1
Managerial Economics (ARE) 155
University of California, Davis
Fall Quarter, 2011
Instructor: John H. Constantine
Homework 3: Due Friday, October 21, 2011 (ONE WEEK AFTER MIDTERM 1)
Problems 1 – 4 should be done prior to the midterm.
Problem 5 should be done after the midterm.
Problem 1
:
A furniture firm produces three outputs chairs, tables, and desks.
Production of the output requires inputs
labor (x
1
) and wood (x
2
).
The firm’s objective is to minimize the cost of meeting the three production
constraints.
min TC = 10x
1
+ 40x
2
s.t.
2x
1
+
8x
2
≤
40
(chairs)
3x
1
+
9x
2
≥
36
(tables)
8x
1
≥
32
(desks)
x
1
, x
2
≥
0
(a)
How many primal slack variables are there?
Why?
(b)
How many dual choice variables and dual slack variables are there?
Explain briefly.
(c)
Solve and graph the primal problem given above.
State the optimal values for x
j
and s
i
, as well as TC.
(d)
Write the dual LP.
(e)
Clearly provide an economic interpretation for dual constraint 1.
(Remember, in this problem the
primal variables (x
j
) are inputs and primal constraints are inputs (b
1
).
This changes how to interpret
the problem relative to a “max” problem where the x
j
are outputs and b
i
are inputs.)
(f)
Using your results from the Primal problem above, state the all ComplementarySlackness
relationships.
State the dual variables that are possibly nonzero.
(g)
Rewrite the dual LP of problem (d), but now only including the potentially nonzero variables as
implied by your answers in parts (a) thru (F).
(f)
Are your dual constraints of part (g) written with equality or inequality constraints?
Explain.
(g)
State the complete dual LP answer by using the answer in part (e).
State the optimal values for all (i)
dual choice variables, (ii) dual slack variables and (iii) TC.
You do not need to graph this problem to
solve it.
(i)
Given the optimal solution to this problem, by how much will the value of the objective function
change if b
2
increases from 36 to 37?
Explain this answer in the context of the problem of the factory
hiring inputs to produce outputs.
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 Fall '08
 Staff
 Linear Programming, Optimization, Dual problem, production plan, Evans Enterprises

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