Download Week 6 - Feb 17

1
Integer Programming
• Each year CrossChek decides which lines of golf clubs
and clothing it will market. Consider that each line of golf
clubs is expected to generate $400K profit for the year,
while each new line of golf clothing generates $300K
profit for the year. Each new offering requires time for
marketing (clubs 50 hrs, clothing 60 hrs), which is limited
to 300 hrs, as well as storage space (clubs 900 sq ft,
clothes 400 sq ft), which is limited to 3600 sq ft.
CrossChek will not consider marketing any more than 4
lines of clothing. How many new lines of golf clubs and
clothing should be offered?
2
Integer Programming
• Solution:
• 2.82 lines of golf clubs
• 2.65 lines of golf clothing
• Profit:
• $1,923,529
3
Integer Programming
• Solution:
• 2.82 lines of golf clubs
• 2.65 lines of golf clothing
• Could round down:
• 2 lines of golf clubs
• 2 lines of golf clothing
• Profit = $1.4 million
• Feasible. But optimal?
4
Integer Programming in LINDO
Max 400x1 + 300x2
st
50x1 + 60x2 < 300
900x1 + 400x2 < 3600
x2 < 4
x1 > 0
x2 > 0
END
GIN x1
GIN x2
5
ILP Notes
• Removing the integer specifications for an integer linear
programming problem gives the LP relaxation
• The value of an LP relaxation for a maximization ILP is an
upper bound on the value of the ILP
• Meaning that the value of the objective function for the optimal
integer solution is no higher than that for the LP relaxation
• For a minimizing ILP, the LP relaxation gives a lower
bound
6
ILP Notes
• An integer linear programming problem in which all
variables must be integers is referred to as an All Integer
Linear Program (AILP)
• An integer linear programming problem in which some
variables must be integers is referred to as a Mixed
Integer Linear Program (MILP)
7
ILP with 0-1 Variables
• CrossChek is considering expanding its retail outlets. Possible
locations for such outlets are Toronto, Vancouver, Montreal and
Fredericton. A maximum of one outlet would exist in each city. Startup
costs, advertising and marketing costs, and time required to get each
outlet in operation are as follows, as well as expected first-year profits
(all dollar values in thousands; all times in number of days). The
objective is to maximize first-year profits. Into which cities should
CrossChek expand?
City
Start-up costs
Time
Advertising costs First-year profits
Toronto
100
30
20
50
Vancouver
80
40
80
45
Montreal
90
35
25
35
Fredericton
70
25
20
5
Limit
260
110
100
8
Defining 0-1 Variables in LINDO
Max 50x1 + 45x2 + 35x3 + 5x4
st
100x1 + 80x2 + 90x3 + 70x4 < 260
30x1 + 40x2 + 35x3 + 25x4 < 110
20x1 + 80x2 + 25x3 + 20x4 < 100
END
INT x1
INT x2
INT x3
INT x4
9
k-out-of-n Alternatives Constraints
• Considering the same problem, add each of the following
constraints:
• CrossChek would like to ensure that:
a) Exactly one outlet is added
b) At least one outlet goes in Montreal or Fredericton
c) Exactly three outlets are added
d) Two of Montreal, Fredericton and Vancouver are added
e) One of Toronto or Vancouver are added but not both
10
Conditional Constraints
• Considering the same problem, add each of the following
constraints:
• CrossChek would like to ensure that:
a) If an outlet in Toronto is added, then an outlet in Fredericton is
added
b) If an outlet in Fredericton is not added, then one in Vancouver
must be added
c) If an outlet in Toronto is added then we cannot add one in
Vancouver
d) If an outlet in Vancouver is added, then at least one outlet in
Montreal or Fredericton must be added
11
Multiple Choice Constraints
• Considering the same problem, consider that there are
three possible sites for the Vancouver option: downtown,
midtown, uptown. We’ve decided to develop in Vancouver
for sure. How do we model this choice as a constraint?
• Let v1 be equal to 1 if the downtown location is chosen
• Let v2 be equal to 1 if the midtown location is chosen
• Let v3 be equal to 1 if the uptown location is chosen
• What if development in Vancouver is not a certainty?
12
Fixed Cost Problems
• Returning to the CrossChek golf clubs/clothing strategy
example. Suppose there is a fixed cost for licensing fees
associated with venturing into the golf clothing market of
$200K. That is, it will cost an extra $200K if it chooses to
sell any golf clothing. What is the optimal course of
action?
• What if the fixed cost were $100K?
13
Solution
Max 400x1 + 300x2 - 200d
st
50x1 + 60x2 < 300
900x1 + 400x2 < 3600
x2 - 4d < 0
x1 > 0
x2 > 0
END
GIN x1
GIN x2
INT d
14
Fixed Cost Problems
• Returning to the original CrossChek golf clubs/clothing
strategy example. Suppose there are two different
marketing companies from which to choose. Marketing
company 1 costs $70K, requires 50 hrs for each line of
clubs and 60 hrs for each line of clothing, which is limited
to 300 hrs, and marketing company 2 costs $80K,
requires 40 hrs for each line of clubs and 45 hrs for each
line of clothing, which is limited to 270 hrs. CrossCheck
will go with either one marketing company or the other.
What is the optimal course of action?
15
Solution
Max 400x1 + 300x2 - 70d1 - 80d2
st
50x1 + 60x2 + 1000000d1 < 1000300
40x1 + 45x2 + 1000000d2 < 1000270
900x1 + 400x2 < 3600
d1 + d2 = 1
x2 < 4
x1 > 0
x2 > 0
END
GIN x1
GIN x2
INT d1
INT d2