Applying Linear Optimization Using GLPK

CCISDevt Sharing Session

Applying Linear Optimization
Using GLPK

Outline

What is Linear Optimization?
What is GLPK?
Available Bindings
GNU MathProg Scripting
Examples:
Trans-shipment (the standard example)
Time Table Scheduling (from school days...)
Allocating HDB Flats by “Combinatorial Auction”
2

Linear Optimization? What?

The minimization of functions of the form
c1 x1 + c2 x2 + …+ cn xn
by varying decision variables xi,
subject to constraints of the form
ai1 x1 + ai2 x2 + …+ ain xn = bi or
ai1 x1 + ai2 x2 + …+ ain xn ≤ bi
where the other coefficients aij, bi and ci are fixed.

3


A typical example: “The Diet Problem”. By
varying the intake of each type of food (e.g.:
chicken rice, durian, cheese cake), minimize
(The Total Cost of Food)
subject to
(Non-negativity of the intakes of each type.)
(Satisfaction of minimum nutritional requirements)

4


A secondary school example. By varying decision
variables x and y, maximize
2x+y
subject to
x ≥ 0, x ≤ 1,
y ≥ 0, y ≤ 1,
x + y ≤ 1.5

5


2x+y
subject to
x ≥ 0, x ≤ 1,
y ≥ 0, y ≤ 1,
x + y ≤ 1.5

6


2x+y
subject to
x ≥ 0, x ≤ 1,
y ≥ 0, y ≤ 1,
x + y ≤ 1.5

7


2x+y
subject to
x ≥ 0, x ≤ 1,
y ≥ 0, y ≤ 1,
x + y ≤ 1.5

8


A more useful sounding example: “Simple
Commodity Flow”. By varying the number of TV
sets being transferred along each road in a
road network, minimize
(Total Distance each TV set is moved through)
subject to
(Non-negativity of the flows)
(Sum of Inflows = Sum of Outflows at each junction where
Stock counts as inflow & demand, outflow.)
9


Perhaps one of the most widely used mathematical
techniques:
Network flow / multi-commodity flow problems
Project Management
Production Planning, etc...
Used in Mixed Integer Linear Optimization for:
Airplane scheduling
Facility planning
Timetabling, etc...
10


Solution methods:
Simplex-based algorithms
Non-polynomial-time algorithm
Performs much better in practice than predicted by
theory
Interior-point methods
Polynomial-time algorithm
Also performs better in practice than predicted by
theory
11

What is GLPK

GLPK: GNU Linear Programming Kit
An open-source, cross-platform software package for
solving large-scale Linear Optimization and Mixed
Integer Linear Optimization problems.
Comes bundled with the GNU MathProg scripting
language for rapid development.
URL: http://www.gnu.org/software/glpk/
GUSEK (a useful front-end):
http://gusek.sourceforge.net/gusek.html

12

Bindings for GLPK

GLPK bindings exist for:
C, C++ (with distribution)
Java (with distribution)
C#
Python, Ruby, Erlang
MATLAB, R
… even Lisp.
(And probably many more.)

13

Scripting with GNU MathProg

For expressing optimization problems in a compact,
human readable format.
Used to generate problems for computer solution.
Usable in production as problem generation is typically a
small fraction of solution time and problem generation
(in C/C++/Java/etc) via the API takes about as long.
Hence, “rapid development” as opposed to “rapid
prototyping”.

14


Parts of a script:
Model: Description of objective function and
constraints to be satisfied
Data: Parameters that go into the model
Data can be acquired from a database (e.g. via
ODBC).
Post processed solution can be written to a
database.

15


Model File (For choosing a meet-up date)
set Peoples;
set Days;
set Meals;
... # PrefInd[...] defined here
var x{d in Days, m in Meals} binary;
maximize Participation_and_Preference: sum{p in Peoples, d in Days, m
in Meals} PrefInd[p,d,m]*x[d,m];
s.t. one_meal: sum{d in Days, m in Meals} x[d,m] = 1;
solve;
... # Post-processing here
end;

16


Data File (for the above model)
data;
set Peoples := JC LZY FYN MJ;
set Days := Mon Tue Wed Thu Fri Sat Sun;
set Meals := Lunch Dinner;
# Last element is 1 if preferred, 0 if otherwise
set Prefs :=
(JC, Mon, Dinner, 1),
(JC, Tue, Dinner, 0),
... # Rest of preference data
;
end;

17


Good “student” practice:
Decouple model from data (separate files)
Write script to generate data file from sources
Reasonable production practice
Decouple model from data
Acquire data from and write output to database
Abuse: Read statements can be used to write to the
database (e.g.: a “currently working” flag)

18

Examples

Trans-shipment (the standard example)
A source to destination network flow problem
Data: Quantity at source, Demand at destination
Time Table Scheduling (from school days...)
Respect requirements, Maximize preference
Support alternate “sessions” and multiple time slots
A “Combinatorial Auction” for HDB Flat Allocation
Allocation and pricing of HDB flats
Efficiently solvable (Surprise!)

