In [1] the following problem is proposed:
Furthermore:
Well, let's see...
There are 106,376 elements in this data structure.
This is a large, but easy MIP model. The size is large:
But it solves very fast:
The solution looks like:
We have really debunked that this is an intractable problem for a good MIP solver.
Note that the age of the universe is 13.7 billion years or 4.3e17 seconds. So we have achieved a speed-up of about 1e17. This is a record for me.
It is noted that this model does not really answer the question: what are the best locations for my stores? It only answers: how many stores do we need? A subsequent optimization model can find good (or optimal) locations for the stores [2].
The question: what is the minimum number of stores needed to service all customers can be answered by formulating and solving a set covering problem. This will give you locations for the stores. However, this model only looks at this objective, so no further attempts are undertaken to minimize the customer-store distances.
Consider a grid with 40,000 cells. We have customers located on this grid (this is data). We want to place stores on the grid such that each customer has at least one store within reach. Finally, we want to minimize the number of stores we need to use.
Furthermore:
so in a situation like mine I could easily end up with a linear system with millions of rows and columns, which with integer/binary variables will take about the age of the universe to solve.
Well, let's see...
Data
The first thing we do is generating some data. We introduce a grid of \(200 \times 200\) with \(40,000\) cells. Then we randomly place \(M=500\) customers on this grid.
---- 22 PARAMETER cloc customer locations
x y
cust1 3575
cust2 16984
cust3 11118
cust4 61163
cust5 59102
cust6 45147
cust7 70165
cust8 17283
cust9 14185
cust10 10179
. . .
cust495 83186
cust496 174159
cust497 196148
cust498 115136
cust499 63101
cust500 9287
Reach
Secondly, we create a large, sparse boolean parameter \(\mathit{reach}_{c,i,j}\) indicating if cell \((i,j)\) is within reach of customer \(c\). I used the rule: if the Manhattan distance between \(c\) and \((i,j)\) is less that 10, we are within reach: \[ |i-\mathit{cloc}_{c,x}|+|j-\mathit{cloc}_{c,y}| \le 10\] This generates a large parameter. A fragment can look like:
---- 29 PARAMETER reach location (i,j) is within reach of customer
j1 j2 j3 j4 j5 j6 j7 j8 j9
cust3 .i110 1
cust3 .i111 11
cust3 .i112 1
cust39 .i116 1
cust39 .i117 111
cust39 .i118 11111
cust39 .i119 111111
cust39 .i120 1111111
cust39 .i121 11111111
There are 106,376 elements in this data structure.
MIP model
With this, we can formulate the set covering model:
| Set covering model |
|---|
| \[\begin{align}\min\>& \color{darkred}{\mathit{numStores}} \\ & \color{darkred}{\mathit{numStores}} = \sum_{i,j} \color{darkred}{\mathit{placeStore}}_{i,j}\\ & \sum_{i,j} \color{darkblue}{\mathit{reach}}_{c,i,j} \cdot \color{darkred}{\mathit{placeStore}}_{i,j} \ge 1 & \forall c \\ &\color{darkred}{\mathit{placeStore}}_{i,j} \in \{0,1\} \end{align}\] |
This is a large, but easy MIP model. The size is large:
MODEL STATISTICS
BLOCKS OF EQUATIONS 2 SINGLE EQUATIONS 501
BLOCKS OF VARIABLES 2 SINGLE VARIABLES 40,001
NON ZERO ELEMENTS 146,377 DISCRETE VARIABLES 40,000
But it solves very fast:
Found incumbent of value 495.000000 after 0.00 sec. (1.89 ticks)
Tried aggregator 8 times.
MIP Presolve eliminated 306 rows and 39764 columns.
MIP Presolve modified 6 coefficients.
Aggregator did 29 substitutions.
Reduced MIP has 166 rows, 208 columns, and 658 nonzeros.
Reduced MIP has 208 binaries, 0 generals, 0 SOSs, and 0 indicators.
Presolve time = 0.36 sec. (72.59 ticks)
Probing time = 0.00 sec. (0.05 ticks)
Tried aggregator 1 time.
Detecting symmetries...
MIP Presolve eliminated 1 rows and 5 columns.
Reduced MIP has 165 rows, 203 columns, and 644 nonzeros.
Reduced MIP has 203 binaries, 0 generals, 0 SOSs, and 0 indicators.
Presolve time = 0.00 sec. (0.42 ticks)
Probing time = 0.00 sec. (0.05 ticks)
Clique table members: 35.
MIP emphasis: balance optimality and feasibility.
MIP search method: dynamic search.
Parallel mode: deterministic, using up to 8 threads.
Parallel mode: deterministic, using up to 2 threads for parallel tasks at root LP.
Tried aggregator 1 time.
No LP presolve or aggregator reductions.
Presolve time = 0.00 sec. (0.09 ticks)
Iteration log . . .
