the left
yields an equiprofit line with maximal profit through vertex A. Hence A = (50, 100) is
the unique solution. The maximum profit is z = 2500.
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Figure 1: Unique optimum at (50, 100) with objective 2500.?
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10 Problem 1.2. Farmer Jane owns 45 acres, on which she is going to plant
wheat or corn. Each acre planted with wheat yields $200 profit; and each acre planted
with corn yields $300 profit. Labor costs are 3W + 2C, the fertilizer costs are 2W + 4C.
One hundred workers and 120 tons of fertilizer are available.
Solve the problem graphically. How many optimal solutions do exist? How do you
see that?
Answer. The unique optimum occurs at C = 20, W = 20. The maximum profit is 1000$.
The land constraint C + W ≤ 45 turns out to be non binding.
Figure 2: The optimal choice is planting 20 acres corn and 20 acres wheet.
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40 Problem 1.3. Solve the problems graphically: By default, all variables are
nonnegative.
• Indicate the feasible region in blue.
• Draw an arbitrary equiprofit line in orange.
• Indicate the optimal solutions in red.
and decide which one of the four possible cases occur:
• empty feasible region
• multiple solutions
• unbounded objective
• unique optimal solution
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Answer.
(a)
max x1 + 2x2
subject to x1 − x2 ≤ 3
x1 + x2 ≤ 4
Figure 3: Problem (a) has the unique solution x1 = 0, x2 = 4 with maximum objective 8.
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(b)
max x1 − x2
subject to x1 − x2 ≤ 3
x1 + x2 ≤ 4
Figure 4: Problem (b) has as solution set the segment (3, 0) to (3.5, 3.5). Maximum objective
is 3.
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(c)
min x1 − x2
subject to x1 − x2 ≤ 3
x1 + x2 ≥ 4
Figure 5: Problem (c) has unbounded objective.
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(d)
max x1 + 2x2
subject to x1 − x2 ≤ −5
x1 + x2 ≤ 4
Figure 6: Problem (d) has empty feasible region.
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(e)
min x1 + x2
subject to x1 − x2 ≤ 3
x1 + x2 ≥ 4
Figure 7: Problem (e) has as solution set the segment (0, 4) to (3.5, 3.5). Minimum objective
is 4.
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Definition 1. A region in a Euclidean space (or any vector space) is called convex iff
it contains the segment between any two of its points.
The extremal points of a region are those points which are not interior points of the
segment between any two points of the region.
10 Problem 1.4. Draw the regions (as closed sets). Tell whether they are convex
or not. Indicate the set of extremal points with a different color.
(a) circular disk
(b) half disk
(c) rectangle semicircles erected on two opposite sides (football stadium)
(d) rectangle
(e) star with four spikes
(f ) annulus
Answer. Interior points are never extremal, but boundary points can be extremal points
or not. The extremal points are the part of the boundary curved convexly, and the
vertices between straight segments which are pointing outside. Here is my hardware
store:
Figure 8: Convex shapes (a)(b)(c)(d) and non-convex shapes (e)(f). The extremal points
are the red parts of the boundary.
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