IntegerProgrammingWithAMPL < OpsRes

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---+ Integer Programming with AMPL 

[[VariablesInAMPL#types][Specifying variables to be integer or binary]] in AMPL will cause the solver, e.g., CPLEX, to use mixed-integer programming. This will often be enough to solve many of the problems you will encounter. However, if your integer programmes are taking a long time to solve you can use some "tricks" to speed up the branch-and-bound process.

---++ A "Simple" Integer Programme

To demonstrate the techniques we can use to control integer programming we will look at a simple integer programming problem:

<blockquote>
Jim has three requests for frozen ice sculptures, his commission is $10000, $7000 and $5000 respectively. He must hire a refrigeration unit to transport each one. The units cost $4000 each. The sculptures will be transported on a truck with capacity 1.7 tonnes and he estimates the total weight of each sculpture (including the refrigeration unit) to be 1 tonne, half a tonne and a quarter of a tonne respectively. Jim must decide which sculptures to make to maximize his profit.
</blockquote>

The AMPL model and data files, [[%ATTACHURL%/ice.mod][<tt>ice.mod</tt>]] and [[%ATTACHURL%/ice.dat][<tt>ice.dat</tt>]] respectively, are attached.

Solving this problem with AMPL and CPLEX is very fast (it is only a small problem):

<img src="%ATTACHURLPATH%/ice_original.jpg" alt="ice_original.jpg" width='669' height='218' />

However, sometimes all the technology behind CPLEX does not work so well and we need to control the branch-and-bound tree. We will use this small example problem to demonstrate the effect of changing the default behaviour of CPLEX.

First, let's remove all the CPLEX technology and re-solve our problem using an AMPL script:
<pre>
reset;

model ice.mod;
data ice.dat;

option solver cplex;

option presolve 0;
option cplex_options ('timing 1 mipdisplay 5 mipinterval 1' &
                      'presolve 0 mipcuts -1 cutpass -1 ' &
                      'heurfreq -1');

solve;

display Fridges, Make;
</pre>

With all CPLEXs "bells and whistles" removed we get a slightly larger branch-and-bound tree:

<img src="%ATTACHURLPATH%/ice_nofrills.jpg" alt="ice_nofrills.jpg" width='669' height='518' />

Let's look at ways to reduce the size of this branch-and-bound tree.

---+++ Looking at the LP Relaxation

Often you can gain insight into the branch-and-bound process by considering the [[LPRelaxation][LP relaxation]]. You can relax integrality without reformulating using =option relax_integrality 1;=.

If we look at the variables we can see where our solution is fractional:

<img src="%ATTACHURLPATH%/ice_relaxation.jpg" alt="ice_relaxation.jpg" width='669' height='362' />

As you can see we are using 2.8 fridge units for our 2.8 sculptures. Also, if we check the =TotalWeight= constraint (<tt>display !TotalWeight.body;</tt>) we can see that the truck is at its weight limit.

It looks likely that we should only use 2 fridges. We can create some new suffixes to experiment with our hypothesis.

---+++ Priorities, Searching and Directions

AMPL and CPLEX allow you to define a _priority_ for your integer variables. This means that if more than one integer variable is fractional in a solution, CPLEX will branch on the highest priority variable first. Let's add the priority _suffix_ to our run file (before solving):
<pre>
suffix priority IN, integer, >= 0, <= 9999;
</pre>
(now we can assign variables priorities ranging from 0 - least - to 9999 - most). Let's give the =Fridges= variable a priority of 100 and the =Make= variables a priority of 0:
<pre>
let Fridges.priority := 100;
let {s in SCULPTURES} Make[s].priority := 0;
</pre>

<img src="%ATTACHURLPATH%/ice_nofrills.jpg" alt="ice_nofrills.jpg" width='669' height='518' />

The branch-and-bound tree appears unchanged, so perhaps CPLEX had already branched on =Fridges= early in the branch-and-bound tree. However, we can try a _breadth-first search_ of the tree, since this will try different values for =Fridges= before performing branching on other variables. Setting =nodeselect= to 2 (best estimate) and =backtrack= to 0 makes CPLEX perform a search very close to breadth-first (see [[https://www.ampl.com/BOOKLETS/amplcplex100userguide.pdf][The AMPL CPLEX User Guide]] for full details).
<pre>
option cplex_options ('timing 1 mipdisplay 5 mipinterval 1 ' &
                      'presolve 0 mipcuts -1 cutpass -1 ' &
                      'heurfreq -1 ' &
                      'nodeselect 2 backtrack 0');
</pre>

<img src="%ATTACHURLPATH%/ice_breadth.jpg" alt="ice_breadth.jpg" width='669' height='506' />

Now the tree has been fathomed earlier (it only has 4 nodes instead of 6). However, we are not sure if CPLEX branched down to 2 fridges first (our hypothetical optimum).

To control the direction of the branches we can create a new suffix for the direction we should branch on each variable (-1 for down, 0 for no preference, 1 for up).
<pre>
suffix direction IN, integer, >= -1, <= 1;
</pre>

We can force a down branch first on =Fridges=:
<pre>
let Fridges.direction := -1;
</pre>

<img src="%ATTACHURLPATH%/ice_breadth.jpg" alt="ice_breadth.jpg" width='669' height='506' />

This doesn't seem to have decreased the size of the branch-and-bound tree.

Let's try one more thing. We have given CPLEX a good branch to try first, but we have not carefully considered what to do next. Let's remove the breadth-first search option and let CPLEX decide how to proceed:
<pre>
reset;

model ice.mod;
data ice.dat;

option solver cplex;

option presolve 0;
option cplex_options ('timing 1 mipdisplay 5 mipinterval 1' &
                      'presolve 0 mipcuts -1 cutpass -1 ' &
                      'heurfreq -1');

suffix priority IN, integer, >= 0, <= 9999;
suffix direction IN, integer, >= -1, <= 1;

let Fridges.priority := 100;
let {s in SCULPTURES} Make[s].priority := 0;

let Fridges.direction := -1;

solve;

display Fridges, Make;
</pre>

<img src="%ATTACHURLPATH%/ice_best.jpg" alt="ice_best.jpg" width='669' height='458' />

Now we have reduced our branch-and-bound tree to a single node by making a good choice about our first variable branch and letting CPLEX proceed from there.

As stated earlier, CPLEX does a lot of good things automatically for you. Often, these "tricks" will be enough to solve your mixed-integer programming problems. However, if your problem is taking a long time to solve, you can experiment with adding some of your own control to the branch-and-bound process. History has shown that problem-specific approaches often work very well for hard integer programmes.

-- Main.MichaelOSullivan - 23 Apr 2008
Attachments
Topic attachments
I	Attachment	History	Action	Size	Date	Who	Comment
dat	ice.dat	r1	manage	0.2 K	2008-04-23 - 07:10	MichaelOSullivan
mod	ice.mod	r1	manage	0.4 K	2008-04-23 - 07:10	MichaelOSullivan
Topic revision: r5 - 2008-05-11 - MichaelOSullivan