Continuous Facility Location: Symmetry, SOCP and grid search

In this post, I consider the following question: 

Given \(m\) consumer locations, what is the minimum number of facilities (warehouses)  needed, \(n\), so each consumer is within \(K\) kilometers of a warehouse. 

A second, related question is:

Where should we place these \(n\) warehouses? 

I looked at this earlier in [1]. I want to add some new experiments: 

• Do something about symmetry: breaking symmetry by simple ordering constraints. These can be highly effective (but can also disappoint).
• See if I can formulate the "min sum distance" model as a convex SOCP (second-order cone programming) problem.
• Compare with a discrete location problem: place facilities on a grid. This makes the model very large but linear. How does it compare to the quadratic models that place facilities anywhere?

Generate consumer locations.

I just use random locations drawn from a uniform distribution: \[\color{darkblue}p_{i,c} \sim U(0,\color{darkblue}S)\] where \(c \in \{x,y\}\) are coordinates and \(\color{darkblue}S=100\). Of course, this is not very realistic. 

Q: What would be a good way to generate more realistic locations? Typically, locations are clustered a bit. Maybe: uniform locations with some random count of consumers at each location.   

For my experiments, I used a dataset \(m=50\) consumer locations. This is both small enough to make the models solve quickly and large enough to show differences in performance.

----     37 PARAMETER p  points

              x           y

p1       17.17584.327
p2       55.03830.114
p3       29.22122.405
p4       34.98385.627
p5        6.71150.021
p6       99.81257.873
p7       99.11376.225
p8       13.06963.972
p9       15.95225.008
p10      66.89343.536
p11      35.97035.144
p12      13.14915.010
p13      58.91183.089
p14      23.08266.573
p15      77.58630.366
p16      11.04950.238
p17      16.01787.246
p18      26.51128.581
p19      59.39672.272
p20      62.82546.380
p21      41.33111.770
p22      31.4214.655
p23      33.85518.210
p24      64.57356.075
p25      76.99629.781
p26      66.11175.582
p27      62.74528.386
p28       8.64210.251
p29      64.12554.531
p30       3.15279.236
p31       7.27717.566
p32      52.56375.021
p33      17.8123.414
p34      58.51362.123
p35      38.93635.871
p36      24.30324.642
p37      13.05093.345
p38      37.99478.340
p39      30.00312.548
p40      74.8876.923
p41      20.2020.507
p42      26.96149.985
p43      15.12917.417
p44      33.06431.691
p45      32.20996.398
p46      99.36036.990
p47      37.28977.198
p48      39.66891.310
p49      11.95873.548
p50       5.54257.630

Model 1: Minimize the number of facilities 

In the following, \(i\) indicates the consumer locations, and \(j\) refers to warehouses (facilities). The number of possible warehouses we can use is encoded in the set \(j \in J\). It should be larger than the number we expect, but not crazy large as this makes the problem much larger. If you choose it too small, the problem will be infeasible.

We restricted the search area to \([0,\color{darkblue}S]^2\). Assuming facilities will be placed within the convex hull of customer points, we could as well require: \[ \min_i \color{darkblue}p_{i,c}\le \color{darkred}w_{j,c} \le \max_i \color{darkblue}p_{i,c}\] For simplicity, I chose \(\color{darkred}w_{j,c} \in [0,\color{darkblue}S]\). 

The distance equation says: for consumer \(i\) there is (at least) one facility \(j\) that is closer than \(K\) km. We do this by assigning a warehouse to each consumer and imposing the distance constraint for that assigned pair. A general nonlinear distance constraint could look like: \[\color{darkred}a_{i,j}=1 \Rightarrow \sqrt{\sum_c \left( \color{darkblue}p_{i,c} - \color{darkred}w_{j,c}\right)^2} \le \color{darkblue} K\] This could be implemented as a quadratic indicator constraint: \[\color{darkred}a_{i,j}=1 \Rightarrow \sum_c \left( \color{darkblue}p_{i,c} - \color{darkred}w_{j,c}\right)^2 \le \color{darkblue} K^2\] Interestingly, solvers that support indicator constraints, seem to allow only linear constraints on the right. (Why? I don't know) Here we have a quadratic constraint, so we have to reformulate this as a big-M constraint \[ \sum_c \left( \color{darkblue}p_{i,c} - \color{darkred}w_{j,c}\right)^2 \le \color{darkblue}K^2 +\color{darkblue}M_1 (1-\color{darkred}a_{i,j})\] A conservative bound gives us the following value \(\color{darkblue}M_1  = 2\color{darkblue}S^2 - \color{darkblue}K^2\).

