Contents - School of Mathematics, Statistics and Operations Research

School of Mathematics, Statistics and Operations Research
Te Kura Mātai Tatauranga, Rangahau Pūnaha
OPRE 355 Model Building with GMPL Weeks 1 and 2
Outline. We revise the GMPL modelling language (from OPRE253) and extend its use on
a number of further applications. The transportation problem is a good example of a problem
class, i.e., it is a generic problem structure of which there are many problem instances with
particular costs, supplies and demands. Hence, we will also show how to separate the general
model of the problem class from the problem instance data in GMPL.
MSOR Student Account Activation. Please follow the “register register” instructions on
the lab door.
Contents
1 Introduction 2
1.1 GMPL: A Modelling Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 GLPSOL: A Model Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Application: Assigning Jobs to Machines 5
2.1 A First GMPL Model using Subscripts . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 A Second GMPL Model using Summations and Families of Constraints . . . . . 8
2.3 A General Model of the “Assignment Problem” . . . . . . . . . . . . . . . . . . . 10
2.4 GMPL: Model + Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Application: A Transportation Problem 13
3.1 GMPL: Model + Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Application: Solving Two-Person Zero-Sum Games 18
4.1 GMPL: Model + Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5 Practice Problems 22
OPRE 355, 2012 1 Lab 1

1 Introduction
Example: A Diet Problem
My diet requires that all the food I eat come from one of the four “basic food groups” (chocolate
cake, ice cream, soft drink, and cheesecake). Each (large) slice of chocolate cake costs 50c, each
scoop of chocolate ice cream costs 20c, each bottle of cola costs 30c, and each piece of pineapple
cheesecake costs 80c. Each day, I must ingest at least 500 calories, 6 oz of chocolate, 10 oz
of sugar, and 8 oz of fat. The nutritional content per unit of each food is shown in the table
below. Formulate a linear programming model that can be used to satisfy my daily nutritional
requirement at minimum cost.
Type of Calories Chocolate Sugar Fat
Food (ounces) (ounces) (ounces)
Chocolate Cake (1 slice) 400 3 2 2
Chocolate ice cream (1 scoop) 200 2 2 4
Cola (1 bottle) 150 0 4 1
Pineapple cheesecake (1 piece) 500 0 4 5
Here is an LP model for this diet problem.
##
## GMPL Model: Diet Problem
##
var x1>=0; # number of slices of chocolate cake eaten daily
var x2>=0; # number of scoops of chocolate ice cream eaten daily
var x3>=0; # number of bottles of cola drunk daily
var x4>=0; # number of pieces of pineapple cheesecake eaten daily
minimize DailyCost: 50*x1 + 20*x2 + 30*x3 + 80*x4;
subject to Calories: 400*x1 + 200*x2 + 150*x3 + 500*x4 >= 500;
subject to Chocolate: 3*x1 + 2*x2 >= 6;
subject to Sugar: 2*x1 + 2*x2 + 4*x3 + 4*x4 >= 10;
subject to Fat: 2*x1 + 4*x2 + x3 + 5*x4 >= 8;
end;
1.1 GMPL: A Modelling Language
The GNU Mathematical Programming Language (GMPL or “GNU MathProg”) is a mathematical
modelling language that is part of the GNU Linear Programming Kit (GLPK). 1 GMPL is a
subset of a more well-known language called A Mathematical Programming Language (AMPL)
which is expensive (see www.ampl.com).
1 GLPK is free software and you can download it from http://www.gnu.org/software/glpk/glpk.html.
OPRE 355, 2012 2 Lab 1

A mathematical modelling language allows you to define or construct a mathematical model. It
is not the same as a programming language (such as Java) or a statistical computing language
(such as R). A separate solver is used to solve the model.
The amazing thing is that the diet problem formulation above is written exactly the input
format that GMPL uses.
Tasks
• Each line ends with a semicolon ‘;’.
• Also, the hash character ‘#’ represents a comment (the rest of the line is ignored).
• Each variable is defined using the keyword ‘var’.
• Note that the objective function and each constraint are named.
• All white space is ignored but is useful for clarity of reading.
• The file ends with the line ‘end;’.
1. Createadirectory calledopre355inyourhomedirectory. Todothisclick onthe“Terminal
program” icon at the bottom of the desktop. Then type
mkdir opre355
cd opre355
2. Type the formulation from diet problem example into a text file and save it as diet.mod
in your opre355 directory. To do this you will need to use a text editor, e.g., use KWrite
from the icons at the bottom of the desktop.
Modelling with Integer and Binary Variables
GMPL can also solve Integer Programming (IP) models.
We can specify that any variable is restricted to integer or binary, i.e., {0,1} values using the
notation below:
var x1 integer;
var x2 integer >= 0;
var x3 binary;
1.2 GLPSOL: A Model Solver
The solver we will be using is called glpsol which is available in the Statistical Computing Lab
(Cotton 535). It is capable of solving LP models and IP models but not nonlinear models.
OPRE 355, 2012 3 Lab 1

Tasks
1. In your “terminal program” (usually just called a “shell”) make sure you change the
directory to the place where your file diet.mod is saved. You probably want to type
something like
cd ~/opre355
ls
and make sure that diet.mod has been listed.
2. To solve a model you simply type
glpsol --model diet.mod
which will produce alot of output something like
Reading model section from diet.mod...
17 lines were read
Generating DailyCost...
Generating Calories...
Generating Chocolate...
Generating Sugar...
Generating Fat...
Model has been successfully generated
lpx_simplex: original LP has 5 rows, 4 columns, 18 non-zeros
lpx_simplex: presolved LP has 4 rows, 4 columns, 14 non-zeros
lpx_adv_basis: size of triangular part = 4
0: objval = 0.000000000e+00 infeas = 1.000000000e+00 (0)
4: objval = 2.200000000e+02 infeas = 0.000000000e+00 (0)
* 4: objval = 2.200000000e+02 infeas = 0.000000000e+00 (0)
* 7: objval = 9.000000000e+01 infeas = 0.000000000e+00 (0)
OPTIMAL SOLUTION FOUND
Time used: 0.0 secs
Memory used: 0.1M (151281 bytes)
The output above tells us that an optimal solution has been found and that the optimal
objective function value is 90. Unfortunately it does not tell us the optimal values of the
decision variables.
3. In your file diet.mod add the following two lines before the end; line.
solve;
display x1,x2,x3,x4;
Running the solver again gives the full optimal solution.
• What is the optimal solution?
• Interpret this in terms of the original word problem.
4. Modify some of the coefficients in the LP model and resolve. What is the effect on the
optimal solution? Can change the problem so that it is (a) infeasible, or (b) unbounded?
OPRE 355, 2012 4 Lab 1

2 Application: Assigning Jobs to Machines
Suggested Reading: Hillier and Lieberman §8.3
We have seen that LP models often belong to a problem class, e.g., production models, diet
models, blending models, multiple period models, distribution models, etc. A problem class is
simply a generic problem structure of which there are many problem instances with particular
costs, resourcesavailable, etc. Thefirstobjectiveofthislabistoshowhowtoseparatethegeneral
model of the problem class from the problem instance data in GMPL. The second objective of
this lab is solve two-person zero-sum games, i.e., to determine the optimal strategies for each
player and the value of the game.
Example
MachineCo has four machines and four jobs to be completed. Each machine must be assigned
to complete one job. The time required to set up each machine for completing each job is shown
in the table below. MachineCo wants to minimize the total setup time needed to complete the
four jobs.
Time (hours)
Machine Job 1 Job 2 Job 3 Job 4
1 14 5 8 7
2 2 12 6 5
3 7 8 3 9
4 2 4 6 10
Formulate this problem as an optimization model.
OPRE 355, 2012 5 Lab 1

