Statistical Consultants Ltd


Operations Research Services

Operations research (also known as management science) is an interdisciplinary subject which combines statistics, mathematics and computational techniques to solve a special class of problems which are common in business.

Constrained Optimisation

Constrained optimisation problems involve making a set of choices which maximise or minimise some quantity, subject to a set of constraints which must be adhered to.  Applications include (but are not limited to):
  • Cost minimisation e.g. Choosing a combination of inputs, which minimises total cost subject to satisfying all product orders.
  • Marketing campaign optimisation e.g. Maximising the net returns from an advertising campaign, subject to an advertising budget.
  • Resource management e.g. Choosing a combination of outputs, which minimises the total amount of wastage of a valuable input resource. 
  • Labour scheduling e.g. Assigning hours to part time workers, such that each worker has a shift which is neither too short nor too long, and is a shift that the worker would be available for.
There are a variety of techniques used for solving constrained optimisation problems including linear programming, nonlinear programming, and simulations. 

Linear Programming

Many constrained optimisation problems can be solved via linear programming.  A linear programming problem would have all of the following features:
  1. The problem’s objective seeks to maximise or minimise some quantity.
  2. Solving the problem would involve making a set of choices.
  3. The set of possible choices would be limited by a set of constraints.
  4. Both the objective and the constraints could be expressed as linear equations or linear inequalities of the set of available choices.
A simple linear program expressed mathematically  A simple linear program displayed graphically.

Queue Systems

Queue process animated gif

Queuing processes (also known as wait-line processes) occur in a variety of situations including:
  • Customers queuing at a shop checkout
  • Parts in line to be assembled at a factory
  • Phone callers waiting for service from a call-centre
  • Patients awaiting treatment at a hospital
  • Cars waiting at traffic lights, or on a congested motorway
Queuing systems can often be modelled either mathematically or through simulations, by taking into account:
  1. The arrival processes e.g. the distribution of times between arrivals.
  2. The queue(s) e.g. the carrying capacity of the queue(s), whether people leave the queue(s), whether there is a single queue or multiple queues etc.
  3. The service processes e.g. the distribution of service times, whether the service operators take a first-in-first-out (FIFO) approach, whether there are multiple service operators (and whether those operators are alternatives or prerequisites to other operators).

The following information can be found by modelling / simulating queue systems:
  • The average waiting time (or distribution of waiting times) for customers or objects in the queue
  • The average queue length (or distribution of queue lengths)
  • The average time (or distribution of times) a customer or object spends in the system (waiting time + service time)
  • The average number (or distribution) of customers or objects in the system
  • The percentage of time a service facility is idle
  • How the service times compare to the arrival rates

Queue processing diagram

Arrival times distribution  Service times distribution

Statistical Tools for Project Management

Network diagram for project management

Statistical techniques have been devised which can help managers plan, schedule, monitor and control large complex projects.  These techniques include the program evaluation and review technique (PERT) and the critical path method (CPM).  The statistical techniques involve estimating the time taken for each activity in the project, and then taking into account which activities are prerequisites of others.  This would allow for the following information to be obtained:
  • The expected total length of time for the project to be completed
  • The expected remaining time for the project (given the completion of particular activities)
  • Which activities are critical i.e. which activities would delay the completion of the project if they run late
  • Which activities are non-critical i.e. which activities can be completed late without delaying the entire project 
  • Whether the project is on, behind or ahead of schedule at a particular date

Sales Forecasting for Inventory Management

forecast graph

Stock shortages are bad, not just in the short term due to a lost profit opportunity, it can also detract customers in the long term.  Over stocking a product can also be bad, if it means less room available for products that are in higher demand.  Sales forecasts, in terms of quantity, can allow for more effective inventory management, where shortages and large surpluses are far less likely.

See the Forecasting Services page for more information.

Statistical Tools for Quality Control

Statistical tools such as acceptance sampling schemes and quality control charts, can be used to assist in the monitoring of product quality.

During a production process, it is often a good idea to check that products are conforming to particular qualities.  For example:
  • Each bottle of drink contains 20% fruit juice.
  • The weight of each machine part is 450 grams.
  • The length of each reel of cable is 12 metres.
  • The lifespan of a battery is three days.

In most situations, quality perfection is neither practical nor possible.  In such cases, a quality characteristic may follow a statistical process, where when controlled in a desirable way:
  • On average the characteristic would equal the desired level of quality.
  • The variation of the characteristic would be low enough, not to have a significantly negative impact on the level of quality.

The application of statistical techniques to quality control, is known as statistical process control.  These techniques include (but are not limited to) acceptance sampling schemes and quality control charts.

Acceptance Sampling Schemes

Sequential sampling scheme graph for quality control

Acceptance sampling involves sampling a batch of products for defects, and then accepting the batch if the rate of defects is small enough, or rejecting the batch if the rate is too high.  Some sampling schemes would have a range of rates deemed inconclusive, with an inconclusive sample being followed by further sampling.

Control Charts

quality control chart

Control charts are graphs with a quantified quality characteristic on the y-axis, and time on the x-axis.  The charts also have a centre line, and upper and lower control limits.  The charts are used to check that a characteristic is staying between the upper and lower control limits (i.e. the desired range of values), and isn’t trending away from the centre line.


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