J E Beasley

OR-Notes are a series of introductory notes on topics that fall under the broad heading of the field of operations research (OR). They were originally used by me in an introductory OR course I give at Imperial College. They are now available for use by any students and teachers interested in OR subject to the following conditions.

A full list of the topics available in OR-Notes can be found here.

Introduction to OR


The British/Europeans refer to "operational research", the Americans to "operations research" - but both are often shortened to just "OR" (which is the term we will use).

Another term which is used for this field is "management science" ("MS"). The Americans sometimes combine the terms OR and MS together and say "OR/MS" or "ORMS". Yet other terms sometimes used are "industrial engineering" ("IE") and "decision science" ("DS"). In recent years there has been a move towards a standardisation upon a single term for the field, namely the term "OR".


OR is a new field which started in the late 1930's and has grown and expanded tremendously in the last 30 years (and is still expanding). As such the academic journals contain many useful articles that reflect state of the art applications of OR. We give below a selection of the major OR journals.

  1. Operations Research
  2. Management Science
  3. European Journal of Operational Research
  4. Journal of the Operational Research Society
  5. Mathematical Programming
  6. Networks
  7. Naval Research Logistics
  8. Interfaces

The first seven of the above are mainly theoretical whilst the eighth (Interfaces) concentrates upon case studies. If you have access to an appropriate library that stocks these journals have a browse through them to see what is happening in state of the art OR.

Note here that my personal view is that in OR, as in many fields, the USA is the country that leads the world both in the practical application of OR and in advancing the theory (for example, the American OR conferences have approximately 2500 participants, the UK OR conference has 300).

One thing I would like to emphasise in relation to OR is that it is (in my view) a subject/discipline that has much to offer in making a real difference in the real world. OR can help you to make better decisions and it is clear that there are many, many people and companies out there in the real world that need to make better decisions. I have tried to include throughout OR-Notes discussion of some of the real-world problems that I have personally been involved with.

History of OR

OR is a relatively new discipline. Whereas 70 years ago it would have been possible to study mathematics, physics or engineering (for example) at university it would not have been possible to study OR, indeed the term OR did not exist then. It was really only in the late 1930's that operational research began in a systematic fashion, and it started in the UK. As such I thought it would be interesting to give a short history of OR and to consider some of the problems faced (and overcome) by early OR workers.

Whilst researching for this short history I discovered that history is not clear cut, different people have different views of the same event. In addition many of the participants in the events described below are now elderly/dead. As such what is given below is only my understanding of what actually happened.

Note: some of you may have moral qualms about discussing what are, at root, more effective ways to kill people. However I cannot change history and what is presented below is essentially what happened, whether one likes it or not.


Early in 1936 the British Air Ministry established Bawdsey Research Station, on the east coast, near Felixstowe, Suffolk, as the centre where all pre-war radar experiments for both the Air Force and the Army would be carried out. Experimental radar equipment was brought up to a high state of reliability and ranges of over 100 miles on aircraft were obtained.

It was also in 1936 that Royal Air Force (RAF) Fighter Command, charged specifically with the air defense of Britain, was first created. It lacked however any effective fighter aircraft - no Hurricanes or Spitfires had come into service - and no radar data was yet fed into its very elementary warning and control system.

It had become clear that radar would create a whole new series of problems in fighter direction and control so in late 1936 some experiments started at Biggin Hill in Kent into the effective use of such data. This early work, attempting to integrate radar data with ground based observer data for fighter interception, was the start of OR.


The first of three major pre-war air-defence exercises was carried out in the summer of 1937. The experimental radar station at Bawdsey Research Station was brought into operation and the information derived from it was fed into the general air-defense warning and control system. From the early warning point of view this exercise was encouraging, but the tracking information obtained from radar, after filtering and transmission through the control and display network, was not very satisfactory.


In July 1938 a second major air-defense exercise was carried out. Four additional radar stations had been installed along the coast and it was hoped that Britain now had an aircraft location and control system greatly improved both in coverage and effectiveness. Not so! The exercise revealed, rather, that a new and serious problem had arisen. This was the need to coordinate and correlate the additional, and often conflicting, information received from the additional radar stations. With the outbreak of war apparently imminent, it was obvious that something new - drastic if necessary - had to be attempted. Some new approach was needed.

Accordingly, on the termination of the exercise, the Superintendent of Bawdsey Research Station, A.P. Rowe, announced that although the exercise had again demonstrated the technical feasibility of the radar system for detecting aircraft, its operational achievements still fell far short of requirements. He therefore proposed that a crash program of research into the operational - as opposed to the technical - aspects of the system should begin immediately. The term "operational research" [RESEARCH into (military) OPERATIONS] was coined as a suitable description of this new branch of applied science. The first team was selected from amongst the scientists of the radar research group the same day.