Iteration: 1 Dual objective = 57.000000
Iteration: 62 Dual objective = 93.000000
Iteration: 124 Dual objective = 109.000000
Iteration: 186 Dual objective = 111.500000
Root relaxation solution time = 0.06 sec. (0.94 ticks)
Nodes Cuts/
Node Left Objective IInf Best Integer Best Bound ItCnt Gap
* 0+ 0495.000056.000088.69%
Found incumbent of value 495.000000 after 0.44 sec. (77.77 ticks)
* 0+ 0 118.0000 56.0000 52.54%
Found incumbent of value 118.000000 after 0.44 sec. (78.15 ticks)
00111.777878118.0000111.77782035.27%
* 0+ 0 116.0000 111.7778 3.64%
Found incumbent of value 116.000000 after 0.45 sec. (79.32 ticks)
00112.500078116.0000 Cuts: 102093.02%
* 0+ 0 113.0000 112.5000 0.44%
Found incumbent of value 113.000000 after 0.52 sec. (85.02 ticks)
00 cutoff 113.0000112.50002090.44%
Elapsed time = 0.55 sec. (85.03 ticks, tree = 0.01 MB, solutions = 3)
Zero-half cuts applied: 5
Lift and project cuts applied: 2
Gomory fractional cuts applied: 3
Root node processing (before b&c):
Real time = 0.55 sec. (86.90 ticks)
Parallel b&c, 8 threads:
Real time = 0.00 sec. (0.00 ticks)
Sync time (average) = 0.00 sec.
Wait time (average) = 0.00 sec.
------------
Total (root+branch&cut) = 0.55 sec. (86.90 ticks)
MIP status(101): integer optimal solution
Solution
The solution looks like:
---- 51 VARIABLE numStores.L = 113 number of stores
---- 51 VARIABLE placeStore.L store locations
j1 j6 j7 j8 j9 j15 j16 j17 j18
i4 1
i18 1
i40 1
i70 1
i79 1
i80 1
i107 1
i118 1
i136 1
i157 1
i167 1
i193 1
+ j21 j23 j26 j28 j29 j31 j32 j36 j38
i10 1
i28 1
i54 1
i72 1
i96 1
i113 1
i147 1
i158 1
i179 1
i184 1
i198 1
+ j39 j44 j45 j46 j49 j50 j56 j58 j59
i5 1
i18 1
i39 1
i62 1
i85 1
i102 1
i104 1
i133 1
i166 1
i195 1
+ j62 j66 j67 j68 j69 j73 j74 j76 j80
i11 1
i16 1
i36 1
i61 1
i76 1
i105 1
i112 1
i117 1
i128 1
i146 1
i190 1
+ j82 j84 j85 j88 j90 j92 j95 j96 j97
i17 1
i26 1
i35 1
i48 1
i68 1
i79 1
i97 1
i136 1
i156 1
i170 1
i183 1
i191 1
+ j98 j102 j107 j111 j112 j114 j115 j116 j118
i4 1
i22 1
i36 1
i56 1
i63 1
i68 1
i88 1
i100 1
i101 1
i111 1
i129 1
i140 1
+ j119 j121 j126 j127 j132 j133 j134 j136 j139
i11 1
i30 1
i53 1
i72 1
i111 1
i129 1
i144 1
i159 1
i183 1
i191 1
+ j140 j147 j149 j150 j152 j153 j154 j156 j158
i14 1
i35 1
i48 1
i83 1
i98 1
i117 1
i158 1
i174 1
i194 1
+ j161 j162 j163 j164 j166 j170 j172 j174 j175
i5 1
i32 1
i42 1
i61 1
i69 1
i103 1
i143 1
i145 1
i158 1
i192 1
i198 1
+ j176 j178 j179 j180 j182 j183 j184 j188 j191
i6 1
i13 1
i23 1
i47 1
i61 1
i81 1
i93 1
i103 1
i125 1
i182 1
i193 1
+ j192 j193 j196
i73 1
i120 1
i138 1
i167 1
We have really debunked that this is an intractable problem for a good MIP solver.
Note that the age of the universe is 13.7 billion years or 4.3e17 seconds. So we have achieved a speed-up of about 1e17. This is a record for me.
It is noted that this model does not really answer the question: what are the best locations for my stores? It only answers: how many stores do we need? A subsequent optimization model can find good (or optimal) locations for the stores [2].
Conclusion
We can solve this problem for this particular problem quite efficiently using a standard MIP solver. Quite often I see that these solution approaches are dismissed, due to a misunderstanding of complexity-theoretic arguments. Complexity theory says very little about the performance of a particular MIP model using a particular solver on a given data set. It is better not to reject a MIP approach even before trying.
References
- Minimize number of shops while reaching all customers, https://stackoverflow.com/questions/62224042/minimize-number-of-shops-while-reaching-all-customers
- Solving a facility location problem as an MIQP, https://yetanothermathprogrammingconsultant.blogspot.com/2018/01/solving-facility-location-problem-as.html (this is essentially a continuous version of the current problem)