Operational Research – Matt Randall

Data on the Lake Poster – Slot Offering for Online Shopping

Matt Randall — Mon, 01 Aug 2022 11:12:44 +0000

In April I was fortunate enough to get to go to the lake district for an event hosted by STOR-i and the from the Republic of Ireland. At this event I got the opportunity to present a poster, this can be found below. The poster is about my PhD project on slot offering for the , and the work I had done up to that point on it.

The results here are interesting, and since the presentation of this poster the reason for this phenomenon has been identified. The heuristic used is biased towards using vehicles which are already in use rather than calling new ones into use, this results in inefficient routes which make it more difficult to insert customers who arrive later in the ordering period. For more information stay tuned for the paper I recently submitted.

Heuristic Search Part 3: Selective Search Methods

Matt Randall — Mon, 06 Apr 2020 10:00:00 +0000

Recently, I published blog post introducing the concept of heuristic search, this can be found here. Which was followed up with a discussion linear space search methods, in particular branch and bound, which can be found here. This blog is a continuation of these, in which some selective search methods will be discussed.

In order to solve very complicated combinatorial optimisation problems, in particular ones in which it is not possible to guarantee the quality of solutions found, can be used. All search methods considered up to this point have been based around a systematic search of the solution space, and the methods involving heuristics increased the speed of the search process in order to reach the goal more quickly. As they all take the complete search space into account, they should all find the optimal solution. In addition to this, the previously addressed approaches have been deterministic, they will always give the same result every time they are performs on the same problem. Search methods discussed here have elements of randomness involved in them. Randomised heuristic search methods can be catagorised as either a Las Vegas algorithm, which always yields the optimal solution, however takes a variable amount of time, and Monte Carlo methods, which only sometimes yield the optimal solution.

Selective search is a generic term, which includes attributed from both local and randomised search. Selective search methods as said to be satisficing, meaning that they may not always output the optimal solution, although they may by chance, yet should still yield a very good when implemented correctly

Hill Climbing

is an example of a local search method. is applied widely in combinatorial optimisation. The aim of local search methods is the optimisation of an objective function in a local neighbourhood of states in a solution space, with the aim of finding a solution which is approximately globally optimal. This type of algorithm does not search the entire solution space, hence they aim to find a solution which is more optimal than the other states in its neighbourhood. Selective search methods can be used to modify the size of steps in the solution space, in order to tune the process and alter the size of the neighbourhood being explored at the cost of optimality.

Hill climbing first selects uses an evaluation function in order to select a neighbourhood in which the optimal solution is likely to be located, and searches the solution space here. Hill climbing (for maximisation) and gradient descent (the equivalent for minimisation) aims to make changes which improve the objective function until it is no longer possible to improve it. This is often done in one of two ways, the first is that all neighbours to the current solution are checked, and the one which offers the greatest improvement is chosen, this is known as simple hill climbing. The other by proposing a change to the solution at random, and accepting the change if it results in the objective function improving, and rejecting the change if it does not, this is known as stochastic hill climbing.

There are issues associated with hill climb methods. First is the issue that some proposed changes may result in an infeasible solution that violates the constraints of problem, which is prevalent when the solution space has regions which are both feasible and infeasible regions as seen in figure (a) below (where F represents the disconnected feasible region and I represents the infeasible region). Second is the obvious issue that the local optimum may not be globally optimal and may in fact be very far way, such as for the solution space shown with the objective function in figure (b) below (where A is the global maximum but B is a local maximum). This places importance on choosing where the hill climb starts, however there is often not much information available about the locations of local maxima, so a random start point is often chosen.

Simulated Annealing

is a search method based on an analogy of the process of annealing in , it can be viewed as a development of a stochastic hill climb. The process of annealing involves the heating a material above a certain threshold temperature, maintaining this temperature for some time, then cooling it in order to alter the properties of the material.

This algorithm is effectively a form of . The algorithm generates a new state by perturbing the current state using a small, random change to move the sate of the solution in the solution space. In metallurgy this algorithm is used to simulate the cooling part of the annealing process, here the objective function, represents the total energy of the material. If the energy of the material in the proposed state is less than that of the current state, then the move is accepted. If it is not, then the move is accepted with a probability according to a . The reason for this is based on the physics of the annealing process. In statistical thermodynamics energy changes at a certain temperature are distributed according to a Boltzmann distribution. Over time, the temperature is decreased, to simulate the material cooling. A high energy material will make bigger jumps in its state space, whereas a cooler one will make smaller jumps, hence the gradual cooling will allow a coarse energy minimisation to start with (as a lot of unfavourable moves will be accepted) as the algorithm effectively selects a neighbourhood of states and the the minimisation gradually becomes more and more local. This allows the material to cool to its minimum energy state whilst attempting to avoid energy traps (local minima in its ). It should be noted however that slower cooling will result in an increased computational cost.

This translates to a general combinatorial optimisation problem by considering an general objective function which is to be minimised, and a tuning parameter (in place of temperature) which is gradually reduced. Application of this method allows for a coarse optimisation at the start of the problem, effectively choosing the neighbourhood to search in, which tends towards being a more and more local search as time goes on. In theory this should avoid local minima, such as the one shown in figure (b) above, however due to the randomness in the method it still may become stuck in a local minimum, but this is considerably less likely than with a stochastic hill climb.

Heuristic Search Part 2: Linear Search Methods

Matt Randall — Thu, 02 Apr 2020 12:46:22 +0000

Recently, I published blog post introducing the concept of heuristic search, this can be found here. This is a continuation of that post, in which I will discuss linear space search methods, in particular branch and bound.

involve the exploration of a search tree in a systematic exploration of the solution space for a problem. Often the search tree analysed is considerably larger than the problem graph for the problem itself. These algorithms often consider the search tree nodes as members in a search space. Search trees are designed to be simple to analyse in comparison to their underlying problem graphs due to each node having a unique path between it and the root. This
means that members of the solution space in the search space are paths.