Solution
Let
Warning
xij =
1 if job j is assigned to machine i
0 otherwise
• Note carefully here that xij can only take the value 0 or 1, so is called a binary variable.
• The resulting optimization model is not an LP model but rather it is an Integer Programming
(IP) model. Fortunately GMPL is very good at solving IP models.
Model Formulation
min 14x11 +5x12 +8x13 +7x14 +2x21 +12x22 +6x23 +5x24
+7x31 +8x32 +3x33 +9x34 +2x41 +4x42 +6x43 +10x44
s.t. x11 +x12 +x13 +x14 ≤ 1 (machine 1)
Explanation of Model Formulation
x21 +x22 +x23 +x24 ≤ 1 (machine 2)
x31 +x32 +x33 +x34 ≤ 1 (machine 3)
x41 +x42 +x43 +x44 ≤ 1 (machine 4)
x11 +x21 +x31 +x41 ≥ 1 (job 1)
x12 +x22 +x32 +x42 ≥ 1 (job 2)
x13 +x23 +x33 +x43 ≥ 1 (job 3)
x14 +x24 +x34 +x44 ≥ 1 (job 4)
xij ∈ {0,1} for all i,j
• The first four constraints ensure that each machine is assigned to at most one job.
• The last four constraints ensure that each job is completed by at least one machine.
• In general, there may be more machines that jobs, so that some machines have no job
assigned, i.e., some machines are not used.
OPRE 355, 2012 6 Lab 1

2.1 A First GMPL Model using Subscripts
###
### GMPL Model: MachineCo Assignment Problem
###
var x{1..4,1..4} binary;
minimize totalcost: 14*x[1,1] + 5*x[1,2] + 8*x[1,3] + 7*x[1,4] +
2*x[2,1] + 12*x[2,2] + 6*x[2,3] + 5*x[2,4] +
7*x[3,1] + 8*x[3,2] + 3*x[3,3] + 9*x[3,4] +
2*x[4,1] + 4*x[4,2] + 6*x[4,3] + 10*x[4,4];
subject to machine1: x[1,1] + x[1,2] + x[1,3] + x[1,4] = 1;
solve;
display x;
end;
New Ideas
• Here var x{1..4,1..4} binary; defines a family of variables with subscripts. We can
then write x[1,1] instead of x11, etc.
• Also, each variable is a binary variable, i.e., the only feasible values are 0 and 1.
Tasks
1. TypetheMachineComodelformulationaboveintoatextfileandsaveitasmachineco1.mod
in your opre355 directory.
2. In your “terminal program” (usually just called a “shell”) make sure you change the
directory to the place where your file machineco1.mod is saved. You probably want to
type something like cd ~/opre355 followed by ls and make sure that machineco1.mod
has been listed.
3. Solve this GMPL model using
glpsol --model machineco1.mod
and interpret your output.
OPRE 355, 2012 7 Lab 1

2.2 A Second GMPL Model using Summations and Families of Constraints
The GMPL model machineco1.mod requires quite a lot of repetitive typing.
Summations
Consider the constraint for machine 1.
x11 +x12 +x13 +x14 ≤ 1
We would be quite happy writing (in summation notation)
4
x1j ≤ 1 (machine 1)
j=1
Similarly, for the other machine constraints we would be happy writing
4
x2j ≤ 1 (machine 2)
j=1
4
x3j ≤ 1 (machine 3)
j=1
4
x4j ≤ 1 (machine 4)
j=1
In GMPL, we can use a summation in a constraint, e.g.
subject to machine1: sum{j in 1..4} x[1,j] = 1;
Families of Constraints
Actually, we can be even more brief by writing
4
xij ≤ 1 for machine i ∈ {1,2,3,4}
j=1
which defines one constraint for each machine i ∈ {1,2,3,4} and
4
xij ≥ 1 for job j ∈ {1,2,3,4}
i=1
which defines one constraint for each machine j ∈ {1,2,3,4}.
OPRE 355, 2012 8 Lab 1

