IMSE881
Fall 2017
8:12 am, Oct 24, 2017
IMSE 881
LINEAR PROGRAMMING
MIDTERM #1
By
Dr. John Wu
Name: ______________________________
eID: ________________________________
Format: Open Book, Take Home
Test Period: 48 hours
Total Points: 120 points
Use computers and any software packages you can.
Scan your report (including this page) and upload to Canvas
Honor’s Pledge: (Must sign to get credits)
I __________________ have not asked and received helps from someone else on this exam.
Signature: ___________________________ Date: _______/_______/_______
Problem #1 (20 points) True (T) or False (F):
(Please provide explanations if you’re not sure T or F answers are not fully cover your points)
____
(1) A two-dimensional solution space with two equality constraints can include an infinity of
feasible point only if the two lines coincide (i.e., the two constraints are dependent).
____
(2) If a nonbasic variable xk has zk − ck = 0 at optimality, then can one claim that alternative optimal
solutions exist.
____
(3) The revised simplex method is designed to reduce the amount of computations and to economize
the use of computer memory.
____
(4) If the number of primal variables is much smaller than the number of constraints, it is more
efficient to obtain the solution of the primal by solving its dual.
____
(5) If the primal is infeasible, the dual always has unbounded optimum.
____
(6) If an optimal dual variable is positive, its associated primal constraint may not be active ( i.e.,
satisfied in its equality form).
____
(7) Changes in any coefficient of the linear program other than the right-hand side of its constraints
can affect only the optimality of the solution.
____
(8) If there exists a direction d such that Ad = 0, d ≥ 0, and cTd > 0, then is the optimal objective
value unbounded?
____
(9) Is it possible for an optimal solution to have more than m positive variables? where m denotes
number of constraints.
____
(10) Converting a minimization problem to a maximization problem changes the sign of the dual
variables.
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Problem #2 (20 points)
In the tableau for the minimization problem below, the values of the seven constants α1, α2, α3,
β1, β2, δ1, and δ2 are unknown (assume there are no artificial variables):
Z
1
0
x1
δ1
x3
0
1
x4
0
0
x5
-3
α2
x6
0
0
RHS
Z
β1
Bs
4
x2
δ2
α1
0
0
-1
α3
-5
-3
0
0
1
0
-1
-4
0
1
β2
a4
a6
4
a3
Clearly state the restrictions or conditions applied on these seven unknowns (α1, α2, α3, β1, β2, δ1,
and δ2) which would make the following statements true about the given tableau:
a)
The current solution is optimal but alternate optimum exists.
b)
The current solution is not primal feasible.
c)
The solution at the next iteration will be degenerate.
d)
The current solution is feasible but the problem has no finite optimum.
e)
The current solution is feasible but the objective function value can be improved by
replacing x6 by x1. What will be the total change in the objective function value after
the pivot?
(Example of restrictions on the unknowns: α1 > 0, β1 = 0 and δ1 ≤ 0.)
Page | 3
Problem #3 (20 points)
Consider the optimal tableau shown in the table below for a minimization problem,
Z
1
0
0
0
x1
0
0
1
0
x2
0
0
0
1
x3
a
15
1
7
x4
0
1
0
0
x5
b
7/2
1/2
2
x6
c
-3/2
-1/2
-1
RHS
g
d
e
f
Bs
a4
a1
a2
In the above tableau, x4, x5, and x6 are the slack variables, respectively, for the constraints 1, 2 and
3 of the ≤ type.
a)
b)
c)
d)
e)
f)
g)
Find B−1.
Find B.
Find original tableau.
From the tableau give identify cbTB−1 and give its interpretation.
What are the limitations (conditions) for a, b, c, d, e, and f?
Give an extreme direction and its conditions that the extreme direction exists.
List the first constraint and what is the current value (left-hand-side) of first constraints
ignoring the slack variable x4?
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Problem 4: (20 points) To improve tax compliance the Texas Comptroller's staff regularly
audits at corporate home offices the records of out-of-state corporations doing business in
Texas. Texas is considering the opening of a series of small offices near these corporate
locations to reduce the travel costs now associated with such out-of-state audits. The
following table shows the fixed cost (in thousands of dollars) of operating such offices at 5
sites i, the number of audits required in each of 5 states j, and the travel cost (in thousands of
dollars) per audit performed in each state from a base at any of the proposed office sites.
Tax
Site
1
2
3
4
5
Audits
Fixed
Cost
160
49
246
86
100
Cost to Audit of Corporate
Location:
1
2
3
4
5
0
0.4
0.8
0.4
0.8
0.7
0
0.8
0.4
0.4
0.6
0.4
0
0.5
0.4
0.6
0.4
0.9
0
0.4
0.9
0.4
0.7
0.4
0
200
100 300 100 200
We seek a minimum total cost auditing plan. (1 point each)
(a) Briefly explain why appropriate decision variables for an optimization model of this problem
are
xij : fraction of audits at j done from i
1,
if office 𝑖𝑖 is opened
𝑦𝑦𝑖𝑖 = �
0,
otherwise
(b) Explain why the yi must be modeled as discrete variables.
(c) Assign suitable symbolic names to the constants in the foregoing table: the fixed cost of office
i, the travel cost for audits done at j from i, and the number of audits at j.
(d) Formulate an objective function minimizing the sum of fixed office operating cost plus travel
costs to audit sites. (Hint: The number of audits done at i from j is xij times the total number
required at j.)
(e) Formulate a system of 5 main constraints requiring that 100% of audits at each j be performed.
(f) Formulate a system of 25 main constraints specifying that no part of the audits at any j can be
done from i unless an office is opened at i.
(g) Complete your model with systems of variable-type constraints for the x and y decision variables.
(h) Is your model best classified as an LP, an NLP, an ILP, or an INLP, and is it single- or multiobjective? Explain.
(i) (2 points) Enter and solve your model with a optimization software.
Page | 5
Problem 5: (20 points) Consider the following Primal-Dual Linear Programming Problem
(P)
Minimize
ctx
Subject to
Ax ≥ b
(D)
⇒
Maximize
btw
Subject to
Atw ≤ c
x≥0
w≥0
a) (10 points) Show that xt(Atw - c) = 0, and wt(Ax - b) = 0.
b) (10 points) Given that x’ and w’ are the feasible solutions for the (P) and (D) respectively,
show that ctx’ ≥ btw’.
Page | 6
Problem 6: (20 points) Consider the canonical primal-dual pair
(P)
max
z = ctx
s.t.
Ax ≤ b
(D)
min
z = btw
s.t.
Atw ≥ c
w≥0
x≥0
After adding a slack vector s to the primal and a surplus vector λ to the dual, the problem can be
rewritten as follows.
(P)
max
z = ctx
s.t.
Ax + s = b
(D)
min
z = btw
s.t.
Atw - λ = c
x ≥ 0, s ≥0
w ≥ 0, λ ≥ 0.
Prove the following (Explain the following the best you can):
a) xj is unbounded in the primal feasible region if and only if its complementary dual variable λj
is bounded above in the dual feasible region.
b) xj is zero in all optimal solutions to the primal if and only if its complementary dual variable
λj is strictly positive in some optimal feasible solution to the dual.
c) xj is strictly positive for some optimal solutions to the primal if and only if its complementary
dual variable λj is bounded above in the set of all optimal dual feasible solutions.
d) xj is unbounded on the set of all optimal primal solutions if and only if its complementary dual
variable λj is zero for all optimal dual feasible solutions.
e) Replace xj by si , replace λj by wi and repeat parts b) and c).
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