In the summer of 1939 Britain held what was to be its last pre-war air defence exercise. It involved some 33,000 men, 1,300 aircraft, 110 antiaircraft guns, 700 searchlights, and 100 barrage balloons. This exercise showed a great improvement in the operation of the air defence warning and control system. The contribution made by the OR team was so apparent that the Air Officer Commander-in-Chief RAF Fighter Command (Air Chief Marshal Sir Hugh Dowding) requested that, on the outbreak of war, they should be attached to his headquarters at Stanmore in north London.

Initially, they were designated the "Stanmore Research Section". In 1941 they were redesignated the "Operational Research Section" when the term was formalised and officially accepted, and similar sections set up at other RAF commands.


On May 15th 1940, with German forces advancing rapidly in France, Stanmore Research Section was asked to analyse a French request for ten additional fighter squadrons (12 aircraft a squadron - so 120 aircraft in all) when losses were running at some three squadrons every two days (i.e. 36 aircraft every 2 days). They prepared graphs for Winston Churchill (the British Prime Minister of the time), based upon a study of current daily losses and replacement rates, indicating how rapidly such a move would deplete fighter strength. No aircraft were sent and most of those currently in France were recalled.

This is held by some to be the most strategic contribution to the course of the war made by OR (as the aircraft and pilots saved were consequently available for the successful air defense of Britain, the Battle of Britain).

1941 onward

In 1941 an Operational Research Section (ORS) was established in Coastal Command which was to carry out some of the most well-known OR work in World War II.

The responsibility of Coastal Command was, to a large extent, the flying of long-range sorties by single aircraft with the object of sighting and attacking surfaced U-boats (German submarines). The technology of the time meant that (unlike modern day submarines) surfacing was necessary to recharge batteries, vent the boat of fumes and recharge air tanks. Moreover U-boats were much faster on the surface than underwater as well as being less easily detected by sonar.

Amongst the problems that ORS considered were:

Here the problem was that in a squadron each aircraft, in a cycle of approximately 350 flying hours, required in terms of routine maintenance 7 minor inspections (lasting 2 to 5 days each) and a major inspection (lasting 14 days). How then was flying and maintenance to be organised to make best use of squadron resources?

ORS decided that the current procedure, whereby an aircrew had their own aircraft, and that aircraft was serviced by a devoted ground crew, was inefficient (as it meant that when the aircraft was out of action the aircrew were also inactive). They proposed a central garage system whereby aircraft were sent for maintenance when required and each aircrew drew a (different) aircraft when required.

The advantage of this system is plainly that flying hours should be increased. The disadvantage of this system is that there is a loss in morale as the ties between the aircrew and "their" plane/ground crew and the ground crew and "their" aircrew/plane are broken.

In one trial (over 5 months) when flying was organised by ORS the daily operational flying hours were increased by 61% over the previous best achieved with the same number of aircraft. Their system was accepted and implemented.

Here the problem was one of deciding, for a particular type of operation, the relative merits of different aircraft in terms of factors such as: miles flown per maintenance man per month; lethality of load; length of sortie; chance of U-boat sighting; etc.

Experience showed that it required some 170 man-hours by maintenance and ground staff to produce one hour of operational flying and more than 200 hours of flying to produce one attack on a surfaced U-boat. Hence over 34,000 man-hours of effort were necessary just to attack a U-boat.

In early 1941 the attack kill probability was 2% to 3% (i.e. between 1.1 million and 1.7 million man-hours were needed by Coastal Command to destroy one U-boat). It is in this area that the greatest contribution was made by OR in Coastal Command and so we shall examine it in more detail. (Note here that we ignore the question of the U-boat being attacked and damaged, but not killed. To include this merely complicates the discussion).

Plainly in the above calculation the "weak link" is the low attack kill probability and it is this that really needs to be improved.

The main weapon of attack against a surfaced (when spotted) U-boat was depth charges dropped in a stick (typically six 250lb (110kg) depth charges) in a more or less straight line along the direction of flight of the attacking aircraft. After hitting the water a depth charge sinks whilst at the same time being carried forward by its own momentum. After a pre-set time delay, or upon reaching a certain depth, it explodes and any U-boat within a certain distance (the lethal radius) is fatally damaged. Six variables were considered as influencing the kill probability:

We consider each in turn.