• In GMPL, we can define a family of constraints
subject to machine{i in 1..4}: sum{j in 1..4} x[i,j] = 1;
In each such constraint, the value of the index i is fixed.
• Similarly,
subject to job{j in 1..4}: sum{i in 1..4} x[i,j] = 1;
In each such constraint, the value of the index j is fixed.
In this way, we follow very closely (and powerfully) the general formulation with the Σ notation
and the indexed constraints.
Full GMPL Model
###
### GMPL Model: MachineCo Assignment Problem
###
var x{1..4,1..4} binary;
minimize totalcost: 14*x[1,1] + 5*x[1,2] + 8*x[1,3] + 7*x[1,4] +
2*x[2,1] + 12*x[2,2] + 6*x[2,3] + 5*x[2,4] +
7*x[3,1] + 8*x[3,2] + 3*x[3,3] + 9*x[3,4] +
2*x[4,1] + 4*x[4,2] + 6*x[4,3] + 10*x[4,4];
subject to machine{i in 1..4}: sum{j in 1..4} x[i,j] = 1;
solve;
display x;
end;
Tasks
1. TypethisnewMachineComodelformulation intoatextfileandsave itasmachineco2.mod
in your opre355 directory. You may wish to copy-and-paste most of the model from your
machineco1.mod text file.
2. Solve this GMPL model using
glpsol --model machineco2.mod
and check that you get the same solution as before.
OPRE 355, 2012 9 Lab 1

2.3 A General Model of the “Assignment Problem”
The MachineCo example above is a particular instance of a general class of problem called the
“Assignment Problem”. It involves the optimal matching or pairing of objects of two distinct
types — jobs to machines, sales personnel to customers, and so on.
• Suppose there are m machines and n jobs, where m ≥ n
• Each job must be done by at least one machine.
• Each machine can do at most one job.
• The cost of machine i doing job j is cij.
• The problem is to assign the machines to the jobs so as to minimize the total cost of
completing all of the jobs.
We introduce binary variables xij, i = 1,...,m, j = 1,...,n
1 if job j is assigned to machine i
xij =
0 otherwise
• Since each machine j can do no more than one job, we have the machine constraints
n
xij ≤ 1 for i = 1,...,m
j=1
• Since at least one machine must do job j, we also have the job constraints
• The objective function is
m
xij ≥ 1 for j = 1,...,n
i=1
In summary, the assignment problem is
min
s.t.
m
n
i=1 j=1
cijxij
min
m
n
i=1 j=1
cijxij
n
xij ≤ 1 for machine i = 1,...,m
j=1
m
xij ≥ 1 for job j = 1,...,n
i=1
xij ∈ {0,1} for all i,j
OPRE 355, 2012 10 Lab 1

2.4 GMPL: Model + Data
We are now ableto separate (in GMPL)the general model of theproblem class fromthe problem
instance data. Rather than using numbers for the machines and the jobs, we can now use any
labels we wish, e.g., people’s names. These will be defined as sets below.
GMPL Model
###
### GMPL Model: Assignment Problem
###
# Set definitions
set machines;
set jobs;
# Parameter definitions
param cost{machines,jobs};
# Variable definitions
var x{machines,jobs} binary;
# Objective function
minimize totalcost:
sum{i in machines,j in jobs} cost[i,j]*x[i,j];
# Constraints
subject to machine_constraint{i in machines}:
sum{j in jobs} x[i,j] = 1;
# Solve and display results
solve;
display x;
end;
Sets and Parameters
The model starts with the reserved word set to identify the constraints and variables of the
problem generically using the user-generated names machines and jobs. Next, we use the
reserved word param to name the elements of the objective function and constraints, again using
user-generated name cost. Thus cost defines the later coefficients in the objective function as
a function of the sets machines and jobs. Once the set and param segments have been defined,
the model itself can be developed.
OPRE 355, 2012 11 Lab 1

