Algorithm Design Techniques: The Assignment Problem
Pretend for a moment that you are writing software for a famous ride sharing application. You might have to solve the assignment problem.
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One of the interesting things about studying optimization is that the techniques show up in a lot of different areas. The "assignment problem" is one that can be solved using simple techniques, at least for small problem sizes, and is easy to see how it could be applied to the real world.
Pretend for a moment that you are writing software for a famous ride sharing application. In a crowded environment, you might have multiple prospective customers that are requesting service at the same time, and nearby you have multiple drivers that can take them where they need to go. You want to assign the drivers to the customers in a way that minimizes customer wait time (so you keep the customers happy) and driver empty time (so you keep the drivers happy).
The assignment problem is designed for exactly this purpose. We start with
m agents and
n tasks. We make the rule that every agent has to be assigned to a task. For each agent-task pair, we figure out a cost associated to have that agent perform that task. We then figure out which assignment of agents to tasks minimizes the total cost.
Of course, it may be true that
m != n, but that's OK. If there are too many tasks, we can make up a "dummy" agent that is more expensive than any of the others. This will ensure that the least desirable task will be left to the dummy agent, and we can remove that from the solution. Or, if there are too many agents, we can make up a "dummy" task that is free for any agent. This will ensure that the agent with the highest true cost will get the dummy task, and will be idle.
If that last paragraph was a little dense, don't worry; there's an example coming that will help show how it works.
There are special algorithms for solving assignment problems, but one thing that's nice about them is that a general-purpose solver can handle them too. Below is an example, but first it will help to cover a few concepts that we'll be using.
Up above, we talked about making "rules" and minimizing costs. The usual name for this is optimization. An optimization problem is one where we have an "objective function" (which tells us what our goals are) and one or more "constraint functions" (which tell us what the rules are). The classic example is a factory that can make both "widgets" and "gadgets". Each "widget" and "gadget" earns a certain amount of profit, but it also uses up raw material and time on the factory's machines. The optimization problem is to determine exactly how many "widgets" and how many "gadgets" to make to maximize profit (the objective) while fitting within the material and time available (the constraints).
If we were to write this simple optimization problem out, it might look like this:
maximize 45g + 40w // Step 3: Profit! subject to 120g + 100w <= 4000 // Raw material 1 (we have 4000 lbs) 80g + 80w <= 2500 // Raw material 2 (we have 2500 lbs) 3.8g + 3.7w <= 200 // Machine time (200 hours available)
In this case, we have two variables:
g for the number of gadgets we make and
w for the number of widgets we make. We also have three constraints that we have to meet. Note that they are inequalities; we might not use all the available material or time in our optimal solution.
Just to unpack this a little: in English, the above is saying that we make 45 dollars / euros / quatloos per gadget we make. However, to make a gadget we need 120 lbs of raw material 1, 80 lbs of raw material 2, and 3.8 hours of machine time. So there is a limit on how many gadgets we can make, and it might be a better use of resources to balance gadgets with widgets.
Of course, real optimization problems have many more than two variables, and many constraint functions, making them much harder to solve. The easiest kind of optimization problem to solve is linear and, fortunately, the assignment problem is linear.
A linear program is a kind of optimization problem where both the objective function and the constraint functions are linear. (OK, that definition was a little self-referential.) We can have as many variables as we want, and as many constraint functions as we want, but none of the variables can have exponents in any of the functions. This limitation allows us to apply very efficient mathematical approaches to solve the problem, even for very large problems.
We can state the assignment problem as a linear programming problem. First, we choose to make "i" represent each of our agents (drivers) and "j" to represent each of our tasks (customers). Now, to write a problem like this, we need variables. The best approach is to use "indicator" variables, where
xij = 1 means "driver i picks up customer j" and
xij = 0 means "driver i does not pick up customer j".
We wind up with:
minimize sum(i,j) Cij * xij for all i,j subject to sum(j) xij = 1 for all i in A sum(i) xij = 1 for all j in T xij >= 0
This is a compact mathematical way to describe the problem, so again let me put it in English.
