Date post: | 03-Jan-2016 |
Category: |
Documents |
Upload: | betty-reynolds |
View: | 283 times |
Download: | 19 times |
1
Chapter 4: Integer and Mixed-Integer Linear Programming Problems
4.1 Introduction to Integer and Mixed-Integer Linear Programming
4.2 Solving Integer and Mixed-Integer Linear Programming Problems Using the OPTMODEL Procedure
4.3 Modeling Using Binary Variables
4.4 Modeling Sudoku as an ILP (Self-Study)
4.5 Mixed-Integer Linear Programming Solver Options (Self-Study)
2
Chapter 4: Integer and Mixed-Integer Linear Programming Problems
4.1 Introduction to Integer and Mixed-Integer 4.1 Introduction to Integer and Mixed-Integer Linear ProgrammingLinear Programming
4.2 Solving Integer and Mixed-Integer Linear Programming Problems Using the OPTMODEL Procedure
4.3 Modeling Using Binary Variables
4.4 Modeling Sudoku as an ILP (Self-Study)
4.5 Mixed-Integer Linear Programming Solver Options (Self-Study)
3
Objectives Understand how, conceptually and geometrically,
the branch-and-bound method solves integer and mixed-integer linear programming problems.
Understand how, geometrically, cutting planes can strengthen the formulation of an integer or mixed-integer linear programming problem.
Understand the role and importance of the presolver and cutting planes in solving integer and mixed-integer linear programming problems using the OPTMODEL procedure.
4
Mixed-Integer Linear Programming Problems
Each of the linear constraints can be either an inequality or an equation.
The bounds can be ±∞, so that xj can be restricted to be non-negative (lj=0 and uj=+∞) or free (lj=-∞ and uj=+∞).
S is a non-empty subset of {1,2,…,n}
{ , , }
( )
( )
1 1 n n
j j j
j
min | max c x +...+c x
subject to
l x u j =1,2,...,n
x integer j S
Ax b
5
Integer Linear Programming Problems
Each of the linear constraints can be either an inequality or an equation.
The bounds can be ±∞, so that xj can be restricted to be non-negative (lj=0 and uj=+∞) or free (lj=-∞ and uj=+∞).
{ , , }
( )
( )
1 1 n n
j j j
j
min | max c x +...+c x
subject to
l x u j =1,2,...,n
x integer j =1,2,...,n
Ax b
6
Two-Dimensional ExampleThe following integer linear programming (ILP) problem has decision variables x and y:
maximize 6x +5y
subject to 2x + 3y ≤ 7
2x - y ≤ 2
x ≥ 0, y ≥ 0
x integer, y integer
7
Two-Dimensional ExampleThe following integer linear programming (ILP) problem has decision variables x and y:
maximize 6x +5y
subject to 2x + 3y ≤ 7
2x - y ≤ 2
x ≥ 0, y ≥ 0
x integer, y integer
The linear constraints and bounds of the ILP determine a feasible region in two dimensions: the feasible region of the linear programming (LP) relaxation.
linear constraints and bounds
19
Feasible Region
y ax
is
x axis
feasible solutions
20
The Branch-and-Bound Method
y ax
is
x axis
feasible solutions
optimal solution to LP relaxation
21
Feasible Region
y ax
is
x axis
feasible solutions
22
Feasible Region
y ax
is
x axis
feasible solutions
optimal solution to LP relaxation
23
Feasible Region
y ax
is
x axis
feasible solutions
24
Feasible Region: Branch y ≥ 2
y ax
is
x axis
feasible solutions
25
Feasible Region: Branch y ≤ 1
y ax
is
x axis
feasible solutions
26
Feasible Region: Branch y ≤ 1
y ax
is
x axis
feasible solutions
optimal solution to LP relaxation
27
Feasible Region: Branch y ≥ 2
y ax
is
x axis
feasible solutions
optimal solution to LP relaxation
28
Feasible Region: Branch y ≥ 2
y ax
is
x axis
feasible solutions
29
Feasible Region: Branch y ≥ 2
y ax
is
x axis
feasible solutions
optimal solution to LP relaxation
30
The Branch-and-Bound Tree
(LP 0): z* = 16 x*=1⅝, y*=1¼
(LP 2): z* = 14⅓ x*=1, y*=1⅔
(LP 3): z* = 11 x*=1, y*=1
(LP 4): z* = 13 x*=½, y*=2
(LP 5): infeasible
(LP 6): z* = 11⅔ x*=0, y*=2⅓
(LP 1): infeasible
x ≤ 1 x ≥ 2
y ≤ 1 y ≥ 2
x ≥ 1x ≤ 0
31
How Branch-and-Bound Works If the LP relaxation of a subproblem has an optimal
solution that is feasible to the MILP, no branching is required.
