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.

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

Jim has three requests for frozen ice sculptures, his commission is $1000, $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.

The AMPL model and data files, `ice.mod` and `ice.dat` respectively, are attached.

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

However, sometimes all the technology behind CPLEX does not work so well and we need to control the branch and bound tree. First, let’s remove all the CPLEX technology and re-solve our problem:

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;<span

style="font-family: monospace;">

With all CPLEX’s “bells and whistles” removed we get a slightly larger branch-and-bound tree:

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

Often you can gain insight into the branch-and-bound process by considering the LP relaxation. You can relax integrality without reformulating using

\begin{verbatim}

option relax_integrality 1;

\end{verbatim}

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

As you can see we are using 2.8 fridge units for our 2.8 sculptures. Also, if we check the {\tt TotalWeight} constraint ({\tt display TotalWeight.body;}) 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.

AMPL and CPLEX allow you to define a

\begin{verbatim}

suffix priority IN, integer, >= 0, <= 9999;

\end{verbatim}

(now we can assign variables priorities ranging from 0 – least – to 9999 – most). Let’s give the {\tt Fridges} variable a priority of 100 and the {\tt Make} variables a priority of 0 (using {\tt let} statements).

\begin{verbatim}

let Fridges.priority := 100;

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

\end{verbatim}

The branch-and-bound tree appears unchanged, so perhaps CPLEX had already branched on {\tt Fridges} first earlier. However, we can try a *breadth-first search* of the tree, since this will try different values for {\tt Fridges} before performing branching on other variables. Setting {\tt nodeselect} to 2 (best estimate) and {\tt backtrack} to 0 makes CPLEX perform a search very close to breadth-first (see The AMPL CPLEX User Guide for full details).

\begin{verbatim}

option cplex_options ('timing 1 mipdisplay 5 mipinterval 1 ' &

'presolve 0 mipcuts -1 cutpass -1 ' &

'heurfreq -1 ' &

'nodeselect 2 backtrack 0');

\end{verbatim}

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).

\begin{verbatim}

suffix direction IN, integer, >= -1, <= 1;

\end{verbatim}

We can force a down branch first on {\tt Fridges}:

\begin{verbatim}

let Fridges.direction := -1;

\end{verbatim}

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:

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;<span

style="font-family: monospace;">

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

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.

-- MichaelOSullivan - 23 Apr 2008

- ice_original.jpg:

- ice_nofrills.jpg:

- ice_relaxation.jpg:

- ice_breadth.jpg:

- ice_best.jpg:

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: r1 - 2008-04-23 - MichaelOSullivan

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