2d bin packing with Google OR-Tools CP-SAT, how to fix?

In [1] I looked at some MIP formulations for a 2d bin packing problem. OR-Tools has a nice model.AddNoOverlap2D constraint function. So, I was excited to try this out. (In these times I get quickly enthusiastic about these things). 

The idea is as follows. ortools has this construct Constraint.OnlyEnforce If(boolean). This is essentially an indicator constraint: \((\color{darkred}x\ge 5)\text{.OnlyEnforceIf($\color{darkred}\delta$)}\) means \[\color{darkred}\delta=1 \Rightarrow \color{darkred}x \ge 5\] This inspires me to develop a model along the lines of: \[\begin{align} \min& \sum_k \color{darkred}u_k\\  & \color{darkred}\beta_{i,j} = 0 \Rightarrow \color{darkred}b_i\ne\color{darkred}b_j \\ & \color{darkred}\beta_{i,j} = 1 \Rightarrow \mathbf{NoOverlap2D}(i,j) \\ & \color{darkred}u_k = 0 \Rightarrow \color{darkred}b_i \lt k \end{align}\] Here \(\color{darkred}b_i\) is the bin number where item \(i\) is placed and \(\color{darkred}\beta_{i,j}\) is true if  \(\color{darkred}b_i=\color{darkred}b_j\).

OK, so I started coding:

An easy variant of 2d bin packing

 In [1] the following problem is posted:

Consider the 2d bin packing problem [2]. Now we have the following additional restrictions: items of a different type or category cannot be below or above each other. In other words, we arrange items of the same type in columns. 

We use again our small data set: 

----     37 PARAMETER itemSize  sizes of bin and items

               w           h

cat1       7.000      12.000
cat2       9.000       3.000
cat3       5.000      14.000
cat4      13.000       9.000
cat5       6.000       8.000
cat6      20.000       5.000
bin       40.000      60.000


----     37 SET itemMap  mapping between items and categories

item1 .cat1,    item2 .cat1,    item3 .cat1,    item4 .cat1,    item5 .cat1,    item6 .cat1,    item7 .cat1
item8 .cat1,    item9 .cat1,    item10.cat1,    item11.cat2,    item12.cat2,    item13.cat2,    item14.cat2
item15.cat2,    item16.cat2,    item17.cat2,    item18.cat2,    item19.cat2,    item20.cat2,    item21.cat3
item22.cat3,    item23.cat3,    item24.cat3,    item25.cat3,    item26.cat3,    item27.cat3,    item28.cat3
item29.cat3,    item30.cat3,    item31.cat4,    item32.cat4,    item33.cat4,    item34.cat4,    item35.cat4
item36.cat4,    item37.cat4,    item38.cat4,    item39.cat4,    item40.cat4,    item41.cat5,    item42.cat5
item43.cat5,    item44.cat5,    item45.cat5,    item46.cat6,    item47.cat6,    item48.cat6,    item49.cat6
item50.cat6

I'll try two different models:

• Assign items to bins, arrange them in columns
• Assign columns to bins

2d Bin Packing

 In [1], a question was posed about a particular type of 2d bin-packing. This is a good reason to play with some models.

First I discuss three formulations with continuous no-overlap constraints and one for discrete data. In the next post, I'll discuss the variant described by the post [1]. This actually leads to a very different, but easy problem.

Problem statement and example data

We consider \(n=50\) rectangular items of different sizes that have to be placed into bins. There are six different categories, each with a different size (height and width). All the bins have the same size. The goal is to use as few bins as possible.

----     37 PARAMETER itemSize  sizes of bin and items

               w           h

cat1       7.000      12.000
cat2       9.000       3.000
cat3       5.000      14.000
cat4      13.000       9.000
cat5       6.000       8.000
cat6      20.000       5.000
bin       40.000      60.000


----     37 SET itemMap  mapping between items and categories

item1 .cat1,    item2 .cat1,    item3 .cat1,    item4 .cat1,    item5 .cat1,    item6 .cat1,    item7 .cat1
item8 .cat1,    item9 .cat1,    item10.cat1,    item11.cat2,    item12.cat2,    item13.cat2,    item14.cat2
item15.cat2,    item16.cat2,    item17.cat2,    item18.cat2,    item19.cat2,    item20.cat2,    item21.cat3
item22.cat3,    item23.cat3,    item24.cat3,    item25.cat3,    item26.cat3,    item27.cat3,    item28.cat3
item29.cat3,    item30.cat3,    item31.cat4,    item32.cat4,    item33.cat4,    item34.cat4,    item35.cat4
item36.cat4,    item37.cat4,    item38.cat4,    item39.cat4,    item40.cat4,    item41.cat5,    item42.cat5
item43.cat5,    item44.cat5,    item45.cat5,    item46.cat6,    item47.cat6,    item48.cat6,    item49.cat6
item50.cat6

Furthermore, we assume items cannot be rotated.