**Course MBA – 2**

^{nd}

**Semester**

**Subject:**

**Operations Research**

**Assignment MB0048 – Set1**

**Q.1 (a) “Operation Techniques is a bunch of mathematical techniques.” Comment.**

Ans:

Operations Research is an interdisciplinary branch of applied mathematics and formal science that uses methods such as mathematical modelling, statistics, and algorithms to arrive at optimal or near optimal solutions to complex problems. It is typically concerned with optimizing the maxima (profit, assembly line performance, crop yield, bandwidth, etc) or minima (loss, risk, etc.) of some objective function. Operations research helps management achieve its goals using scientific methods. The terms operations research and management science are often used synonymously. When a distinction is drawn, management science generally implies a closer relationship to the problems of business management. The field of operations research is closely related to Industrial engineering. Industrial engineers typically consider Operations Research (OR) techniques to be a major part of their toolset. Some of the primary tools used by operations researchers are statistics, optimization, probability theory, queuing theory, game theory, graph theory, decision analysis, and simulation. Because of the computational nature of these fields, OR also has ties to computer science, and operations researchers use custom-written and off-the-shelf software. Operations research is distinguished by its frequent use to examine an entire management information system, rather than concentrating only on specific elements (though this is often done as well). An operations researcher faced with a new problem is expected to determine which techniques are most appropriate given the nature of the system, the goals for improvement, and constraints on time and computing power. For this and other reasons, the human element of OR is vital. Like any other tools, OR techniques cannot solve problems by themselves.

Scope of operation Research:

Examples of applications in which operations research is currently used include:

- Critical path analysis or project planning: identifying those processes in a complex project which affect the overall duration of the project.
- Designing the layout of a factory for efficient flow of materials.
- Constructing a telecommunications network at low cost while still guaranteeing QoS (quality of service) or QoS (Quality of Experience) if particular connections become very busy or get damaged.
- Road traffic management and 'one way' street allocations i.e. allocation problems.
- Determining the routes of school buses (or city buses) so that as few buses are needed as possible.
- Designing the layout of a computer chip to reduce manufacturing time (therefore reducing cost) Managing the flow of raw materials and products in a supply chain based on uncertain demand for the finished products.
- Efficient messaging and customer response tactics.
- Robotizing or automating human-driven operations processes.
- Globalizing operations processes in order to take advantage of cheaper materials, labour, land or other productivity inputs Managing freight transportation and delivery systems (Examples: LTL Shipping, intermodal freight transport).
- Scheduling.
- Personnel staffing.
- Manufacturing steps.
- Project tasks.
- Network data traffic: these are known as queuing models or queueing systems.
- Sports events and their television coverage blending of raw materials in oil.
- Refineries determining optimal prices, in many retail and B2B settings, within the disciplines of pricing science.

Operations research is also used extensively in government where evidence-based policy is used.

**Q.1- b) “Operation Research is an aid for the executive in making his decisions based on scientific methods analysis”. Discuss the above statement in brief.**

Operation Research is a scientific method of providing executive departments with a quantitative basis for decisions regarding the operations under their control. Morse & Kimball Operations research is a scientific approach to problem solving for executive management. – H.M. Wagner Operations research is an aid for the executive in making these decisions by providing him with the needed quantitative information based on the scientific method of analysis. The mission of Operations Research is to serve the entire Operations Research (OR) community, including practitioners, researchers, educators, and students. Operations Research, as the flagship journal of our profession, strives to publish results that are truly insightful. Each issue of Operations Research attempts to provide a balance of well-written articles that span the wide array of creative activities in OR. Thus, the major criteria for acceptance of a paper in Operations Research are that the paper is important to more than a small subset of the OR community, contains important insights, and makes a substantial contribution to the field that will stand the test of time. Operational research, also known as operations research, is an interdisciplinary branch of applied mathematics and formal science that uses advanced analytical methods such as mathematical modelling, statistical analysis, and mathematical optimization to arrive at optimal or near-optimal solutions to complex decision-making problems. It is often concerned with determining the maximum (of profit, performance, or yield) or minimum (of loss, risk, or cost) of some real-world objective. Originating in military efforts before World War II, its techniques have grown to concern problems in a variety of industries. Operational research, also known as OR, is an interdisciplinary branch of applied mathematics and formal science that uses advanced analytical methods such as mathematical modelling, statistical analysis, and mathematical optimization to arrive at optimal or near-optimal solutions to complex decision-making problems. It is often concerned with determining the maximum (of profit, performance, or yield) or minimum (of loss, risk, or cost) of some real world objective. Originating in military efforts before World War II, its techniques have grown to concern problems in a variety of industries.

Operational research encompasses a wide range of problem-solving techniques and methods applied in the pursuit of improved decision-making and efficiency. Some of the tools used by operational researchers are statistics, optimization, probability theory, queuing theory, game theory, graph theory, decision analysis, mathematical modelling and simulation.

