#### Problem

A post in [1] describes an interesting problem:

I have read about cutting stock problem, but this is a bit different.

We have orders in different variants, and a maximum sized machine to produce it.

Variants would be X,S,XL,L and ordered quantities are 100,40,40,80.

Say, the machine width is 6. This means, we can put 6 different variants

together and produce it.

We can put 2 X,1 S,1 XL,2 L, this means if we produce it 50 times,

output is :

X = 100 (0 waste)

S = 50 (10 waste)

XL = 50 (10 waste)

L = 100 (20 waste)

Total of 40 waste in 300 produced.

Another aproach to reduce waste would be creating 2 different variation. We

can put 4 X, 2 S and produce it 25 times, with 10 waste and make another

setup and put 2 XL,4 L and produce it 20 times with no waste. With total

10 waste we handled this production in 2 setups.

Since setup has a price, we would prefer first setup, or depending on

quantities, we may choose the second one.

I have read about cutting stock and it looks similar to this one, but

ability to divide quantities between different setups, this has more

potential to optimize and therefore more complex.

#### High level model

We can formulate this as a mixed-integer quadratic model. The data looks like:

The optimization model is as follows:

The optimization model is as follows:

Notes:

- The variables \(run_r\)
- We have \(run_r=0 \Rightarrow pattern_{v,r}=0, runlen_r=0\). We could drop one of these constraints but not both.
- Equation \(edemand\) has a nasty quadratic term, It causes the model to be non-linear and non-convex. Of course we can solve this with a global solver.
- The \(order\) constraint makes sure all active runs (ie. \(run_r=1\)) are before the inactive ones (\(run_r=0\)). This makes the solution look nicer and will reduce some symmetry.
- The funny \(\forall r-1\) means we generate the \(order\) constraint for all \(r\) except the first one.
- The model type is actually MIQCP (Mixed Integer Quadratically Constrained Program). There are commercial solvers for convex problems of this type, but for this non-convex problem I used a global MINLP solver (Baron, Couenne, Antigone).

#### Results

The input data can be summarized as:

---- 62 PARAMETER costwaste cost of waste X 1.000, S 2.000, XL 3.000, L 4.000 ---- 62 PARAMETER costsetup = 100.000 fixed cost of a run ---- 62 PARAMETER demand X 100, S 40, XL 40, L 80

The solution looks like:

---- 62 VARIABLE runlen.L length of a run run1 20, run2 40 ---- 62 VARIABLE pattern.L production pattern of a run run1 run2 X 1 2 S 1 XL 2 L 2 1 ---- 62 VARIABLE waste.L waste per variant ( ALL 0. ) ---- 62 VARIABLE wastecost.L = 0.000 cost of waste VARIABLE setupcost.L = 200.000 cost related to setups VARIABLE cost.L = 200.000 total cost

We see we use two production runs. We are under-utilizing the machine (a pattern of 5 and 4 items). It is noted that this solution is not unique. A different solver will likely give a different solution.

#### Shorter run lengths

I believe we are missing some objective in this model. Suppose we have a demand of X=60 only (zero demand for the other variants). We can organize in a single run in at least the following ways:

- Pattern X=1. This gives a run length of 60. Total cost = 100 (cost of one setup).
- Pattern X=6. This gives a run length of 10. Total cost = 100 (cost of one setup).

I would guess the second configuration is better. Probably we should add some cost for the total sum of the run lengths.

If we add a unit cost of 1 for operating the machine, the total operating cost can be expressed as \(\sum_r runlen_r\). If we add this to the total cost we get the following solution:

---- 68 VARIABLE runlen.L length of a run run1 4, run2 40 ---- 68 VARIABLE pattern.L production pattern of a run run1 run2 X 5 2 S 1 XL 1 L 2 ---- 68 VARIABLE waste.L waste per variant ( ALL 0. ) ---- 68 VARIABLE wastecost.L = 0.000 cost of waste VARIABLE setupcost.L = 200.000 cost related to setups VARIABLE opcost.L = 44.000 operating cost VARIABLE cost.L = 244.000 total cost

A visualization of these two production schedules can look like:

Production Schedules |

#### Linearization

If we want to use a high-performance MIP solver, we need to linearize equation \(edemand\). I don't know how to linearize an integer variable times an integer variable. However, if we can replace an integer variable by a series of binary variables, then we are in more favorable territory.