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Example: Trans-shipment

Model File
set I; /* canning plants */

param a{i in I}; /* capacity of plant i */

set J; /* markets */

param b{j in J}; /* demand at market j */

param d{i in I, j in J}; /* distance in thousands of miles */

param f; /* freight in dollars per case per thousand miles */

param c{i in I, j in J} := f * d[i,j] / 1000; /* transport cost */

var x{i in I, j in J} >= 0; /* shipment quantities in cases */

minimize cost: sum{i in I, j in J} c[i,j] * x[i,j]; /* total costs */

s.t. supply{i in I}: sum{j in J} x[i,j] <= a[i];/* supply limits */

s.t. demand{j in J}: sum{i in I} x[i,j] >= b[j];/* satisfy demand */

solve;

# < Post-processing code >

end;

20


Data File
data;
set I := Seattle San-Diego;
param a := Seattle 350
San-Diego 600;
set J := New-York Chicago Topeka;
param b := New-York 325
Chicago 300
Topeka 275;
param d : New-York Chicago Topeka :=
Seattle 2.5 1.7 1.8
San-Diego 2.5 1.8 1.4 ;
param f := 90;
end;
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Sample Output
GLPK Simplex Optimizer, v4.47

6 rows, 6 columns, 18 non-zeros

...

OPTIMAL SOLUTION FOUND

************************************************

***POST SOLVE (Post-processed Output)

************************************************

Flow[ Seattle -> New-York ] = 50.000000

Flow[ Seattle -> Chicago ] = 300.000000

Flow[ Seattle -> Topeka ] = 0.000000

Flow[ San-Diego -> New-York ] = 275.000000

Flow[ San-Diego -> Chicago ] = 0.000000

Flow[ San-Diego -> Topeka ] = 275.000000

22


Using ODBC as a Data Source
table tbl_plants IN 'ODBC' 'dsn=demo_tpt' 'plants' :
I <- [name], a~capacity;
table tbl_markets IN 'ODBC' 'dsn=demo_tpt' 'markets' :
J <- [name], b~demand;
table tbl_freight IN 'ODBC' 'dsn=demo_tpt' 'freight' :
[plant,market], d~cost;

Sending Output to a Database (via ODBC)
table result{i in I, j in J: x[i,j]} OUT 'ODBC' 'dsn=demo_tpt'
'UPDATE freight SET flow=? WHERE (plant=?) and (market=?)' :
x[i,j], i, j;

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Example: Time Tabling

Objective: Maximize “preference points”
Constraints:
Pick exactly one appointment from appointment groups
marked compulsory
Pick at most one appointment any appointment group
No timing clash between appointments

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Example: Time Tabling

Sample (Post-Processed) Output
Selected Appointment Groups:
SMA_IS4241C_
FM_BMA5008_
...
Selected Appointments:
SMA_IS4241: Fri 1400 - 1730
FM_BMA5008_B: Mon 1800 – 2130
...

Mon:
1000 - 1230: APT_ST5214
1800 - 2130: FM_BMA5008_B
...
25

Example: A “Combinatorial”
Auction for HDB Flat Allocation
Objective: Maximize “Allocative Efficiency” (via bids)
Constraints:
All flats allocated (Balance allocated to “HDB”)
At most one flat per “actual applicant”
Allocation limits for applicant categories (Second-timers, Racial
Quotas according to Ethnic Integration Policy)
Modified data used to price allocated flats (allocated
bidder excluded; VCG Mechanism: optimal objective
function value compared with that of full problem).

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Example: A “Combinatorial”
Auction for HDB Flat Allocation
Sample Output (Synthetic Data)
...

Solution by Bidder:

Bidder 1 (2 bids): 1 x C_8
Bidder 2 (1 bids):
...
...

Bidders in group [SecondTimers, EIP_Limit_Chinese] allocated C_8 at price 88 (reserve
price: 53, % of reserve price: 166.0%)
Bidders in group [EIP_Limit_Malay] allocated C_3 at price 218 (reserve price: 180, % of
reserve price: 121.1%)
...

Bidder 1 allocated C_8 at price 88 [SecondTimers, EIP_Limit_Chinese] (bid: 95, reserve
price: 53, savings: 7, possible savings: 42, % savings: 16.67%)
Bidder 6 allocated C_3 at price 218 [EIP_Limit_Malay] (bid: 319, reserve price: 180,
savings: 101, possible savings: 139, % savings: 72.66%)

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Thanks Everybody,
Thanks.

28

Applying Linear Optimization Using GLPK

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