In the first two years of the war depth charges were mainly set for explosion at a depth of 30/45 metres [this figure having being set years ago and never altered since]. Analysis of pilot reports by ORS showed that in 40% of attacks the U-boat was either still visible or had been submerged less than 15 seconds (these are the U-boats that we would expect to have most chance of killing as we have a good idea of their position). Since the lethal radius of a depth charge was around 5-6 metres it was clear that a shallower setting was necessary. Explosion at a depth of 15 metres was initiated and as new fuses became available at 10 metres and then 8 metres.

Here we have the issue of historical inertia in decision making - in the dim and distant past someone decided that the standard depth setting should be 30/45 metres and this historical decision has been carried forward - never being questioned/re-examined until ORS came on the scene.

As mentioned above the standard 250lb depth charge was believed to have a lethal radius of only 5-6 metres. Plainly to increase this radius (within the 250lb limit) the chemical explosive inside the depth charge should be more powerful (e.g. increasing the lethal radius by just 20% increases the lethal volume (sphere) around the depth charge by 72.8%). The best chemical explosive currently available was therefore introduced.

Note here that it could be argued (and was) that since a 250lb depth charge had too small a lethal radius a bigger charge (600lb (270kg) was prescribed by the Air Staff) was needed. ORS suggested 100lb (45kg) on the basis that it would be more effective to have many small explosions rather than one large explosion. (As an analogy would you prefer to throw many small balls at a small target or one large ball?). In fact neither alternative ever really preceded past the trial stage due to increasing success with the 250lb depth charge.

This illustrates the concept of "tradeoff" which often appears in OR in that, for a given total bomb load we have to make a choice (tradeoff) between bomb size and number of bombs (from one big bomb to many small ones).

By the end of 1942 it had become clear that too many pilots were reporting having had "straddled" a target U-boat with a stick of depth charges without sinking it. Either their claims were unduly optimistic (the ORS view) or the lethal radius of a depth charge was much less than currently believed (the Air Staff view).

To settle the issue cameras were installed for recording U-boat attacks. Analysis of 16 attacks indicated that ORS were right. This analysis also showed that pilots were following tactical instructions and "aiming off" (aiming ahead of the U-boat to allow for its forward travel during fall of the depth charges). However analysis also revealed that had they not aimed off 50% more kills would have been recorded. Pilots were therefore instructed not to aim off.

Here the question was whether to attack from the beam, quarter or along the U-boat track. No definite answer was really reached until 1944 when it was concluded that track attacks were more accurate (probably due to the pilot using the U-boat wake to help him line the plane up).

In the early part of the war this spacing was specified at 12 metres. ORS calculated that increasing this to 33 metres would increase kills by 35% and this was done.

For much of the war all low level attacks on U-boats were by the pilot acting as bomb aimer/release. Although pilots (and Air Staff) believed they were accurate photographic evidence did not support this belief and ORS pressed for bombsights to be provided. By late 1943 a low level (Mk.III) sight came into use increasing kills per attack by 35%.

The overall effect of all the measures discussed above was such that by 1945 the attack kill probability had risen to over 40% (remember it started out at 2-3%).


Although scientists had (plainly) been involved in the hardware side of warfare (designing better planes, bombs, tanks, etc) scientific analysis of the operational use of military resources had never taken place in a systematic fashion before the Second World War. Military personnel, often by no means stupid, were simply not trained to undertake such analysis.

These early OR workers came from many different disciplines, one group consisted of a physicist, two physiologists, two mathematical physicists and a surveyor. What such people brought to their work were "scientifically trained" minds, used to querying assumptions, logic, exploring hypotheses, devising experiments, collecting data, analysing numbers, etc. Many too were of high intellectual calibre (at least four UK wartime OR personnel were later to win Nobel prizes when they returned to their peacetime disciplines).

By the end of the war OR was well established in the armed services both in the UK and in the USA.

Following the end of the war OR took a different course in the UK as opposed to in the USA. In the UK many of the distinguished OR workers returned to their original peacetime disciplines. As such OR did not spread particularly well, except for a few isolated industries (iron/steel and coal). In the USA OR spread to the universities so that systematic training in OR for future workers began.

A short personal account relating to the history of OR can be found here.


OR started just before World War II in Britain with the establishment of teams of scientists to study the strategic and tactical problems involved in military operations. The objective was to find the most effective utilisation of limited military resources by the use of quantitative techniques.

Following the end of the war OR spread, although it spread in different ways in the UK and USA.

You should be clear that the growth of OR since it began (and especially in the last 35 years) is, to a large extent, the result of the increasing power and widespread availability of computers. Most (though not all) OR involves carrying out a large number of numeric calculations. Without computers this would simply not be possible.