A Greedy Heuristic Wins

 A problem posted on OR Stack Exchange starts as follows: "I need to find two distinct values to allocate, and how to allocate them in a network of stores." There are $n$ stores (where, according to the poster, $n$ can be close to 1,000). The two values (let's call them $x_1$ and $x_2$) must be integer, with $x_1 \in \lbrace 1, \dots, k_1 \rbrace$ and $x_2 \in \lbrace k_1, \dots, k_2 \rbrace$ for given parameters $k_1 < k_2$. Additionally, there is an additional set of parameters $s_{i3}$ and a balance constraint saying $$0.95 g(k_1 e) \le g(x_1, x_2) \le 1.05 g(k_1 e)$$ where $$g(y) = \sum_{i=1} \frac{s_{i3}}{y_i}$$ for any allocation $y$ and $e = (1,\dots, 1).$

The cost function (to be minimized) has the form $$f(x_1, x_2) = a\sum_{i=1}^n \left[ s_{i1}\cdot \left( \frac{s_{i2}}{y_i} \right)^b \right]$$with $a$, $s_{i1}$, $s_{i2}$ and $b$ all parameters and $y_i \in \lbrace x_1, x_2 \rbrace$ is the allocation to store $i$. There are two things to note about $f$. First, the leading coefficient $a (> 0)$ can be ignored when looking for an optimum. Second, given choices $x_1$ and $x_2>x_1$, the cheaper choice at all stores will be $x_1$ if $b < 0$ and $x_2$ if $b > 0$.

It's possible that a nonlinear solver might handle this, but I jumped straight to metaheuristics and, in particular, my go-to choice among metaheuristics -- a genetic algorithm. Originally, genetic algorithms were intended for unconstrained problems, and were tricky to use with constrained problems. (You could bake a penalty for constraint violations into the fitness function, or just reject offspring that violated any constraints, but neither of those approaches was entirely satisfactory.) Then came a breakthrough, the random key genetic algorithm [1]. A random key GA uses a numeric vector $v$ (perhaps integer, perhaps byte, perhaps double precision) as the "chromosome". The user is required to supply a function that translates any such chromosome into a feasible solution to the original problem.

I did some experiments in R, using the GA package to implement a random key genetic algorithm. The package requires all "genes" (think "variables") to be the same type, so I used a double-precision vector of dimension $n_2$ for chromosomes. The last two genes have domains $(1, k_1 + 1)$ and $(k_1, k_2 + 1)$; the rest have domain $(0, 1)$. Decoding a chromosome $v$ proceeds as follows. First, $x_1 = \left\lfloor v_{n+1}\right\rfloor $ and $x_2 = \left\lfloor v_{n+2}\right\rfloor $, where $\left\lfloor z \right\rfloor$ denotes the "floor" (greatest lower integer) of $z$. The remaining values $v_1, \dots, v_{n}$ are sorted into ascending order, and their sort order is applied to the stores. So, for instance, if $v_7$ is the smallest of those genes and $v_{36}$ is the largest, then store $7$ will be first in the sorted list of stores and store $36$ will be last. (The significance of this sorting will come out in a second.)

 

Armed with this, my decoder initially assigns every store the cheaper choice between $x_1$ and $x_2$ and computes the value of $g()$. If $g()$ does not fall within the given limits, the decoder runs through the stores in their sorted order, switching the allocation to the more expensive choice and updating $g()$, until $g()$ meets the balance constraint. As soon as it does, we have the decoded solution. This cheats a little on the supposed guarantee of feasibility in a decoded solution, since there is a (small?) (nearly zero?) chance that the decoding process will fail with $g()$ jumping from below the lower bound to above the upper bound (or vice versa) after some swap. If it does, my code discards the solution. This did not seem to happen in my testing.

 

The GA seemed to work okay, but it occurred to me that I might be over-engineering the solution a bit. (This would not be the first time I did that.) So I also tried a simple greedy heuristic. Since $k_1$ and $k_2$ seem likely to be relatively small in the original poster's problem (whereas $n$ is not), my greedy heuristic loops through all valid combinations of $x_1$ and $x_2$. For each combination, it sets $v_1$ equal to the cheaper choice and $v_2$ equal to the more expensive choice, assigns the cheaper quantity $v_1$ to every store and computes $g()$. It also computes, for each store, the ratio \[ \frac{|\frac{s_{i3}}{v_{2}}-\frac{s_{i3}}{v_{1}}|}{s_{i1}\left(\left(\frac{s_{i2}}{v_{2}}\right)^{b}-\left(\frac{s_{i1}}{v_{1}}\right)^{b}\right)} \]in which the numerator is the absolute change in balance at store $i$ when switching from the cheaper allocation $v_1$ to the more expensive allocation $v_2$, and the denominator is the corresponding change in cost. The heuristic uses these ratios to select stores in descending "bang for the buck" order, switching each store to the more expensive allocation until the balance constraint is met.

Both the GA decoder and the greedy heuristic share the approach of initially allocating every store the cheaper choice and then switching stores to the more expensive choice until balance is attained. My R notebook generates a random problem instance with $n=1,000$ and then solves it twice, first with the GA and then with the greedy heuristic. The greedy heuristic stops when all combinations of $x_1$ and $x_2$ have been tried. Stopping criteria for the GA are more arbitrary. I limited it to at most 1,000 generations (with a population of size 100) or 20 consecutive generations with no improvement, whichever came first.

 

The results on a typical instance were as follows. The GA ran for 49 seconds and got a solution with cost 1065.945. The greedy heuristic needed only 0.176 seconds to get a solution with cost 1051.735. This pattern (greedy heuristic getting a better solution in much less time) repeated across a range of random number seeds and input parameters, including switching between positive and negative values of $b$.

If you are interested, you can browse my R notebook (which includes both code and results).

 

[1] Bean, J. C. (1994). Genetic Algorithms and Random Keys for Sequencing and Optimization. ORSA Journal on Computing, 6, 154-160.