Operations Research has had undertakings as early as the third century when Hieron, King of Syracuse, asked Archimedes to devise means for breaking the Roman naval siege of his city but more clear examples of OR were evident during WWI when political figures made attempts to analyze military operations mathematically. World War II marked the beginning of Operations Research as an organized research. Britain was the first to utilize OR because they already had an OR organization in existence. OR finally crossed the Atlantic to the America’s a few months later as both the Army Air Forces and the Navy began work to analyze different war time situations. After the war, OR proved to be very successful that many military operations were faced with considerations on how to continue their field of study and groups were formed within the military structure to continue research. The economic situation that the war had caused began conversion of industries from wartime production to production of goods for the private consumer. The transition from war to peace created a highly competitive economy. Businesses needed to be efficient and profitable. This caused business to analyze their business operations and thus OR found another home.

As discussed above, OR began in the military and then emerged in all aspects of government. It then spread to the banking and financial districts, airlines, and car manufactures. Now it is present in every industry. Operations Research involves many facets of discipline and tends to consider new courses of action to take that make changes to an organization over long periods of time. As one knows, change doesn’t happen over-night.

Since the operating systems studied by OR specialists arose in a wide variety of industries and government environments. It follows that the results of their research frequently make important contributions to the solution of problems of choice, policy, and planning. It can be determined that OR is a flexible and powerful tool to management in improving their operations. The methods of Operations Research were rooted in the basics of science. . An OR team (in some companies this may be a single individual) consists of trained researchers that incorporate their own techniques and methods from their fields to the basics of science. The first step is to formulate the problem, after which a mathematical model is generally constructed to represent the system being studied. Mathematical models, or conceptual models, are usually equations or formulae developed to relate important factors of the operations studied. The factors can then be tested, and operated on mathematically to determine the effects of changing the values of the variables. Basically, OR can be characterized with the following statements:

• Research on the operations of the whole organization to determine the problem and then optimize the operations to improve the organization.
• Application of the newest methods and techniques.
• Application of the methods and techniques of the older management sciences.
• Development and use of analytical models in accordance to the basic sciences.
• Design and use experimental operations that give an insight into behavior of actual operations.
• Use of integrated and creative multi-disciplinary team research to solve complex operational problems.

OR utilizes modeling as a way to recognize the problem and then ultimately solve the problem. From recognition to the decision; the OR process is made up of eight distinct phases:

1. Recognition of a Need: This phase is defined as the perceptions of needing some sort of resolution or need for improvement.
2. Problem Formulation: This phase takes the problem, defines it, and assigns inputs to be used in the next phase. These inputs are variables, parameters and constraints.
3. Model Construction: This phase takes the variables, parameters and constraints and constructs a mathematical representation of the problem defined in phase 2. The purpose of this phase is to create the best model so that the variables can be changed and observed.
4. Data Collection: This phase utilizes the model and the different inputs to the model, which reflect actual problem conditions. Data that is hard to collect include: preferences, opinions, and quality. These items are known as soft data.
5. Model Solution: This phase actually utilizes the input data and mathematical algorithms to produce results.
6. Model Validation and Analysis: This phase is a self-checking phase. It makes sure that the first four phases are free of errors and that the model accurately represents the problem hand. If there are errors in any of the processes, phase 2 to phase 5 must be redone.
7. Interpretation of Results: This phase is concerned with examining the results and makes sure that the solution is the optimum or best solution. Tradeoffs can occur and this is known as sub-optimization.
8. Decision Making and Implementation: Once a project reaches this phase, the OR is done and a decision must be made. The outcome of the decision may not directly be related to the results. Other existing external factors play a part in the decision process. These factors can be personal, ethical or political. Whatever the reason, the OR process has been accomplished and this gives management another tool for making a decision on the problem.

Now that the steps of OR have been identified, the tools to solve the problem must be described. These models or algorithms are utilized in phase 3-5 to produce results and data collection. Models can be broken into two categories. They are the deterministic and stochastic. Each model is created to specify a certain problem or application needed to be solved. Deterministic models are the simplest types. There are no uncertain or probabilistic variables and no optimization. They are straightforward and utilize formulas and graphs to represent the data. Within the deterministic category are:

• Linear Programming: Problems involving the allocation of scarce resources such as materials, manpower, machine time or space. Extensively used in problems of blending ingredients, scheduling, manpower planning and economic planning.
• Transportation: This is a special case of Linear Programming. This involves problems where one resource is stored or made in several locations and needed in several other locations, e.g. warehousing and distribution.
• Assignment: This is a special case of Linear Programming. This involves problems such as assigning n drivers to n cars in order to minimize cost or n operators to n tasks to minimize time.
• Integer Programming: : This is a special case of Linear Programming. This involves problems of resource allocation with the restriction that certain resources are only available in fixed-size units, which can’t be split up. Also used in problems of route selection and other problems involving 0-1 variables.
• Goal Programming: Problems of resource allocation.