Because of the computational nature of these fields, OR also has strong ties to computer science. Operational researchers faced with a new problem must determine which of these techniques are most appropriate given the nature of the system, the goals for improvement, and constraints on time and computing power.

Work in operational research and management science may be characterized as one of three categories:

Fundamental or foundational work takes place in three mathematical disciplines: probability, optimization, and dynamical systems theory.

Modelling work is concerned with the construction of models, analyzing them mathematically, implementing them on computers, solving them using software tools, and assessing their effectiveness with data. This level is mainly instrumental, and driven mainly by statistics and econometrics. Application work in operational research, like other engineering and economics' disciplines, attempts to use models to make a practical impact on real-world problems.

• The major sub disciplines in modern operational research, as identified by the journal Operations Research, are:

• Computing and information technologies

• Decision analysis

• Environment, energy, and natural resources

• Financial engineering

• Manufacturing, service sciences, and supply chain management

• Policy modelling and public sector work

• Revenue management

• Simulation

• Stochastic models

• Transportation

**Q. 2 Comment on the following statements:**

**a) Operation Research advocates a system approach and is concerned with optimization.**

1. Systems approach:

The term system approach implies that each problem should be examined in its entirely to the extent possible and economically feasible from the point of view of the overall system of which the problem under consideration is one part. Under those approaches a manager makes conscious attempt to understand the relationships among various parts of the organisation and their role in supporting the overall performance of the organisation. Operations objective of operations research is to provide managers of the organisation with a scientific basis for solving problems involving the interaction of components of the organisation as a whole. The decision which is best for the organisation as a whole is called an optimal decision. Operations research tries to find the best decision relative to a large portion of the total organisation. Hence in operations research every problem is considered in its totality, i.e. O.R. adopts systems approach for solving the problem. In other words, “Operations Research is the scientific study of large systems with a view to identify problem areas and provide the mangers with a quantitative basis for decisions which will enhance their effectiveness in achieving the specified objectives.”

2. Inter-disciplinary Team Approach:

It is an important characteristic of O.R. According to this characteristic, no single individual can be an expert on all aspects of a problem under consideration. Thus, O.R. utilizes the inter disciplinary team approach. Under this approach, a team comprising experts from different disciplines such as mathematics, statistics, economics, management, computer science, engineering and psychology, etc. is constituted. Such a team when confronted with a problem determines its solution by utilizing the diverse background and skills of the teammates. Every expert of the team, while solving the problem, tries to abstract the essence of the problem and then determines whether a similar type of problem has been dealt by his team or not. If the answer is yes, then it is solution of the current problem. In this way, each member of the team, by utilizing his experience and expertise, may be in a position to suggest an approach to overcome a problem that otherwise may not be possible for an individual to tackle.

3. Methodological Approach:

O.R. utilizes scientific methods for solving a problem. Specifically, the process begins with the careful observation and formulation of the problem. The next step is to construct a scientific model (typically a mathematical model) that attempts to abstract the essence of the real problem. From this model, conclusions or solutions are obtained which are also valid for the real problem. In an interactive fashion, the model is then verified through appropriate experiments to determine the best or optional solution to the problem under consideration.

4. Operations economy:

O.R. is a problem solving and a decision making science. Whenever we have conflicts, uncertainty and complexity in any situation, O.R. can help in the end to reduce costs and improve profits and effects substantial “Operations Economy”. Once the old approach of management by intuit is buried, a scientific approach to decision making is bound to help. Often the conflicts are so tangled that they defy any intuitive solution, viz., the marketing function frequently caught up in recoiling the following conflicting objectives: i) product innovation, ii) high scale volume, iii) increasing market share, iv) flexibility in the market place, and v)entry into new markets and revenue markets. It is here that O.R. is likely to convincingly optimize the total effectiveness.

From all above areas of applications, one may conclude that operations research can be widely advocate a systems approach for making timely management decisions and also used as a corrective measure. O.R. encourages systems approach which concerned with the cost optimization, and hence we can say:

*Operation Research advocates a system approach and is concerned with optimization.***Q.2- b). Operation Research replaces management by personality. Comment.**

Operations research is today recognised as an applied science concerned with large number of diverse human activities. To be precise an operation uses some valuable resources like men, money, machines, time, effort, etc. The outcome of the operation has also some value. An operations research worker is required: i) to minimize the input value for a specific output, or /and ii) to maximize the output value for a specific input, or /and iii) maximize some function of these values, e. g. the profit function (difference between output & input values) or return-on-investment function (ratio of output and input values), etc.

Some of the areas of management where techniques of operations research are applied are listed below:

1. Finance, Budgeting and Investments:

a) cash flow analysis, long range capital requirements, investment portfolios, dividend policies, etc.

b) Credit policies, credit risks and delinquent account procedures.

c) Claim and complaint procedures.

d) Dividend policies, investment and portfolio management, balance sheet and cash flow analysis.

