Transcript Spreadsheet Modeling & Decision Analysis:

Introduction to
Mathematical Programming
OR/MA 504
Chapter 5
Integer Linear Programming
Introduction
• When one or more variables in an LP problem
must assume an integer value we have an
Integer Linear Programming (ILP) problem.
• ILPs occur frequently…
– Scheduling workers
– Manufacturing airplanes
• Integer variables also allow us to build more
accurate models for a number of common
business problems.
6-2
Integrality Conditions
MAX: 350X1 + 300X2
S.T.: 1X1 + 1X2 <= 200
9X1 + 6X2 <= 1566
12X1 + 16X2 <= 2880
X1, X2>= 0
X1, X2 must be integers
}
}
}
}
}
}
profit
pumps
labor
tubing
nonnegativity
integrality
Integrality conditions are easy to state but make the
problem much more difficult (and sometimes impossible)
to solve.
6-3
Relaxation
• Original ILP
MAX:
S.T.:
2X1 + 3X2
X1 + 3X2 <= 8.25
2.5X1 + X2 <= 8.75
X1, X2 >= 0
X1, X2 must be integers
• LP Relaxation
MAX:
S.T.:
2X1 + 3X2
X1 + 3X2 <= 8.25
2.5X1 + X2 <= 8.75
X1, X2 >= 0
6-4
Integer Feasible vs. LP Feasible Region
X2
Integer Feasible Solutions
3
2
1
0
0
1
2
3
4
X1
6-5
Solving ILP Problems
• When solving an LP relaxation, sometimes
you “get lucky” and obtain an integer feasible
solution.
• This was the case in the original Blue Ridge
Hot Tubs problem in earlier chapters.
• But what if we reduce the amount of labor
available to 1520 hours and the amount of
tubing to 2650 feet?
See file Fig5-1.xls
6-6
Bounds
• The optimal solution to an LP relaxation of an ILP
problem gives us a bound on the optimal objective
function value.
• For maximization problems, the optimal relaxed
objective function values is an upper bound on the
optimal integer value.
• For minimization problems, the optimal relaxed
objective function values is a lower bound on the
optimal integer value.
6-7
Rounding
• It is tempting to simply round a
fractional solution to the closest integer
solution.
• In general, this does not work reliably:
– The rounded solution may be infeasible.
– The rounded solution may be suboptimal.
6-8
How Rounding Down Can Result
in an Infeasible Solution
X2
3
2
optimal relaxed solution
infeasible solution obtained
by rounding down
1
0
0
1
2
3
4
X1
6-9
Branch-and-Bound
• The Branch-and-Bound (B&B) algorithm
can be used to solve ILP problems.
• Requires the solution of a series of LP
problems termed “candidate problems”.
• Theoretically, this can solve any ILP.
• Practically, it often takes LOTS of
computational effort (and time).
6-10
Stopping Rules
• Because B&B can take so long, most ILP packages
allow you to specify a suboptimality tolerance factor.
• This allows you to stop once an integer solution is
found that is within some % of the global optimal
solution.
• Bounds obtained from LP relaxations are helpful
here.
– Example
• LP relaxation has an optimal obj. value of $64,306.
• 95% of $64,306 is $61,090.
• Thus, an integer solution with obj. value of
$61,090 or better must be within 5% of the
6-11
optimal solution.
Using Solver
Let’s see how to specify integrality conditions
and suboptimality tolerances using Solver…
See file Fig5-2.xls
6-12
An Employee Scheduling Problem:
Air-Express
Day of Week
Sunday
Workers Needed
18
Shift
Days Off
1
Sun & Mon
Wage
$680
Monday
27
2
Mon & Tue
$705
Tuesday
22
3
Tue & Wed
$705
Wednesday
26
4
Wed & Thr
$705
Thursday
25
5
Thr & Fri
$705
Friday
21
6
Fri & Sat
$680
Saturday
19
7
Sat & Sun
$655
6-13
Defining the Decision Variables
X1 = the number of workers assigned to shift 1
X2 = the number of workers assigned to shift 2
X3 = the number of workers assigned to shift 3
X4 = the number of workers assigned to shift 4
X5 = the number of workers assigned to shift 5
X6 = the number of workers assigned to shift 6
X7 = the number of workers assigned to shift 7
6-14
Defining the Objective Function
Minimize the total wage expense.