Stochastic models incorporate uncontrolled variation in some way. The widest use of these models is to the use of statistical forecasting. Within the stochastic category are:

• Statistical Forecasting Methods: Making short to medium forecasts of future values of a time series of data. Typically used for predicting sales, spare parts demands, etc.
• Simulation: Problems that involve uncertainty in the system – e.g. in production lines, shops, transport services, manpower modeling. This model is to mimic real life systems and used to evaluate the real system.
• Game and Hypergame Theory: Competitive problems where there is some kind of opponent. This tends to be more theoretical than practical.
• Decision Analysis: Problems involving a decision or a series with a small number of options and a small number of outcomes. Examples include decisions on whether or not to test market new products, bidding strategies for contracts and diagnosis of medical condition.
• Replacement Theory: Problems involving the failure of components and/or machines.
• Search Theory: Problems involving something with the use of limited resources.
• Queuing Theory: Problems where customers are queuing for service of some sort. Examples include shops, banks, telephone exchange, repair of machines, and job-shop scheduling.

• Dynamic Programming: Problems involving a series of similar linked decisions, differing only in time or space, such as the choice of the shortest route between two points or decisions involving monthly production or storage.
• Project Network Analysis: Problems involving sequencing and scheduling a collection of identifiably separate jobs subject to various logical constraints. Used almost everywhere in construction industry.

• Stock Control: Problems involving the holding of stocks, raw materials, spare parts, finished goods or stationary.

Two examples of Operations Research are enclosed in this report. Case study A utilized the Stochastic Gaming Theory technique to solve a problem of competition between two companies. Case study B uses the Deterministic Linear Programming to solve a problem of production.

The gaming theory is utilized in problems where competition is evident. In most cases, this theory tends to be more theoretical than practical. There are few assumptions that can be made when using this theory to combat competitive problems. They are:

1. All courses of action for both you and your opponents are known.
2. All decisions are made simultaneously.
3. Given the decisions, the outcome is determined.
4. Decision-maker can place an order of preference on the outcomes.

Like any problem there must be some inputs. They are:

• Data on all possible courses of action and the resulting outcomes.
• Value or ranking of the outcomes
• Overall criteria for judging between decisions. - E.g. best average payoff, highest minimum payoff, highest joint payoff, etc.

The output one is looking for in this theory is:

• Best strategy to adopt. This will not necessarily be one action, but could be a random choice of courses of action.

This theory is very easy to understand and it is quite rapidly done without the need of a computer. Plus this is the only technique that can be used in competition problems. Some disadvantages to consider is that assumption 1 may be wrong or believed right, but it is wrong. Assumption 2 and 3 are not valid. A third disadvantage is that the decision-makers sometimes find ranking the outcomes difficult and lastly the analysis may only be possible after the event.

Knowing the above inputs, outputs and assumptions of the Gaming Theory, let us use the characteristics of a gaming theory to solve this case study below.

Two competing ice cream chains, Kool Ice and Ice Kold, want to hold ice cream sales in order to capture some extra business from each other. They each have the option of having a sale either on their most popular flavors or on all their ice cream.

They found the following payoff matrix to show which way the business is turning for each of their choices. Number of customers of Kool Ice switching to Ice Kold

(In hundreds of people). Note that a negative number means they are switching from Ice Kold to Kool Ice. Find the best strategy for each of them.

Ice Kold Sale on All Sale on Popular

Sale an All 4 -3

Sale on Popular -3 2

Kool Ice wants to sell to the most people thus they want the smallest number (negative means consumers switching to Kool Ice). Ice Kold also wants the most people, thus they want the largest number. If the minimax procedure is used, one will end up with two different points. Kool Ice will want -3 and Ice Kold will want 2. The best strategy to get the best results is to choose one approach with a certain probability. Assuming Ice Kold has a sale on all the ice cream with probability p, then it has a sale on the popular ice cream with probability 1-p. To find p, find the expected values of the columns. Expected value is the value of the box times its probability, summed up.

