Clock problem

From [1]:

The hour, minute and second hands of this clock are all the same length and move smoothly in a circle. The dial contains hour and minute markers, but the numbers are missing. Therefore, it’s impossible to tell which one of the 12 hour markers belongs to the 12. The two hands on the left are positioned exactly on hour markers, and the hand on the right is positioned between a minute and an hour marker. What time does the clock show?

It is possible to solve this without really any math, but, of course, here I try to model this as a mathematical programming model.

diag(x)

When using \({\bf diag}(x)\) in a text, there is always the nagging feeling that there must be a nice way to express this in standard matrix algebra (i.e., some combination of identity matrices, all-ones vectors, and standard matrix multiplications). To remind ourselves, the \({\bf diag}(x)\) function creates a diagonal matrix with \(x_i\) as diagonal elements: \[{\bf diag}(x) = \begin{bmatrix} x_1 & & & \\ & x_2 & & \\ & & \ddots & \\ & & & x_n \end{bmatrix} \] Obviously, my intuition is wrong here, as I have never seen such formulation, and \({\bf diag}(x)\) is used all over the place. I have seen some attempts, but they require exotic notation, and, as a result, don't really improve upon straight use of \({\bf diag}(x)\).

Graph connectivity as constraints

I was generating, for an example model, a random, sparse, directed graph. Unfortunately, when sparse enough, this is likely to yield a graph that is not connected. Here is my first attempt.

Nodes

I generate simply \(n=25\) nodes with \((x,y)\) coordinates from a uniform distribution \(U(0,100)\). This looks like:

\(n=25\) nodes randomly placed

Revisiting a continuous facility location problem

I am revisiting here a problem from [1]:

We have \(n\) demand points and their locations. How many facilities do we need to service these customers? And where do we place them? We have a restriction: there is a maximum distance between customer and facility.   

Data

We randomly generated 75 demand points inside the \([0,1]\times[0,1]\) square. The maximum allowed distance between a demand point and a facility is 0.25. 

Random Demand Points

Nonconvex problem: local vs multistart vs global

In [1] a somewhat abstract non-convex problem is given:

\[\begin{align}\min_x & - x_1^2 - x_2^2 - x_3^2 - x_4^2\\ & Ax \le b \end{align}\]

This is a nonconvex objective with some linear constraints. Of course, the easiest way to gain some insight into how to solve this is to perform some quick and dirty tests with a global QP solver designed for these types of models. As the constraints are linear, a solver like Cplex or Gurobi may be a good starting point (pun intended).

If you have access to a local NLP solver, it may not always be easy to find a good starting point. One possible approach is to use a multistart algorithm (some NLP solvers, like Baron and Knitro, have this built-in). This will not guarantee a global solution (and most likely, it doesn't give you one), but at least we can prevent really bad solutions.  

Towers of Hanoi: inventory and network formulation

The towers of Hanoi problem [1] is a famous puzzle demonstrating recursion. The task is to move a stack of disks from one peg to another. We can only move one disk at a time, and we need to obey the rule that larger disks can never be on top of a smaller disk. The moves for the 3 disk problem are:

Wolf, goat and cabbage problem: MIP and network model

The Wolf, goat, and cabbage problem can be stated as [1]:

A farmer with a wolf, a goat, and a cabbage must cross a river by boat. The boat can carry only the farmer and a single item. If left unattended together, the wolf would eat the goat, or the goat would eat the cabbage. How can they cross the river without anything being eaten?

A long transcontinental flight was a good opportunity to try to attack this problem. Here are some approaches to model this. Not at all a very useful or practical model, but still interesting, I think (although I may be in a small minority here). 

Inventory model

In this model, we keep track of the inventory of items (wolf, goat, cabbage). We assume the starting inventory is: all items are on the left bank of the river. The final inventory should be: all items are on the right bank. In this model, I assume the following numbering scheme:

----     31 SET trip  trips

trip1 ,    trip2 ,    trip3 ,    trip4 ,    trip5 ,    trip6 ,    trip7 ,    trip8 ,    trip9 ,    trip10


----     31 SET dir  direction of trip

L->R,    R->L


----     31 SET tripDir  trip direction combos

              L->R        R->L

trip1          YES
trip2                      YES
trip3          YES
trip4                      YES
trip5          YES
trip6                      YES
trip7          YES
trip8                      YES
trip9          YES
trip10                     YES

Small MIP, proving optimality is difficult

This is a simple, smallish MIP model that is difficult to solve to proven optimality. The problem is stated as [1]:

I have grid of dimensions H and W of boolean variables. The only constraint is that if a variable is false then at least one of the adjacent variables (top, right, left, bottom, diagonals don't count) must be true. The goal is to minimize the number of true values in the grid.

A high-level mathematical representation of this can be:

High-level model
\[\begin{align}\min&\sum_{i,j}\color{darkred}x_{i,j}\\ & \color{darkred}x_{i,j}=0 \implies \color{darkred}x_{i-1,j} + \color{darkred}x_{i+1,j} + \color{darkred}x_{i,j-1} + \color{darkred}x_{i,j+1} \ge 1 \\ & \color{darkred}x_{i,j}\in \{0,1\} \end{align}\]

Programming vs Modeling

I found another interesting GAMS model. Here, I wanted to demonstrate the difference between programming (implementing algorithms in a programming language) and modeling (building models and implementing them in a modeling tool). The problem is solving a shortest path problem given a sparse network with random costs. We compare three strategies:

1. Implement Dijkstra's shortest path algorithm in GAMS,
2. implement Dijkstra's shortest path algorithm in Python and
3. use a simple LP model.  

One of the practical problems is how to represent and display the optimal shortest path in a meaningful and compact way. In some cases, this is more difficult than implementing the pure shortest path algorithm or model. 

Network models with node capacities

Min-cost flow network model

A standard min-cost flow network model, formulated as a linear programming model, can look like:

Model 1: standard min-cost flow network model
\[\begin{align}\min& \sum_{(i,j)\in A}\color{darkblue}c_{i,j}\cdot\color{darkred}f_{i,j}\\ & \sum_{j|(j,i)\in A}\color{darkred}f_{j,i} + \color{darkblue}{\mathit{supply}}_i = \sum_{j|(i,j)\in A}\color{darkred}f_{i,j} + \color{darkblue}{\mathit{demand}}_i && \forall i \\ & \color{darkred}f_{i,j} \in [0,\color{darkblue}{\mathit{capacity}}_{i,j}] \end{align}\]