MIN: 680X1 +705X2 +705X3 +705X4 +705X5 +680X6 +655X7
6-15
Defining the Constraints
• Workers required each day
0X1+
0X1+
1X1+
1X1+
1X1+
1X1+
1X1+
1X2+
0X2+
0X2+
1X2+
1X2+
1X2+
1X2+
1X3+
1X3+
0X3+
0X3+
1X3+
1X3+
1X3+
1X4+
1X4+
1X4+
0X4+
0X4+
1X4+
1X4+
1X5+
1X5+
1X5+
1X5+
0X5+
0X5+
1X5+
1X6+
1X6+
1X6+
1X6+
1X6+
0X6+
0X6+
0X7
1X7
1X7
1X7
1X7
1X7
0X7
>=
>=
>=
>=
>=
>=
>=
18
27
22
26
25
21
19
} Sunday
} Monday
}Tuesday
} Wednesday
} Thursday
} Friday
} Saturday
• Nonnegativity & integrality conditions
Xi >= 0 and integer for all i
6-16
Implementing the Model
See file Fig5-3.xls
6-17
Binary Variables
• Binary variables are integer variables that
can assume only two values: 0 or 1.
• These variables can be useful in a number
of practical modeling situations….
6-18
A Capital Budgeting Problem:
CRT Technologies
Capital (in $000s) Required in
Expected NPV
Project (in $000s) Year 1 Year 2 Year 3 Year 4 Year 5
1
$141
$75 $25
$20
$15
$10
2
3
4
5
6
$187
$121
$83
$265
$127
$90
$60
$30
$100
$50
$35
$15
$20
$25
$20
$0
$15
$10
$20
$10
$0
$15
$5
$20
$30
$30
$15
$5
$20
$40
• The company has $250,000 available to invest in new projects.
It has budgeted $75,000 for continued support for these
projects in year 2 and $50,000 per year for years 3, 4, and 5.
• Unused funds in any year cannot be carried over.
6-19
Defining the Decision Variables
1, if project i is selected
X 
i  1,2,...,6
i  0, otherwise
6-20
Defining the Objective Function
Maximize the total NPV of selected projects.
MAX: 141X1 + 187X2 + 121X3
+ 83X4 + 265X5 + 127X6
6-21
Defining the Constraints
• Capital Constraints
75X1 + 90X2 + 60X3 + 30X4 + 100X5 + 50X6 <= 250
25X1 + 35X2 +15X3 + 20X4 + 25X5 + 20X6 <= 75
20X1 + 0X2 + 15X3 + 10X4 + 20X5 + 10X6 <= 50
15X1 + 0X2 + 15X3 + 5X4 + 20X5 + 30X6 <= 50
10X1 +30X2 +15X3 + 5X4 + 20X5 + 40X6 <= 50
}
}
}
}
}
year
year
year
year
year
1
2
3
4
5
• Binary Constraints
All Xi must be binary
6-22
Implementing the Model
See file Fig5-4.xls
6-23
Binary Variables & Logical Conditions
• Binary variables are also useful in modeling a
number of logical conditions.
– Of projects 1, 3 & 6, no more than one
may be selected
• X1 + X3 + X6 <= 1
– Of projects 1, 3 & 6, exactly one must be
selected
• X1 + X3 + X6 = 1
– Project 4 cannot be selected unless project
5 is also selected
• X4 – X5 <= 0
6-24
The Fixed-Charge Problem
• Many decisions result in a fixed or lump-sum
cost being incurred:
–
–
–
–
The cost to lease, rent, or purchase a piece of
equipment or a vehicle that will be required if a
particular action is taken.
The setup cost required to prepare a machine or
to produce a different type of product.
The cost to construct a new production line that
will be required if a particular decision is made.
The cost of hiring additional personnel that will
be required if a particular decision is made.
6-25
Example Fixed-Charge Problem :
Remington Manufacturing
Operation
Machining
Grinding
Assembly
Unit Profit
Setup Cost
Hours Required By:
Prod. 1 Prod. 2 Prod. 3 Hours Available
2
3
6
600
6
3
4
300
5
6
2
400
$48
$55
$50
$1000
$800
$900
6-26
Defining the Decision Variables
Xi = the amount of product i to be produced, i = 1, 2, 3

1, if X i  0
Yi  
i = 1, 2, 3
0, if Xi  0


6-27
Defining the Objective Function
Maximize total profit.