C1 c2

ICE KOLD p 4 -3

1-p -3 2

Thus the expected value of column c1 is 4p + (-3)(1-p)= 7p –3

The expected value against a sale of popular flavors by Kool ICE is column c2:

(-3)p + 2(1-p) = -5p + 2

To solve for p set the two equations equal to each other:

1. 7p – 3 = -5p + 2
2. 12p – 3 = 2
3. 12p = 5
4. p = 5/12

The two equations are set equal to each other because we want the same result to occur independently of the other company’s choice. Therefore, Ice Kold should have a sale on their most popular ice cream 7/12 of the time and a sale on all of their ice cream 5/12 of the time. This gives an expected value of :

1. –5p + 2
2. –5(5/12) + 2 = -1/12

This means that they have lost 100*(-1/12) = 8 people. Kool Ice acquires 8 new clients. On the other hand, the probability for ICE Kold is the same. So they should have a sale on all of their ice cream 5/12 of the time and the rest of the time a sale on their popular ice cream. This also yields an expected loss of 8 people to Kool Ice.

These type of problems involving the allocation of scarce resources such as materials, manpower, machine time or space. It is also used extensively in problems of blending ingredients, scheduling, manpower planning and economic planning. This technique is often cited as the "classic" example of an OR technique. Assumptions need to be made for this type of problem.

• The problems are expressed in terms of maximizing or minimizing a single linear objective function subject to restrictions; which are linear constraints.
• That each parameter in the problem can be measured or estimated.

Certain requirements are needed such as data on cost, availability of scarce resource, and all the limitations of the problem, whether logical or technological, in quantitative terms.

There are a number of advantages and disadvantages to the LP techniques. The first advantage is the size of the problem is almost immaterial. Secondly, LP guaranteed best solution, if any solution exist. Thirdly, results can be presented in an easy-to-use format because much practical experience of the use of computer software packages exists. The disadvantage for example is LP can only use one objective function. Secondly, the real-life problems are often non-linear. And finally, there may not be a solution to the problem.

Let us use the technique of Linear Programming to solve the problem below.

The Marvel Toy Company wishes to make three models of boats for the most profit. They found that a model of a steamship takes the cutter one hour, the painter 2 hours, and the assembler 4 hours of work. It produces $6 of profit. Their model of a four-mast sailboat takes the cutter 3 hours, the painter 3 hours, and the assembler 2 hours. It produces $3 of profit. Their model of a two-mast sailboat takes the cutter one hour, the painter three hours, and the assembler one hour. It produces $2 of profit. The cutter is only available for 45 hours, the painter for 50 hours, and the assembler for 60 hours. Assuming that they can sell all the models that are built, find the constraints of the problem and describe how the solution is obtained.

To find the constraints for this problem, let x be the number of models of steamships, Y the number of models of four-mast sailboats, Z the number of two-mast sailboats. The constraints are:

1. X >= 0, There cannot be a negative number of models of steamships.

 

2. Y >= 0, There cannot be a negative number of models of four-mast sailboats.

 

 

3. Z >= 0, There cannot be a negative number of models of two-mast sailboats.
4. A steamship takes one hour of the cutter’s time, a four-mast sailboat takes three hours of the cutter’s time, and the two-mast sailboat takes one hour of the cutter’s time. Since the cutter has only 45 hours available there is the constraint:

 

X + 3Y + Z <= 45

 

5. A Steamship takes two hours of the painter’s time, a four-mast sailboat takes three hours of the painter’s time, and a two-mast sailboat takes three hours of the painter’s time. Since the painter has only 50 hours of time available, there is the constraint:

2X + 3Y + 3Z <= 50

 

6. A Steamship takes four hours of the assembler’s time, a four-mast sailboat takes 2 hours of the assembler’s time, and a two-mast sailboat takes one hour of the assembler’s time. Since the assembler can work only 60 hours, there is the constraint:

4X + 2Y + Z <= 60

Since there are three unknowns, a three dimensional graph is required to plot the constraints. However, it is known from a theorem of Linear Programming that the point of maximum profit lies on the intersection of three lines of constraints (the same number of constraints as the number of unknowns). Thus, to solve this, break up the six constraints into all groups of three that are possible. Then solve each group of three equations for the three unknowns. There is now a group of points. Take from that group of points those that meet the conditions set by all of the constraints. Then take from that new group the point which yields the most profit. That point is the solution to the problem. To find which point yields the most profit, use the following equation for the profit:

Profit = $6X + $3Y + $2Z

After following the above procedure it is found that, in order to make the most profit, the company should produce 13 steamships, no four-mast sailboats, and 8 two-mast sailboats. This will yield a profit of $94.

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Budnick, F.S., McLeavey, D., Mojena, R., Principles of Operations Research for Management, pp. 2-31, 1988.

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