Choosing boxes: set covering model

The data looks like:

We have 1166 records. We see there are items with the same length. These duplicates will always go in the same box type. So we don't have to distinguish them in the optimization model. As a result, we can reorganize the data into unique values and their count:

---    392 PARAMETER data  

            size       count

i1       156.0001.000
i2       162.0001.000
i3       168.0001.000
i4       178.0002.000
i5       180.0001.000
i6       185.0002.000
...
i379     806.0001.000
i380     820.0001.000
i381     823.0001.000
i382     827.0002.000
i383     855.0002.000
i384     864.0001.000

This table has 384 records. A big improvement over 1166 items! This will make our optimization models much smaller. Note that we make sure things are ordered by size (increasing). We will exploit this in one of the models below.

Assignment formulation

It is tempting to start with an assignment variable such as: \[x_{i,j} = \begin{cases} 1 & \text{if item $i$ is assigned to box type $j=1,\dots\,T$}\\ 0 & \text{otherwise}\end{cases}\] Here \(T\) is the number of differently sized boxes. Furthermore let \[a_j = \text{area of box $j$}\] This can lead to a model like: \[\bbox[lightcyan,10px,border:3px solid darkblue] {\begin{align}\min & \sum_{i,j} \> \mathit{count}_i \> a_j \> x_{i,j} \\ & \sum_j x_{i,j} = 1 & \forall i  & \> \> \text{(assignment)} \\ & x_{i,j} = 1 \Rightarrow a_j \ge \mathit{size}_i^2  & &\>\>\text{(fit)} \\ & x_{i,j} \in \{0,1\} \\ & a_j \ge 0 && \>\>\text{(area of box type $j$)}\end{align}}\] This is problematic: we have a non-convex quadratic objective. This model can be linearized quite easily: \[\bbox[lightcyan,10px,border:3px solid darkblue] {\begin{align}min & \sum_{i,j} \> \mathit{count}_i \>  y_{i,j} \\ & a_j \ge \mathit{size}_i^2 \> x_{i,j} \\ & y_{i,j} \ge a_j - \mathit{Amax} (1-x_{i,j}) \\ & a_1 = \mathit{Amax} \\ & a_{j+1} \le a_j - 1 \end{align}}\] where \(\mathit{Amax}=864^2\).

This model is very difficult to solve. After 20 minutes it still had a gap of 30%.

Covering formulation

This model is organized around the variable \[x_{i,j} = \begin{cases} 1 & \text{if items $i$ through $j$ are put in the same size box (with size $j$)} \\ 0 & \text{otherwise}\end{cases}\] Here we see why the ordering is useful: if \(x_{i,j}=1\) then all items \(i,i+1,\dots,j-1,j\) are placed in the size box.

The basic idea is to cover each item exactly once by a variable \(x_{i,j}\). In addition we want exactly \(T\) variables that have a value of one. Here is how we organize our variables \(x_{i,j}\):

Variables x(i,j)

A feasible solution has two properties:

• Each item (column) selects exactly one variable (row),
• The total number of selected variables is \(T\). 

Feasible solution for T=3

Indeed the total cost (area) goes up. Interestingly the smallest box size stays the same.

GAMS implementation

Notes:

• Intermediate sets are used to simplify things. I use this approach a lot, Sets are easier to debug than equations.
• Reporting is important. I try to collect meaningful information in the 3d parameter results. Hopefully the results can be interpreted even when not knowing the model or even when not knowing GAMS.
• The set xij(i,j) will indicate where we have x.l(i,j)=1. In practice, values for a binary variable can be 0.99999 or 1.00001. So I don't test against 1.0 exactly.
• I left out the data step where I combine duplicate item sizes. That was done in a separate piece of GAMS code.

Network model

This problem can also be modeled as a network problem. See [2] for more details.

References