The machine can handle up 6 variants at the time. This means \(pattern_{v,r}\le 6\). This makes \(pattern\) a good candidate for a binary expansion:

\[\begin{align}&pattern_{v,r} = \sum_k k \cdot \delta_{v,r,k}\\&\sum_k \delta_{v,r,k} \le 1\\& \delta_{v,r,k} \in \{0,1\}\end{align}\]

where \(k\) runs from 1 to \(cap=6\). Now we can write the \(edemand\) equation as:

\[\sum_{r,k} k \cdot ( runlen_r \cdot \delta_{v,r,k}) = demand_v + waste_v\]

The multiplication of a binary variable and a continuous variable can be written as linear inequalities [2]. This results in:

\[\begin{align} & \sum_{r,k} k \cdot w_{v,r,k} = demand_v + waste_v\\ & w_{v,r,k} \le \delta_{v,r,k} \cdot maxrunlen\\ & w_{v,r,k} \le runlen_r\\ & w_{v,r,k} \ge runlen_r - maxrunlen \cdot (1-\delta_{v,r,k}) \\ &w_{v,r,k}\ge 0 \end{align} \]

We can also go the other route: expand \(runlen_r\). This would lead to:

\[\begin{align}

&\sum_{r,\ell} \ell \cdot v_{v,r,\ell} = demand_v + waste_v\\

&\sum_{\ell} \ell \cdot \rho_{r,\ell} = runlen_r\\

&\sum_{\ell} \rho_{r,\ell} \le 1\\

& v_{v,r,\ell} \le cap \cdot \rho_{r,\ell}\\

& v_{v,r,\ell} \le pattern_{v,r}\\

& v_{v,r,\ell} \ge pattern_{v,r} - cap \cdot (1-\rho_{r,\ell})\\

& \rho_{r,\ell} \in \{0,1\}\\

& v_{v,r,\ell} \ge 0

\end{align}\]

With these lengthy reformulations we end up with a linear mixed-integer program. This means we can use standard MIP solvers instead of a global nonlinear solver. In addition, MIP formulations often perform better than their non-linear counterparts. The first MIP formulation seems to work much better than the second.

The linearization is somewhat convoluted, so having available a smaller nonlinear model can help in constructing and debugging these models. As is the case in database design (logical vs. physical design) and programming (premature optimization), starting with a simpler model that focuses on concepts is often helpful in developing optimization models.

The machine can handle up 6 variants at the time. This means \(pattern_{v,r}\le 6\). This makes \(pattern\) a good candidate for a binary expansion:

\[\begin{align}&pattern_{v,r} = \sum_k k \cdot \delta_{v,r,k}\\&\sum_k \delta_{v,r,k} \le 1\\& \delta_{v,r,k} \in \{0,1\}\end{align}\]

where \(k\) runs from 1 to \(cap=6\). Now we can write the \(edemand\) equation as:

\[\sum_{r,k} k \cdot ( runlen_r \cdot \delta_{v,r,k}) = demand_v + waste_v\]

The multiplication of a binary variable and a continuous variable can be written as linear inequalities [2]. This results in:

\[\begin{align} & \sum_{r,k} k \cdot w_{v,r,k} = demand_v + waste_v\\ & w_{v,r,k} \le \delta_{v,r,k} \cdot maxrunlen\\ & w_{v,r,k} \le runlen_r\\ & w_{v,r,k} \ge runlen_r - maxrunlen \cdot (1-\delta_{v,r,k}) \\ &w_{v,r,k}\ge 0 \end{align} \]

We can also go the other route: expand \(runlen_r\). This would lead to:

\[\begin{align}

&\sum_{r,\ell} \ell \cdot v_{v,r,\ell} = demand_v + waste_v\\

&\sum_{\ell} \ell \cdot \rho_{r,\ell} = runlen_r\\

&\sum_{\ell} \rho_{r,\ell} \le 1\\

& v_{v,r,\ell} \le cap \cdot \rho_{r,\ell}\\

& v_{v,r,\ell} \le pattern_{v,r}\\

& v_{v,r,\ell} \ge pattern_{v,r} - cap \cdot (1-\rho_{r,\ell})\\

& \rho_{r,\ell} \in \{0,1\}\\

& v_{v,r,\ell} \ge 0

\end{align}\]

With these lengthy reformulations we end up with a linear mixed-integer program. This means we can use standard MIP solvers instead of a global nonlinear solver. In addition, MIP formulations often perform better than their non-linear counterparts. The first MIP formulation seems to work much better than the second.

---- 136 PARAMETER report obj time nodes minlp.baron 244.000 2.120 58.000 mip1.cplex 244.000 1.218 673.000 mip2.cplex 244.000 12.063 3382.000

The linearization is somewhat convoluted, so having available a smaller nonlinear model can help in constructing and debugging these models. As is the case in database design (logical vs. physical design) and programming (premature optimization), starting with a simpler model that focuses on concepts is often helpful in developing optimization models.

#### References

- Similar to cutting stock problem but not quite, https://math.stackexchange.com/questions/2557776/similar-to-cutting-stock-problem-but-not-quite
- Multiplication of a continuous and a binary variable, http://yetanothermathprogrammingconsultant.blogspot.com/2008/05/multiplication-of-continuous-and-binary.html
- Gislaine Mara Melega, Silvio Alexandre de Araujo and Raf Jans,
*Comparison of MIP Models for the Integrated Lot-Sizing and One-Dimensional Cutting Stock Problem*, Pesquisa Operacional (2016) 36(1): 167-196.