Problem Instance and GMPL Data
A particular problem instance would be specified by supplying the cij cost data for the general
model. The equivalent in GMPL is to specify the values that the sets and parameters take.
For the MachineCo example, the data we require is as follows. Note that this is very readable.
###
### GMPL Data: MachineCo Assignment Problem
###
set machines := m1 m2 m3 m4;
set jobs := j1 j2 j3 j4;
param cost : j1 j2 j3 j4 :=
m1 14 5 8 7
m2 2 12 6 5
m3 7 8 3 9
m4 2 4 6 10;
Observe the use of := to flag the start of the data elements of a set or param. The data is
stored in a separate file, in this case machineco.dat (see the tasks below).
Tasks
1. Type the GMPL model formulation of the assignment problem into a text file and save
it as assignment.mod in your opre355 directory. You may wish to copy-and-paste some
of the model from your machineco2.mod text file.
2. Type the GMPL data into a text file and save it as machineco.dat in your opre355
directory.
3. We can solve the GMPL model in assignment.mod together with the GMPL data in
machineco.dat using the following command.
glpsol --model assignment.mod --data machineco.dat
Compare the solution with solutions to previous tasks.
In this way the same GMPL model can be used with many different problem
instances, each of which has its own GMPL data file.
OPRE 355, 2012 12 Lab 1

3 Application: A Transportation Problem
Example
The Fly-by-Night Airline must decide on the amounts of jet fuel to buy from three possible
vendors. The airline refuels its aircraft regularly at the four airports it serves. Theoil companies
have said that they can supply up to the following amounts of fuel during the coming month:
275,000 gallons for BP; 350,000 gallons for Caltex; and 345,000 gallons for Shell. The required
amount of jet fuel is 110,000 gallons at Dunedin; 235,000 gallons at Christchurch; 185,000 gallons
at Wellington; and 440,000 gallons at Auckland.
When transportation costs are added to the price per gallon supplied, the combined cost per
gallon for jet fuel from each vendor supplying a specific airport is shown in the table below.
GMPL Model
Dunedin Christchurch Wellington Auckland
BP 10 10 9 11
Caltex 7 11 12 13
Shell 8 14 4 9
##
## GMPL: Fly-by-Night Airlines model formulation
##
var x{1..3,1..4} >= 0;
minimize total_cost: 10*x[1,1] + 10*x[1,2] + 9*x[1,3] + 11*x[1,4]
+ 7*x[2,1] + 11*x[2,2] + 12*x[2,3] + 13*x[2,4]
+ 8*x[3,1] + 14*x[3,2] + 4*x[3,3] + 9*x[3,4];
subject to supply_BP: sum{j in 1..4} x[1,j] = 185000;
subject to demand_Auckland: sum{i in 1..3} x[i,4] >= 440000;
solve;
display x;
end;
This model has 12 variables and 7 constraints. Note that you can download flybynight.mod
from the OPRE355 webpage, i.e., you do not need to type this in yourself.
OPRE 355, 2012 13 Lab 1

Subscripts and Summation
In the GMPL model above we have introduce a few new things.
var x{1..3,1..4} >= 0;
defines a family of variables with subscripts. We can then write x[1,1] instead of x11, etc.
Also, we have summations in the constraints, e.g.
sum{j in 1..4} x[1,j]

GMPL Model
###
### GMPL Model: Transportation Problem
###
### Set definitions
set sources;
set destinations;
### Parameter definitions
param supply{sources};
param demand{destinations};
param cost{sources,destinations};
### Variable definitions
var x{sources,destinations} >= 0;
### Objective function
minimize total_cost:
sum {i in sources,j in destinations} cost[i,j]*x[i,j];
### Constraints
supply_constraint{i in sources}:
sum{j in destinations} x[i,j] = demand[j];
solve;display x;
end;
Sets and Parameters
The model starts with the reserved word set to identify the constraints and variables of the
problem generically using the user-generated names sources and destinations. Next, we use
the reserved word param to name the elements of the objective function and constraints, again
using user-generated names supply, demand and cost. Thus supply defines the later righthand-sides
of the supply constraints as a function of the set sources. Once the set and param
segments have been defined, the model itself can be developed.
Families of Constraints
Notice also that we can specify whole families of constraints using the notation:
supply_constraint{i in sources}:
sum{j in destinations} x[i,j]