This is a convex quadratic constraint, so we can use well-known solvers for this.  

We have two ordering constraints. The first one says: the used warehouses are the first ones. I.e. \(u_j\) will look like \([1,1,1,0,0,0,0]\). This actually helped me in the implementation of the subsequent models. The first \(n\) should be reused in those models. But it also imposes an ordering that can help against a lot of symmetry. To further reduce symmetry we also order the facilities by their \(x\)-coordinate. 

The addition of these symmetry-breaking constraints is not a guaranteed success. We make the search space much smaller, so this is a big advantage. But we also may make finding a feasible solution much more difficult. Before doing some experimentation, it is very hard to predict how these trade-offs will play out.

The solution can look like:

We see we need 3 facilities, but the locations and the assignments are not optimal. We will use a second model to address this.

The distance constraint implements the implication: \[\color{darkred}a_{i,j} = 1 \Rightarrow \color{darkred}\Delta_{i,j} \ge \sum_c \left( \color{darkblue}p_{i,c} - \color{darkred}w_{j,c}\right)^2\] Again, indicator constraints are only for linear constraints, so we need a big-M formulation. A conservative value for the big-M constant is \(\color{darkblue}M_2 = \color{darkblue}S^2\). This will allow \(\color{darkred}\Delta_{i,j}=0\) when \(\color{darkred}a_{i,j} = 0\).

The result can be:

This picture is much better. We see that the location of warehouses and the assignments make much more sense. 

We can look at this as a multi-objective problem where we have the objectives (1) minimize the number of facilities and (2) minimize the sum of (squared) distances. Here we used a lexicographic approach: solve for the first objective and add this as a "constraint" to the second objective problem. In this case, the constraint turns out just a question of dimensioning the second problem.  

Model 3: Minimize the sum of the distances

It looks like converting model 2 from minimizing the sum of squared distances to the sum of distances should be an easy task. However, this is not exactly the case. A first attempt leads to an MINLP model:

The data and the variables are the same as for model 2. The objective makes this problem a general MINLP model which is more difficult to solve and requires different solvers. 

MISOCP Model 3c: min squared distances
Variables	\[\begin{align} &\color{darkred}z_{i,j}\in [0,\color{darkblue}K]&& \text{distance between assigned facility and customer (or zero)} \\ &\color{darkred}d_{i,j}\in [0,\sqrt{\color{darkblue}M_2}]&& \text{distance between any facility and customer}\\ & \color{darkred} y_{i,j,c} && \text{intermediate free variable} \\ &\color{darkred}w_{j,c} \in [0,\color{darkblue}S] && \text{location of facility}\\ &\color{darkred}a_{i,j} \in \{0,1\} && \text{assignment of consumer to facility} \end{align}\]
Equations	\[\begin{align}\min\>&\sum_{i,j} \color{darkred} z_{i,j} \\ & \color{darkred}z_{i,j} \ge \color{darkred}d_{i,j} - \sqrt{\color{darkblue} M_2 }(1-\color{darkred}a_{i,j}) && \forall i,j && \text{distance is zero when not assigned}\\ & \color{darkred}d^2_{i,j} \ge \sum_c \color{darkred}y^2_{i,j,c} && \forall i,j && \text{SOCP constraint}  \\ & \color{darkred}y_{i,j,c} =  \color{darkblue}p_{i,c} - \color{darkred}w_{j,c} && \forall i,j,c && \text{intermediate}\\ & \sum_j \color{darkred}a_{i,j} = 1 && \forall i && \text{assignment}\\ & \color{darkred}w_{j,x} \le \color{darkred}w_{j+1,x}&& && \text{ordering (optional)} \end{align} \]

We needed to add a lot of extra variables and constraints to make this model work for an SOCP solver. 