The branching variable can be any integral variable that has a fractional value in the optimal solution to the LP relaxation.
An infeasible subproblem can be pruned. A subproblem whose bound is no better than
BestInteger (the objective value of a known feasible solution to the MILP) can be pruned (or fathomed).
The MILP is solved as soon as all active subproblems (or active nodes of the branch-and-bound tree) have been solved.
32
Branch-and-Bound Flow Chart
Choose a branching
variable
Update BestInteger
Choose an active node
Solve LP
Infeasible?
Integral?
Fathom?
Create newactive nodes
yes
nono
no
yes
yes
33
The OPTMILP PresolverBefore solving a mixed-integer linear programming problem, the MILP presolver does the following: identifies and removes redundant constraints and
variables substitutes some variables that are defined by
equations modifies constraint coefficients to strengthen the
formulation of the problem
34
Cutting Planes for ILPs and MILPsSuppose that you solve the LP relaxation of an MILP and find a fractional optimal solution.
Cutting planes are inequalities that cut off (are not satisfied by) the optimal solution to
the LP relaxation are satisfied by all feasible solutions to the MILP.
Cutting planes strengthen the LP relaxation of an MILP and make it easier to solve by the branch-and-bound method.
35
Cutting Planes: Example (ILP)
y ax
is
x axis
feasible solutions
optimal solution to LP relaxation
x+2y ≤ 4
36
How Cutting Planes Are GeneratedCutting planes can be generated from the following: individual inequalities together with the variable
bounds from the MILP the inequalities satisfied by the optimal solution
to the LP relaxation of the MILP stronger relaxations involving all of the inequalities
from the MILP logical relations between 0–1 variables in the MILP
37
These exercises reinforce the concepts discussed previously.
Exercises 1–4
38
Chapter 4: Integer and Mixed-Integer Linear Programming Problems
4.1 Introduction to Integer and Mixed-Integer Linear Programming
4.2 Solving Integer and Mixed-Integer Linear 4.2 Solving Integer and Mixed-Integer Linear Programming Problems Using the OPTMODEL Programming Problems Using the OPTMODEL ProcedureProcedure
4.3 Modeling Using Binary Variables
4.4 Modeling Sudoku as an ILP (Self-Study)
4.5 Mixed-Integer Linear Programming Solver Options (Self-Study)
39
Objectives Be able to solve integer and mixed-integer linear
programming problems using the OPTMODEL procedure.
Learn which information is written to the SAS log file during the solution of an integer or mixed-integer programming problem.
40
Declaring Integer Variables in PROC OPTMODELVariables can be declared to be either integer variables or binary (0–1) variables:
var x >= 0 <= 2 integer; var y{Arcs} >= 0 integer; var z{Nodes} binary;
41
Declaring Integer Variables in PROC OPTMODELVariables can be declared to be either integer variables or binary (0–1) variables:
The integrality of a variable can be relaxed using a nonzero, nonmissing value of the .relax suffix:
x.relax = 1; y['New York','Paris'].relax = 1; for {k in Nodes} z[k].relax = 1;
var x >= 0 <= 2 integer; var y{Arcs} >= 0 integer; var z{Nodes} binary;
42
Termination Criteria for MILPsThe MILP solver will usually terminate when the relative objective gap falls below a certain value:
BestInteger =
BestBound =
-4-10
| | 10
| | 10
BestInteger BestBound
BestBound
Active Nodes
objective value of the best feasible solution found
“best” bound at an active node of the branching tree
43
Termination Criteria for MILPsThe MILP solver has other termination criteria that can be set using OPTMILP solver options:
Solver Option Description
ABSOBJGAP= Absolute difference between BestBound and BestInteger values (default 1E-6)
CUTOFF= Cut off all nodes with a worse bound
MAXNODES= Maximum number of nodes in the tree
MAXSOLS= Maximum number of solutions found
MAXTIME= Maximum real time (seconds) allowed
RELOBJGAP= Relative difference (default 1E-4)
TARGET= Target value for BestInteger
44
This demonstration illustrates the solution of an integer linear programming problem using PROC OPTMODEL.
Solving the Integer Linear McDonald’s Diet Problem Using PROC OPTMODEL mcdonalds_int.sas
45
MILP Solver OptionsThe MILP solver has a number of options in addition to termination criteria:
Solver Option Description
PRINTFREQ= Frequency of printing node log (default 100 = print to log every 100 nodes)
PRIMALIN Current values input a feasible solution
PRESOLVER= Similar to other presolver options
HEURISTICS= This level controls both the frequency and number of iterations allowed in searching for feasible solutions
46
Global Cutting Plane OptionCutting planes are added automatically at the root node and subsequent nodes of the tree.
The ALLCUTS= option sets the level for all classes of cutting planes, but it can be overwritten by setting cut options for individual families.
ALLCUTS= # Description
AUTOMATIC -1 Strategy determined by solver (default)
NONE 0 Disable cutting plane generation
MODERATE 1 Moderate cut strategy
AGGRESSIVE 2 Aggressive cut strategy
47
These exercises reinforce the concepts discussed previously.
Exercises 5–9
48
Chapter 4: Integer and Mixed-Integer Linear Programming Problems
4.1 Introduction to Integer and Mixed-Integer Linear Programming
4.2 Solving Integer and Mixed-Integer Linear Programming Problems Using the OPTMODEL Procedure
4.3 Modeling Using Binary Variables4.3 Modeling Using Binary Variables
4.4 Modeling Sudoku as an ILP (Self-Study)
4.5 Mixed-Integer Linear Programming Solver Options (Self-Study)
49
Objectives Learn how to model logical constraints using binary
(0-1) variables. Learn how to use binary variables with continuous
variables to model fixed costs.
50
Modeling Fixed CostsFor a continuous variable x with 0 ≤ x ≤ u, you can use a binary variable z to model a fixed cost.
cost = m*x + b if x>0
0 if x=0
x axis
cost = m*x+b*z
0 ≤ x ≤ u
x ≤ u*z
z binary
51
This demonstration illustrates the solution of an mixed-integer linear programming problem with set-up costs in PROC OPTMODEL.
Solving the Furniture-Making Problem with Set-Up Costs in PROC OPTMODEL furniture_setup.sas
52
Modeling Logical ConstraintsIn the furniture-making problem with set-up costs, suppose that either bedframes or bookcases can be made, but not both. How can you model this?
53
Modeling Logical ConstraintsIn the furniture-making problem with set-up costs, suppose that either bedframes or bookcases can be made, but not both. How can you model this?
z['bedframes'] + z['bookcases'] <= 1;
54
Modeling Logical ConstraintsIn the furniture-making problem with set-up costs, suppose that either bedframes or bookcases can be made, but not both. How can you model this?
Other logical constraints can also be modeled using binary variables.
z['bedframes'] + z['bookcases'] <= 1;
55
Modeling Logical ConstraintsIn the furniture-making problem with set-up costs, suppose that either bedframes or bookcases can be made, but not both. How can you model this?
Other logical constraints can also be modeled using binary variables. For example, what restriction does the following constraint model?
z['bedframes'] + z['bookcases'] <= 1;
z['chairs'] <= z['desks'];
56
Modeling Logical ConstraintsSuppose that chairs cannot be made if two (or more) of the other products are made. How can you model this?
57
Modeling Logical ConstraintsSuppose that chairs cannot be made if two (or more) of the other products are made. How can you model this?
2*z['chairs'] + z['bedframes'] + z['bookcases'] + z['desks'] <= 3;
58
Modeling Logical ConstraintsSuppose that chairs cannot be made if two (or more) of the other products are made. How can you model this?
This can also be modeled using three inequalities:
2*z['chairs'] + z['bedframes'] + z['bookcases'] + z['desks'] <= 3;
z['chairs'] + z['bedframes'] + z['bookcases'] <= 2; z['chairs'] + z['bedframes'] + z['desks'] <= 2; z['chairs'] + z['bookcases'] + z['desks'] <= 2;
59
Example: Discount Pricing a Product MixRecall the organic potting soil product mix:
Discount Pricing Strategy:
[1] x
classic_1 84.066 new mex 189.970 packs 17.310
planting 45.045 prick-out 116.564 rec5
92.500 0-50 cubic yards: full price 50-100 cubic yards: 10% discount 100-200 cubic yards: 20% discount
60
Example: Discount Pricing a Product MixRecall the organic potting soil product mix:
Discount Pricing Strategy:
[1] x
classic_1 84.066 new mex 189.970 packs 17.310
planting 45.045 prick-out 116.564 rec5
92.500 0-50 cubic yards: full price 50-100 cubic yards: 10% discount 100-200 cubic yards: 20% discount
61
Example: Discount Pricing a Product MixDiscount Pricing Strategy:
0-50 cubic yards: full price (k=0) 50-100 cubic yards: 10% discount (k=1) 100-200 cubic yards: 20% discount (k=2)
x[p] = total sold
y[p,k] = total sold at 10*k% discount
z[p,k] = indicator for 10*k% discount
How can we link these variables?
62
Example: Discount Pricing a Product MixDiscount Pricing Strategy:
0-50 cubic yards: full price (k=0) 50-100 cubic yards: 10% discount (k=1) 100-200 cubic yards: 20% discount (k=2)
x[p] = total sold
y[p,k] = total sold at 10*k% discount
z[p,k] = indicator for 10*k% discount
z[p,0]+z[p,1]+z[p,2] = 1; y[p,0]+y[p,1]+y[p,2] = x[p]; 0 <= y[p,0] <= 50*z[p,0]; 50*z[p,1] <= y[p,1] <= 100*z[p,1]; 100*z[p,2] <= y[p,2] <= 200*z[p,2];
63
This demonstration illustrates the solution of an mixed-integer linear programming problem with piecewise-linear pricing in PROC OPTMODEL.
Solving the Organic Potting Soils Problem with Discount Pricing in PROC OPTMODEL discount.sas
64
Example: Bank Credit Card OffersA bank has three different credit card offers it can make to customers. For each customer, the bank estimates the expected profit and credit loss for each offer.
Each credit card will be offered to at least 50 customers, but at most one credit card can be offered to a customer.
65
Example: Bank Credit Card OffersA bank has three different credit card offers it can make to customers. For each customer, the bank estimates the expected profit and credit loss for each offer.
Each credit card will be offered to at least 50 customers, but at most one credit card can be offered to a customer.
Memo
To: Optimization Department From: Credit Department
Maximum credit loss: $10,000. No exceptions. No excuses.
66
This class participation exercise formulates and solves an ILP with binary variables.
Solving the Bank Credit Card Offer Problem in PROC OPTMODEL credit_offers.sas
67
These exercises reinforce the concepts discussed previously.
Exercises 10–14