MAX: 48X1 + 55X2 + 50X3 – 1000Y1 – 800Y2 – 900Y3
6-28
Defining the Constraints
• Resource Constraints
2X1 + 3X2 + 6X3 <= 600
6X1 + 3X2 + 4X3 <= 300
5X1 + 6X2 + 2X3 <= 400
} machining
} grinding
} assembly
• Binary Constraints
All Yi must be binary
• Nonnegativity conditions
Xi >= 0, i = 1, 2, ..., 6
• Is there a missing link?
6-29
Defining the Constraints
(cont’d)
• Linking Constraints (with “Big M”)
X1 <= M1Y1
or
X1 - M1Y1 <= 0
X2 <= M2Y2
or
X2 - M2Y2 <= 0
X3 <= M3Y3
or
X3 - M3Y3 <= 0
• If Xi > 0 these constraints force the
associated Yi to equal 1.
• If Xi = 0 these constraints allow Yi to equal 0
or 1, but the objective will cause Solver to
choose 0.
• Note that Mi imposes an upper bounds on Xi.
• It helps to find reasonable values for the Mi.
6-30
Finding Reasonable Values for M1
• Consider the resource constraints
2X1 + 3X2 + 6X3 <= 600
6X1 + 3X2 + 4X3 <= 300
5X1 + 6X2 + 2X3 <= 400
} machining
} grinding
} assembly
• What is the maximum value X1 can assume?
Let X2 = X3 = 0
X1 = MIN(600/2, 300/6, 400/5)
= MIN(300, 50, 80)
= 50
• Maximum values for X2 & X3 can be found
similarly.
6-31
Summary of the Model
MAX: 48X1 + 55X2 + 50X3 - 1000Y1 - 800Y2 - 900Y3
S.T.:
2X1 + 3X2 + 6X3 <= 600 } machining
6X1 + 3X2 + 4X3 <= 300 } grinding
5X1 + 6X2 + 2X3 <= 400 } assembly
X1 - 50Y1 <= 0
X2 - 67Y2 <= 0
linking constraints
X3 - 75Y3 <= 0
All Yi must be binary
Xi >= 0, i = 1, 2, 3
6-32
Potential Pitfall
• Do not use IF( ) functions to model the
relationship between the Xi and Yi.
– Suppose cell A5 represents X1
– Suppose cell A6 represents Y1
– You’ll want to let A6 = IF(A5>0,1,0)
– This will not work with Solver!
• Treat the Yi just like any other variable.
– Make them changing cells.
– Use the linking constraints to enforce the
proper relationship between the Xi and Yi.
6-33
Implementing the Model
See file Fig5-5.xls
6-34
Minimum Order Size Restrictions
Suppose Remington doesn’t want to
manufacture any units of product 3 unless
it produces at least 40 units...
Consider,
X3 <= M3Y3
X3 >= 40 Y3
6-35
Quantity Discounts
• Assume…
– If Blue Ridge Hot Tubs produces more
than 75 Aqua-Spas, it obtains discounts
that increase the unit profit to $375.
– If it produces more than 50 HydroLuxes, the profit increases to $325.
6-36
Quantity Discount Model
MAX: 350X11 + 375X12 + 300X21 + 325X22
S.T.: 1X11 + 1X12 + 1X21 + 1X22 <= 200 } pumps
9X11 + 9X12 + 6X21 + 6X22 <= 1566 } labor
12X11+ 12X12+ 16X21+16X22 <= 2880 } tubing
X12<=M12Y1
X11>=75Y1
X22<=M22Y2
X21>=50Y2
Xij >= 0
Xij must be integers, Yi must be binary
6-37
A Contract Award Problem
• B&G Construction has 4 building projects and
can purchase cement from 3 companies for the
following costs:
Co. 1
Co. 2
Co. 3
Needs
(tons)
Max.