Problem Instance and GMPL Data
A particular problem instance would be specified by supplying the cij, si and dj data for the
general model. The equivalent in GMPL is to specify the values that the sets and parameters
take.
For the Fly-By-Night Airlines example, the data we require is as follows. Note that this is very
readable.
###
### GMPL Data: Fly-By-Night Airlines transportation problem
###
### Set values
set sources := BP Caltex Shell;
set destinations := Dunedin Christchurch Wellington Auckland;
### Parameter values
param supply :=
BP 275000
Caltex 350000
Shell 345000;
param demand :=
Dunedin 110000
Christchurch 235000
Wellington 185000
Auckland 440000;
param cost : Dunedin Christchurch Wellington Auckland :=
BP 10 10 9 11
Caltex 7 11 12 13
Shell 8 14 4 9 ;
Observe the use of := to flag the start of the data elements of a set or param. The data is
stored in a separate file, in this case flybynight.dat. In this way the same GMPL model can
be used with many different problem instances, each of which has its own data file.
Solution of the Problem Instance
Task
WecansolvetheGMPLmodelintransport.modtogetherwiththeGMPLdatainflybynight.dat
using the following command.
glpsol --model transport.mod --data flybynight.dat
OPRE 355, 2012 16 Lab 1

Practice Problem
Tom would like 3 pints of home brew today and an additional 4 pints of home brew tomorrow.
Dick is willing to sell a maximum of 5 pints total at a price of $3.00 per pint today and $2.70
per pint tomorrow. Harry is willing to sell a maximum of 4 pints total at a price of $2.90 per
pint today and $2.80 per pint tomorrow. Tom wishes to know what his purchases should be to
minimize his cost while satisfying his thirst requirements.
Task
Formulate this problem as a transportation problem by writing a GMPL data file and call it
tomdickharry.dat. Solve your model using
glpsol --model transport.mod --data tomdickharry.dat
and interpret your results.
OPRE 355, 2012 17 Lab 1

4 Application: Solving Two-Person Zero-Sum Games
Suggested Reading: Hillier and Lieberman Chapter 14
We will see that a two-person zero-sum game can be solved by solving two LP models, one for
the row player and one for the column player.
Example. The Birthday
Frank is hurrying home late, after a particularly grueling day, when it pops into his mind that
today is Kitty’s birthday! Or is it? Everything is closed except the florist.
If it is not her birthday and he brings no gift, the situation will be neutral, i.e., payoff 0. If it is
not her birthday and he comes in bursting with roses, and obviously confused, she may be very
suspicious, a payoff of say −1. If it is her birthday and he has, clearly, remembered it, that is
worth say 2. If he has forgotten it, he is down like a stone, say, −10. So Frank mentally forms
the payoff matrix.
Nature
Not Birthday Birthday
Frank Nothing 0 −10
Flowers −1 2
OPRE 355, 2012 18 Lab 1

###
### GMPL: Frank’s LP Model
###
var x{1..2} >=0;
var v;
maximize valueofgame: v;
subject to col1: v = (0)*y[1] + (-10)*y[2];
subject to row2: w >= (-1)*y[1] + (2)*y[2];
subject to prob: y[1]+y[2] = 1;
solve;
display w;
display y;
end;
Tasks
1. Type Frank’s LP model above into a text file and save is as frank.mod in your opre355
directory.
2. Solve Frank’s LP model using glpsol --model frank.mod
3. Type the Nature’s LP model above into a text file and save is as nature.mod in your
opre355 directory.
4. Solve Nature’s LP model using glpsol --model nature.mod
5. Interpret the optimal strategies and the value of the game.
OPRE 355, 2012 19 Lab 1