First, we need to figure out the cost of having each driver pick up each customer. Then, we can calculate the total cost for any scenario by just adding up the costs for the assignments we pick. For any assignment we don't pick, xij will equal zero, so that term will just drop out of the sum.
Of course, the way we set up the objective function, the cheapest solution is for no drivers to pick up any customers. That's not a very good business model. So we need a constraint to show that we want to have a driver assigned to every customer. At the same time, we can't have a driver assigned to mutiple customers. So we need a constraint for that too. That leads us to the two constraints in the problem. The first just says, if you add up all the assignments for a given driver, you want the total number of assignments for that driver to be exactly one. The second constraint says, if you add up all the assignments to a given customer, you want the total number of drivers assigned to the customer to be one. If you have both of these, then each driver is assigned to exactly one customer, and the customers and drivers are happy. If you do it in a way that minimizes costs, then the business is happy too.
Solving With Octave and GLPK
Given some made-up sample data, the program looks like this:
% Assignment Problem Example % Cost information (to minimize) c = [20 10 12 11 15 14 12 24 18 11 9 5 0 0 0 0]'; % Right-hand side (constraint) b = [1 1 1 1 1 1 1 1]; % Coefficients (for constraints) a = [ 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 ]; lb = [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]; ub = [1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]; ctype = "SSSSSSSS"; vartype = "IIIIIIIIIIIIIIII"; s = 1; [xmin, fmin, status, extra] = glpk(c, a, b, lb, ub, ctype, vartype, s)
Start with the definition of "c", the cost information. For this example, I chose to have four drivers and three customers. There are sixteen numbers there; the first four are the cost of each driver to get the first customer, the next four are for the second customer, and the next four are for the third customer. Because we have an extra driver, we add a "dummy" customer at the end that is zero cost. This represents one of the drivers being idle.
The next definition is "b", the right-hand side of our constraints. There are eight constraints, one for each of the drivers, and one for each of the customers (including the dummy). For each constraint, the right-hand side is 1.
The big block in the middle defines our constraint matrix "a". This is the most challenging part of taking the mathematical definition and putting it into a form that is usable by GLPK; we have to expand out each constraint. Fortunately, in these kinds of cases, we tend to get pretty patterns that help us know we're on the right track.
The first line in "a" says that the first customer needs a driver. To see why, remember that in our cost information, the first four numbers are the cost for each driver to get the first customer. With this constraint, we are requiring that one of those four costs be included and therefore that a driver is "selected" for the first customer. The other lines in "a" work similarly; the last four ensure that each driver has an assignment.
Note that the number of rows in "a" matches the number of items in "b", and the number of columns in "a" matches the number of items in "c". This is important; GLPK won't run if this is not true (and our problem isn't stated right in any case).
Compared to the above, the last few lines are easy.
- "lb" gives the lower bound for each variable.
- "ub" gives the upper bound.
- "ctype" tells GLPK that each constraint is an equality ("strict" as opposed to providing a lower or upper bound).
- "vartype" tells GLPK that these variables are all integers (can't have half a driver showing up).
- "s" tells GLPK that we want to minimize our costs, not maximize them.
We push all that through a function call to GLPK, and what comes back are two values (along with some other stuff I'll exclude for clarity):
fmin = 27 ans = 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0
The first item tells us that our best solution takes 27 minutes, or dollars, or whatever unit we used for cost. The second item tells us the assignments we got. (Note for pedants: I transposed this output to save space.)
This output tells us that customer 1 gets driver 2, customer 2 gets driver 3, customer 3 gets driver 4, and driver 1 is idle. If you look back at the cost data, you can see this makes sense, because driver 1 had some of the most expensive times to the three customers. You can also see that it managed to pick the least expensive pairing for each customer. (Of course, if I had done a better job making up cost data, it might not have picked the least expensive pairing in all cases, because a suboptimal individual pairing might still lead to an overall optimal solution. But this is a toy example.)
Of course, for a real application, we would have to take into consideration many other factors, such as the passage of time. Rather than knowing all of our customers and drivers up front, we would have customers and drivers continually showing up and being assigned. But I hope this simple example has revealed some of the concepts behind optimization and linear programming and the kinds of real-world problems that can be solved.
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