Cost per Delivered Ton of Cement
Project 1 Project 2 Project 3 Project 4 Supply
$120
$115
$130
$125
525
$100
$150
$110
$105
450
$140
$95
$145
$165
550
450
275
300
350
6-38
A Contract Award Problem
• Side constraints:
– Co. 1 will not supply orders of less than 150 tons
for any project
– Co. 2 can supply more than 200 tons to no more
than one of the projects
– Co. 3 will accept only orders that total 200, 400,
or 550 tons
6-39
Defining the Decision Variables
Xij = tons of cement purchased
from company i for project j
6-40
Defining the Objective Function
Minimize total cost
MIN:
120X11 + 115X12 + 130X13 + 125X14
+ 100X21 + 150X22 + 110X23 + 105X24
+ 140X31 + 95X32 + 145X33 + 165X34
6-41
Defining the Constraints
• Supply Constraints
X11 + X12 + X13 + X14 <= 525
X21 + X22 + X23 + X24 <= 450
X31 + X32 + X33 + X34 <= 550
} company 1
} company 2
} company 3
X11
X12
X13
X14
}
}
}
}
• Demand Constraints
+
+
+
+
X21
X22
X23
X24
+
+
+
+
X31
X32
X33
X34
=
=
=
=
450
275
300
350
project
project
project
project
1
2
3
4
6-42
Implementing the Transportation
Constraints
See file Fig5-6.xls
6-43
Defining the Constraints-I
• Company 1 Side Constraints
X11<=525Y11
X12<=525Y12
X13<=525Y13
X14<=525Y14
X11>=150Y11
X12>=150Y12
X13>=150Y13
X14>=150Y14
Yij binary
6-44
Defining the Constraints-II
• Company 2 Side Constraints
X21<=200+250Y21
X22<=200+250Y22
X23<=200+250Y23
X24<=200+250Y24
Y21 + Y22 + Y23 + Y24 <= 1
Yij binary
6-45
Defining the Constraints-III
• Company 3 Side Constraints
X31 + X32 + X33 + X34 = 200Y31 + 400Y32 + 550Y33
Y31 + Y32 + Y33 <= 1
6-46
Implementing the Side Constraints
See file Fig5-7.xls
6-47
The Branch-And-Bound Algorithm
MAX: 2X1 + 3X2
S.T.
X1 + 3X2 <= 8.25
2.5X1 + X2 <= 8.75
X1, X2 >= 0 and integer
6-48
Solution to LP Relaxation
X2
3
Feasible Integer Solutions
Optimal Relaxed Solution
X1 = 2.769, X2=1.826
Obj = 11.019
2
1
0
0
1
2
3
4
X1 6-49
The Branch-And-Bound Algorithm
Problem I MAX: 2X1 + 3X2
S.T.
X1 + 3X2 <= 8.25
2.5X1 + X2 <= 8.75
X1 <= 2
X1, X2 >= 0 and integer
Problem II MAX: 2X1 + 3X2
S.T.
X1 + 3X2 <= 8.25
2.5X1 + X2 <= 8.75
X1 >= 3
X1, X2 >= 0 and integer
6-50
Solution to LP Relaxation
X2
Problem I
3
X1=2, X2=2.083, Obj = 10.25
2
Problem II
1
0
0
1
2
3
4
X1 6-51
The Branch-And-Bound Algorithm
Problem III MAX:
S.T.
2X1 + 3X2
X1 + 3X2 <= 8.25
2.5X1 + X2 <= 8.75
X1 <= 2
X2 <= 2
X1, X2 >= 0 and integer
Problem IV
MAX:
2X1 + 3X2
S.T.
X1 + 3X2 <= 8.25
2.5X1 + X2 <= 8.75
X1 <= 2
X2 >= 3
X1, X2 >= 0 and integer
6-52
Solution to LP Relaxation
X2
Problem III
3
X1=2, X2=2, Obj = 10
2
Problem II
X1=3, X2=1.25, Obj = 9.75
1
0
0
1
2
3
4
X1
6-53
B&B Summary
Original Problem
X1<=2
X1=2.769
X2=1.826
Obj = 11.019
X1>=3
Problem II
Problem I
X1=2
X2=2.083
Obj = 10.25
X2<=2
X1=3
X2=1.25
Obj = 9.75
X2>=3
Problem III
X1=2
X2=2
Obj = 10
Problem IV
infeasible
6-54
End of Chapter 5
6-55