4.1 GMPL: Model + Data
As in Section 2.4 we can formulate a general model in a GMPL Model file (with sets and
parameters) and give the problem instance data in a GMPL Data file.
General GMPL Model
###
### GMPL Model: Two-Person Zero-Sum Game
###
### Model and solve the row player’s LP
###
set ROWS;
set COLS;
param payoff {ROWS,COLS};
var x {ROWS} >=0;
var v;
maximize valueofgame: v;
subject to constraint {j in COLS}:
v

Tasks
1. Download the file rowplayer.mod from the OPRE355 Labs webpage and save it in your
opre355 directory.
2. Type the GMPL Data above into a text file and save is as birthday.dat in your opre355
directory.
3. Solve Frank’s LP model using glpsol --model rowplayer.mod --data birthday.dat
and compare with the previous solution.
4. Adapt the GMPL model given above to solve the column player’s LP. Save it in a
different file called columnplayer.mod in your opre355 directory. You may wish to copyand-paste
some of the model from rowplayer.mod.
5. Solve the Nature’s LP usingglpsol --model columnplayer.mod --data birthday.dat
6. Suppose Frank is in a whole lot more trouble than he expected if he does not buy flowers
on Kitty’s birthday, i.e., supposethe payoff is now −100. What is the effect on the optimal
strategy and the value of the game?
OPRE 355, 2012 21 Lab 1