Model 4: grid search

If we only consider a grid of possible locations for the facilities, things simplify significantly. This now becomes a discrete facility location problem. If we indicate our grid points by \(g\), then we can calculate the distances between customer locations \(\color{darkblue}p_{i,c}\) and possible facility locations \(\color{darkblue}q_{g,c}\) in advance (i.e. outside the optimization model). So these nonlinear expressions are now reduced to just data manipulation: \[\color{darkblue}{\mathit{dist}}_{i,g} = \sqrt{\sum_c \left( \color{darkblue}p_{i,c} - \color{darkblue}q_{g,c}\right)^2}\] This is great, as we can now develop a linear model. We again use two models to capture the multi-objective nature of this problem. 

In the models below we only consider admissible assignments, that is only assignments of customer to warehouse where the distance is less than \(\color{darkblue}K\). We do this by introducing a two dimensional set \(\color{darkblue}{\mathit{ok}}_{i,g}\) with \[\color{darkblue}{\mathit{ok}}_{i,g} = \color{darkblue}{\mathit{dist}}_{i,g} \le \color{darkblue}K\] In the model we only consider \((i,g)\) combinations in the set \(\color{darkblue}{\mathit{ok}}\).

 

MIP Model 4: locations on a grid
Data	\[\begin{align}&g&&\text{grid points}\\ &\color{darkblue}{\mathit{dist}}_{i,g}&&\text{distances}\\ &\color{darkblue}{\mathit{ok}}_{i,g}&&\text{admissible assignments} \end{align}\]
Variables	\[\begin{align}&\color{darkred}{\mathit{place}}_g \in \{0,1\}&&\text{whether to place a facility at $g$}\\ &\color{darkred}{\mathit{ag}}_{i,g} \in \{0,1\}&&\text{assign customer $i$ to $g$} \end{align}\]
Model 4a	\[\begin{align}\min\>&\color{darkred}n\\&\sum_{g|\color{darkblue}{\mathit{ok}}(i,g)} \color{darkred}{\mathit{ag}}_{i,g} = 1 &&\forall i &&\text{assignment} \\ & \color{darkred}{\mathit{ag}}_{i,g} \le \color{darkred}{\mathit{place}}_g &&\forall i,g|\color{darkblue}{\mathit{ok}}(i,g) &&\text{use location}\\ &\color{darkred}n = \sum_g \color{darkred}{\mathit{place}}_g && && \text{number of facilities}\end{align}\]
Model 4b	\[\begin{align} \min & \sum_{i,g|\color{darkblue}{\mathit{ok}}(i,g)} \color{darkblue}{\mathit{dist}}_{i,g} \cdot \color{darkred}{\mathit{ag}}_{i,g} && && \text{sum distances} \\ & \sum_{g|\color{darkblue}{\mathit{ok}}(i,g)} \color{darkred}{\mathit{ag}}_{i,g} = 1 &&\forall i &&\text{assignment} \\ & \color{darkred}{\mathit{ag}}_{i,g} \le \color{darkred}{\mathit{place}}_g &&\forall i,g|\color{darkblue}{\mathit{ok}}(i,g) &&\text{use location}\\ & \sum_g \color{darkred}{\mathit{place}}_g = \color{darkblue}n^* && && \text{fix number of facilities} \end{align} \]

Both models 4a and 4b are linear. I set up a grid for \(x\) and \(y\) coordinates: \(1,3,5,\dots,99\). This results in \(50\times50=2500\) grid points.

The pictures look very much the same.

This fine grid leads to a very large model. The number of binary variables is 46,542 for our data set which is very large. The number of equations is similarly large: 44,094. These numbers depend on the grid and on how many admissible assignments with a distance less than \(K=40\) there are.

Performance

We have produced a number of models here. Let's see how they behave.

Model	Type	Objective	Sum dist^2	Sum dist	Time	Nodes	Iterations
1	MIQCP	min num fac	42,127	1,380	20.27	15,487	490,331
1 + order	MIQCP	min num fac	39,288	1,313	6.375	1,230	28,284
2	MIQCP	min sum dist^2	21,105	937.8	49.23	255,858	976,174
2 + order	MIQCP	min sum dist^2	21,105	937.8	15.31	87,319	328,474
3	MISOCP	min sum dist	21,186	935.1	2,981	5,896,361	66,465,832
3 + order	MISOCP	min sum dist	21,186	935.1	431.9	933,002	14,612,297
4a	MIP	min num fac	38,134	1,273	31.17	0	21
4b	MIP	min sum dist	21,313	944.7	15.94	0	17

A lot to digest here. But what we see is: