Economics 172A: Introduction to Operations Research
Winter 2008
Problem set 1 Answers
Due Thursday, January 31 at start of class (no late papers)
Instructions
Unless otherwise noted on homework assignments and on examinations, you are
required to supply complete answers and explain how you got them. Simply stating a
numerical answer is insufficient.
For this assignment, attach printouts of Excel spreadsheets when requested and
indicate where to find the answers for each question the spreadsheet covers. This
assignment asks you to solve many linear programming problems, but most are variations
on the same basic problem. Set up one template for the Excel computations and then
make changes to get the answers for variations of the problem. You need not include a
separate printout for every simplex computation as long as you provide a clear
description of how you got the answers. You are responsible for using the notes on Excel
on the website to figure out how to get Excel answers yourself (I won’t lecture on it).
For this assignment there is no need to provide answer reports and sensitivity
reports, but please do indicate which cells on your spreadsheet have the solution. For
graphs, clearly label the graph and show where the objective function is and how you
identified a solution. If I ask you to solve a problem, please give both the solution (the
optimal x) and the value (the objective function value for the optimal x).
1. Consider the linear programming problem:
Choose x
1
,x
2
≥
0
to solve max y
subject to
x
1
+ 2x
2
≤
5
3x
1
+ x
2
≤
3,
where the objective function y is a function of x
1
and x
2
to be specified.
(a) Graph the feasible region. Put x
1
on the vertical axis and x
2
on the horizontal axis.
[When x
1
is on the vertical axis and x
2
is on the horizontal axis, the feasible region is
a quadrilateral with the west and south edges going along the x
1
and x
2
axes, and
with a point sticking out to the east at (x
1
,x
2
) = (1/7, 18/7). It’s the intersection of the
nonnegative quadrant (x
1
,x
2
≥
0), the area northwest of the line from (x
1
,x
2
)
= (5,0)
to (0,5/2), and the area southwest of the line from (x
1
,x
2
)
= (1,0) to (0,3). The corners,
starting from the southwest and going clockwise, are (0,0), (1,0), (1/7,18/7), (0,5/2).]
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(b) Solve the problem graphically when:
(i) y = x
2
.
[The solution when y = x
2
is at the point in the feasible region farthest to the east:
the intersection of the two lines x
1
+ 2x
2
= 5 and 3x
1
+ x
2
= 3, easily found by algebra
to be (x
1
,x
2
) = (1/7, 18/7).]
(ii) y = x
1
+ x
2
.
[The solution when y = x
1
+ x
2
is at the point in the feasible region farthest to the
northeast (because the slope of the objective function contour, 1, is between the
slopes of the constraints at this corner): again the intersection of the two lines at
(x
1
,x
2
) = (1/7, 18/7).]
(iii) y = x
1
 x
2
.
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 Winter '08
 Crawford
 Economics, Operations Research, Linear Programming, Optimization, objective function, Constraint

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