# Lecture 15 (Nov. 5)

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```November 5, 2012
AGEC 352-R. Keeney

Recall
 With 2000 total units (maximum) at harbor and 2000 units
(minimum) demanded at assembly plants it is not possible for
slack constraints
 Supply &lt;=2000
 Demand &gt;=2000
 Supply = Demand

Total movement of 2000 motors is the only feasible
combination, leading all constraints to bind
 One binding constraint is trivial
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Transportation problems do not have to be balanced
Real world problems are rarely balanced
If you have an unbalanced model, might want to balance it
with other activities
If Supply &gt; Demand introduce a storage destination that
takes up the excess
 What is the cost of holding excess supply?
▪ Storage costs or waste/spoil
If Demand &gt; Supply introduce a penalty source that deals
with the imbalance
 What is the cost of shipping less than required?
▪ Lost customers or contract penalties

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If the constraints have integers on RHS the optimal
solution will have transport quantities in integers
 This can be shown mathematically
 Convenient for solving smaller problems by hand
 Choose a route to enter the model, then keep adding
until you hit the supply or demand constraint
In a balanced problem, one constraint is mathematically
redundant
 This is the trivial constraint and it is the one with the
constraint that binds (LHS=RHS) but has a zero shadow
price

The assignment problem is the mathematical allocation of
‘n’ agents or objects to ‘n’ tasks
 The agents or objects are indivisible
▪ Each can be assigned to one task only

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Example using Autopower Company:
Auditing the Assembly Plants @
 Leipzig, Nancy, Liege, Tilburg
A VP is assigned to visit and spend two weeks conducting
the audit
 VP’s of Finance, Marketing, Operations, Personnel
Considerations…
 Expertise to problem areas at plants
 Time demands on VP
 Language ability
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VP
Leipzig Nancy Liege Tilburg
Finance
24
10
21
11
Marketing
14
22
10
15
Operations
15
17
20
19
Personnel
11
19
14
13
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How do you get those costs?
Clearly when you are talking about opportunity costs and
the additional cost of having someone out of their specialty
or who is not a native speaker being assigned the problem a
solution is heavily dependent on how reliable the
opportunity cost information is
 Perhaps the cost of having a full-time translator or
additional support staff for a VP who is dealing with a lot
of problems that are not her specialty
 Other ways--think of skill/aptitude tests
▪ ASVAB

Enumeration is a way of solving a small
problem by hand
 Enumeration means check all possible
combinations…

Combinations for an ‘n’ valued assignment
problem are just n factorial (n!)
▪ n = 4  n!=4*3*2*1=24
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That’s still a lot to check
There are other tempting methods
 Start with the lowest costs and work your way
up?
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Tempting and seems logical but does not guarantee you an
optimal solution
 for a small problem we can find the best solution using
tradeoffs
Think of the destinations as demanding VP with the
lowest cost VP being the preference
 Leipzig prefers Personnel
 Nancy prefers Finance
 Liege prefers Marketing
 Tilburg prefers Finance
 Two locations have Finance as a first preference, this is
the only thing that makes this problem interesting
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Tradeoff 1: 1000 improvement
Tradeoff 2: 6000 worse
Tradeoff 3: 2000 improvement
 Hopefully this convinces you that LP might be
easier for solving these types of problems than
wrangling all of the potential tradeoffs that
occur
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Setup is the same as the Balanced Transportation problem
from last week
Destinations are the locations or assignments with &gt;=1
constraints
Sources are the persons or objects to be assigned with &lt;= 1
constraints
What is different?
 Number of rows and columns are the same (i.e.
square and balanced)
▪ Not the case for transportation problems
Let X i , j be the amount of i assigned to j
min
 C
i
i, j
X i, j
j
s.t.
Assignees :
X
i, j
1  i
j
Assignment s :  X i , j  1  j
i
Non - neg. : X i , j  0  i and j
Recall the problem from Monday’s lecture of assigning VP’s to
plants to be audited
 Objective
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 We want to minimize the cost of sending Vice Presidents to assembly
plants given the per unit costs matrix C(i,j)

Assignees
 The sources
 For any assignee i, that assignee can be placed in a maximum of one
assignment
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Assignments
 The destinations
 For any destination j, the assignment requires that at least one assignee
be put in place
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Non-negativity
 Decision variables must be zero or positive
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Since the problem is balanced and
assignments are 1 to 1 (1 person to 1 place)
 Decision variables will all have an ending value
of either 1 or 0
 Recall that balanced transport problems have
integer solutions if RHS are integer values
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In general, the assignment model can be
formulated as a transportation model in
which supply at each source and demand at
each destination is equal to one
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