Branch and bound

is a method often used in operational research in order to find solutions to complicated . Here, the set of all possible solutions are represented as a tree with the complete set of all possible solutions at the root. Branching refers to the creation of sub-problems, and bounding refers to the dismissal of partial solutions which are worse than the currently found optimal solution. Hence, in order to achieve this, the upper and lower bounds for optimal solutions, U and L, must be calculated at each branch on the tree. In order to apply the branch and bound method to problems with a general solution space, depth first search is used with U and L applied at each stage. Here, the principle of depth first search can be applied, thus creating a branch and bound search tree.

Consider an example where a problem has a solution space representing all possible configurations of a system, and the aim is to find the configuration which minimises some objective function. The branch and bound algorithm progresses by iterating the
following steps at each of a set of predetermined branching points.

Firstly, the problem is branched using a branching rule specific to the problem in order to produce two or more (usually disjoint) subsets of the solution space.
Second, a heuristic is used to estimate a lower bound on the value of the objective function for any candidate solution contained in the first branch from the branching point, which is at that point the lowest value for L found. The algorithm then does the same for each of set of solutions for each branch sequentially and determines whether or not its lower bound is greater than the current lowest found lower bound for all branches which have already been checked. If this is the case, then the that branch is pruned and that set of possible solutions discarded.

This results in only a single branch remaining, on which the process can now be repeated so as to repeatedly branch the set of possible solutions until the optimal solution has been found. An advantage of this method is the ability to keep track of the upper bound as well as the lower bound and stop branching when the gap between lower and upper bounds reaches a certain threshold and so solutions can be considered good enough.

Heuristic Search Part 1: Introduction and Basic Search Methods

Matt Randall — Wed, 01 Apr 2020 14:14:37 +0000

Heuristic search

�� in the most common sense is a method involving adapting the approach to a problem based on previous solutions to similar problems. These approaches aim to be easily and quickly applicable to a range of problems, so as to find approximate solutions quickly without using the time and resources to develop and execute a precise approach. The principle of a heuristic can be to various problems in mathematics, science and optimsation by applying heuristics computationally.

Heuristic search is class of method which is used in order to search a solution space for an optimal solution for a problem. The heuristic here uses some method to search the solution space while assessing where in the space the solution is most likely to be and focusing the search on that area. It is often possible to model the process of solving a problem as a search through a solution space starting from an initial possible solution, with a method or set of rules dictating how to move from one possible solution to another. This method or rule set must be applied repeatedly in order to eventually satisfy some goal condition which indicates that a solution has been found. Often, this means assessing which path through the solution space improves the solution the most at each iteration of the applied method and choosing this. For more information, see .

Searching solution spaces has been integral for the development of . Since then, the use of heuristic search algorithms has expanded considerably to include a variety of applications, such as , and .

A selection of heuristic search methods will be outlined in future posts, while a brief overview of some basic search methods can be seen below.

Basic search methods

Here, some basic introductory search methods which do not use heuristics will be discussed, based on content from book. In general, the exploration of a discrete solution space can be visualised as searching a graph with each vertex representing a possible solution, and an edge representing that possible solutions are adjacent to each other. For an illustration, see the figure below. Here, a simple problem is presented, in which a person is trying to search for treasure, which is at node H, starting from node A, in as few moves as possible, as shown in figure (a). Here, the problem is considered to have been solved once node H has been searched.

At times there is no graph representation available for the problem at the start of the process of finding a solution. In this case, whilst search is performed, a partial picture of the graph is gradually built up from nodes which have been explored. Here, for every iteration, all nodes adjacent to the one being explored according to set of transition rules are added to the graph (along with edges connecting them to the node being explored), this is called expanding the node. Practically, this is the application of all possible actions to that state. When doing this, it is important to keep track of nodes which have already been generated in the search. Here, every node must be explored at least once in order for it to be present. As a result of this, it is possible to sort the set of all nodes which have been reached into one set of nodes which have been expanded, and another set which have been added to the graph and not yet expanded. The rest of these is known as the open list or search frontier, and the second set is known as the closed list. The set of all paths from the node at which search began up to the set of open nodes gives the the search tree of problem, which gives an illustration of the part of the solution space which has been explored by the search algorithm at a given time. Possible search roots for up to three moves on the graph in figure (a) above are shown on the search tree for the solution of the problem in figure (b). There are multiple different methods for performing searches on graphs, one distinction between methods is the order in which the solution space is searched, this is commonly either breadth first or depth first which shall be discussed further here.

When performing breadth first search, the open set is considered to be a first in first out queue. So when a node is added to the open set, it is added to the bottom of the list. When a node is explored, the node at the top of the list is expanded and moved from the open set to the closed set. The result of this is that nodes are explored layer by layer and the graph is expanded from the start node. An illustration of this can be seen in the table to the left, which shows how the open and closed sets of nodes change as breadth first search is performed on the problem with the graph shown in figure (a).

When performing depth first search, the open set is considered to be a last in first out queue. This means that when a node is added to the open set, it is added to the top of the list. It also means that when a node is explored, the node at the top of the list is expanded and moved from the open set to the closed set. The result of this is that every iteration generates a node connected to the last node expanded (unless there are none, in this situation the algorithm backtracks to the last node with an unexplored node adjacent to it). This means the search effectively explores a path until that path has been completely explored, then goes back to the point at which that path branches off and explores other paths from here, repeating this process until all paths have been explored. An illustration of this can be seen in the table to the right, which shows how the open and closed sets of nodes change as depth first search is performed on the problem with the graph shown in figure (a).