5 Practice Problems
1 Metalworking shop. A metalworking shop needs to cut at least 37 large disks and 211 small
ones from sheet metal rectangles of a standard size. Three cutting patterns are available. One
yields 2 large disks with 34% waste, the second gives 5 small disks with 22% waste, and the last
produces 1 large and 3 small disks with 27% waste. The shop seeks a minimum waste way to
fulfill its requirements.
(a) Formulate an LP model to choose an optimal cutting plan.
(b) Enter and solve your model with GMPL.
(c) What happens if we insist that all variables are integers?
2 Knights, knaves and werewolves. Suppose you are visiting a forest in which every inhabitant
is either a knight or a knave. Knights always tell the truth and knaves always lie. In addition
some of the inhabitants are werewolves and have the annoying habit of sometimes turning into
wolves at knight and devouring people. A werewolf can be either a knight or a knave.
You are interviewing three inhabitants, A, B, and C, and it is known that exactly one of them
is a werewolf. They make the following statements:
A: I am a werewolf.
B: I am a werewolf.
C: At most one of us is a knight.
Formulate(andsolve) anIPmodelinGMPLtodetermineacompleteclassification ofinhabitants
A, B and C.
3 Swim team. The coach needs to assign swimmers to a 50-metre medley relay team to send
to the University Games. Since most of his best swimmers are very fast in more than one
stroke, it is not clear which swimmer should be assigned to each of the four strokes. The five
fastest swimmers and the best time (in seconds) they have achieved in each of the strokes (for
50 metres) are given below.
Stroke Danyon Anthony Moss Paul Malcolm
Backstroke 37.7 32.9 33.8 37.0 35.4
Breaststroke 43.4 33.1 42.2 34.7 41.8
Butterfly 33.3 28.5 38.9 30.4 33.6
Freestyle 29.2 26.4 29.6 28.5 31.1
The coach wishes to determine how to assign four swimmers to the four different strokes to
minimize the sum of the corresponding best times.
OPRE 355, 2012 22 Lab 1
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(a) Formulate the coach’s assignment problem by writing a suitable GMPL data file (please
call it swimteam.dat) to be used with the GMPL model assignment.mod.
Hint. First decide what are the machines and what are the jobs.
(b) Solve the problem using glpsol and interpret your solution.
4 School Buses. The city of Springfield is taking bids from six bus companies (A, B, C, D, E
and F) on the eight routes (R1, R2, R3, R4, R5, R6, R7 and R8) that must be driven in the
surrounding school district. Each company enters a bid (in $) on how much it will charge to
drive selected routes, although not all companies bid on all routes. Blank cells indicate routes
on which a company does not bid.
Company R1 R2 R3 R4 R5 R6 R7 R8
A 8200 7800 5400 3900
B 7800 8200 6300 3300 4900
C 3800 4400 5600 3600
D 8000 5000 6800 6700 4200
E 7200 6400 3900 6400 2800 3000
F 7000 5800 7500 4500 5600 6000 4200
Thecity needs to select which companies to assign to which routes with the following conditions.
(1) If a company does not bid on a route, it cannot by assigned to that route.
(2) Exactly one company must be assigned to each route.
(3) A company can be assigned to at most two routes.
The objective is to minimize the total cost of covering all routes.
Tasks
(a) Briefly explain why Springfield’s problem is not an assignment problem.
(b) Adapt theGMPL modelassignment.modto make thechanges required in part (a). Please
call your new model doubleassignment.mod.
(c) Write a suitable GMPL data file called springfield.dat.
(d) Solve Springfield’s problem using
glpsol --model doubleassignment.mod --data springfield.dat
and interpret your solution.
OPRE 355, 2012 23 Lab 1
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5 Two armies are advancing on two cities. The first army is commanded by General Custard
and has four regiments; the second army is commanded by General Peabody and has three
regiments. At each city, the army that sends more regiments to the city captures both the city
and the opposing army’s regiments. If both armies send the same number of regiments to a city,
then the battle at the city is a draw. Each army scores 1 point per city captured and 1 point
per captured regiment. Assume that each army wants to maximize the difference between its
reward and its opponent’s reward.
(a) Formulate this situation as a two-person zero-sum game.
(b) Determine whether this game is strictly determined.
(c) Solve (using GMPL or R) for the value of the game and each player’s optimal strategies.
game001
6 Morra. Each player extends some fingers and, simultaneously, guesses how many the enemy
is extending. Thenumberhemay extend is 1, 2, or 3. If only one player guesses the enemy digits
successfully, the payoff to him is the total number of digits extended on the play. Otherwise,
the payoff is zero. (Payments are usually in coin, rather than fingers.) Thus if Blue holds out 3
fingers and guesses 1, while Red holds out 1 and guesses 2, Blue will win because he is correct;
he will win $4.
Each player may extend 1, 2, or 3 fingersand may guess, reasonably, that the enemy is extending
1, 2, or 3. So there are nine possibilities open to him, i.e., nine pure strategies. Letting 〈i,j〉
represent the strategy of putting out i fingers and guessing the opponent has put out j fingers,
the appropriate reward matrix is shown below.
Blue’s Red’s Strategy
Strategy 〈1,1〉 〈1,2〉 〈1,3〉 〈2,1〉 〈2,2〉 〈2,3〉 〈3,1〉 〈3,2〉 〈3,3〉
〈1,1〉 0 2 2 −3 0 0 −4 0 0
〈1,2〉 −2 0 0 0 3 3 −4 0 0
〈1,3〉 −2 0 0 −3 0 0 0 4 4
〈2,1〉 3 0 3 0 −4 0 0 −5 0
〈2,2〉 0 −3 0 4 0 4 0 −5 0
〈2,3〉 0 −3 0 0 −4 0 5 0 5
〈3,1〉 4 4 0 0 0 −5 0 0 −6
〈3,2〉 0 0 −4 5 5 0 0 0 −6
〈3,3〉 0 0 −4 0 0 −5 6 6 0
(a) Solve the game by writing a GMPL Data file and solving both the row player’s and the
column player’s LP models.
(b) Interpret the optimal solution. game020a
OPRE 355, 2012 24 Lab 1