2. Purchasing, procurement and Exploration:

a) Determining the quality and timing of purchase of raw materials, machinery, etc.

b) Rules for buying and supplies under varying prices.

c) Bidding policies.

d) Equipment replacement policies.

e) Determination of quantities and timings of purchases.

f) Strategies for exploration and exploitation of new material source.

3. Production Management:

a)

*Product planning:* i) Location and size of warehouses, distribution centres, retail outlets, etc.

ii) Distribution policy

b)

*Manufacturing & facility planning:* i) Production scheduling and sequencing

ii) Product scheduling and allocation of resources

iii) Selection & location of factories, warehouses and their sizes

iv) Determining the optimal production mix.

v) Maintenance policies & preventive maintenance.

vi) Scheduling & sequencing the production run by proper allocation of machines.

4. Marketing Management:

a) Product selection, timing, competitive actions.

b) Advertising strategy & choice of different media of advertising.

c) Number of salesman, frequency of calling of accounts, etc.

d) Effectiveness of market research.

e) Size of the stock to meet the future demand.

5. Personnel Management:

a) Recruitment policies & assignment of jobs.

b) Selection of suitable personnel with due consideration for age and skills, etc.

c) Establishing equitable bonus systems.

6. Research & Development:

a) Determination of areas of concentration of research and development.

b) Reliability & evaluation of alternative designs.

c) Control of development projects.

d) Coordination of multiple research projects.

e) Determination of time & cost requirements.

From all above areas of applications, one may conclude that operations research can be widely used in taking timely management decisions and also used as a corrective measure. The application of this tool involves certain data and not merely a personality of decision maker, and hence we can say:

*Operations Research has replaced management by personality.* Q.3.

**Explain how the profit maximization transportation problem can be converted to an equivalent cost minimization transportation problem.** How to convert profit maximization transportation problem to an equivalent cost minimization transportation problem can be understood by following Illustration as:

A firm has three factories located in city A, B & C and supplies goods to four dealers, dealer 1, 2, 3 & 4, spread all over the country. The production capacities of these factories are 1000, 700 & 900 units per month respectively. The monthly orders from the dealers are 900, 800, 500 & 400 units respectively. Per unit return (excluding transportation costs) are Rs. 8, 7 & 9 at the three factories. Unit transportation costs from the dealers are given below:

Factory | Dealers | |||

1 | 2 | 3 | 4 | |

City - A | 2 | 2 | 2 | 4 |

City - B | 3 | 5 | 3 | 2 |

City - C | 4 | 3 | 2 | 1 |

Optimal distribution system to maximize the total r eturn to be determined.

From the given data, we compute a matrix of net returns as done in table below;

*(Transportation matrix (Net return) for the Maximization problem)*

Factory | Dealers | Factory capacity | |||

1 | 2 | 3 | 4 | ||

City - A | 6 | 6 | 6 | 4 | 1000 |

City - B | 4 | 2 | 4 | 5 | 700 |

City - C | 5 | 6 | 7 | 8 | 900 |

To convert the given maximization problem to an equivalent minimization problem, we identify the cell (element) which has the highest contribution per unit (in this problem C-4 has highest per unit contribution, Rs.8), and subtract all elements from this highest element. The resultant matrix is a transportation problem with minimizing objective function. This has been given in the following table.

*(Transportation matrix for the Minimization problem)*

Factory | Dealers | Factory capacity | |||

1 | 2 | 3 | 4 | ||

City - A | 2 | 2 | 2 | 4 | 1000 |

City - B | 4 | 6 | 4 | 3 | 700 |

City - C | 3 | 2 | 1 | 0 | 900 |

Dealer requirement | 900 | 800 | 500 | 400 | 2600 |

The minimization problem is solved as a usual transportation problem. The resulting optimal solution is also the optimal solution to the original (maximization) problem. The value of the objective function is computed by referring the matrix of the maximization problem. It should be noted that the converted minimization problem will have at least one element with zero value.

**Q 4. Write the difference in the simplex solution procedure for a maximization problem and a minimization problem of linear programming.**

The difference in the simplex solution procedure for a maximization problem and a minimization problem of linear programming can be explained by the steps followed to solve the minimization/ minimization problem as follows ;

1. Introduce stack variables (S

_{i}’s) for “£” type of constraint. 2. Introduce surplus variables (S

_{i}’s) and artificial variables (A_{i}) for “³” type of constraint. 3. Introduce only Artificial variable for “=” type of constraint.

4. Cost (C

_{j}) of slack and surplus variables will be zero and that of artificial variable will be “M” 5. Find Z

_{j}– C_{j}for each variable. 6. Slack and artificial variables will form basic variable for the first simplex table. Surplus variable will never become basic variable for the first simplex table.

7. Z

_{j}= sum of [cost of variable x its coefficients in the constraints – Profit or cost coefficient of the variable]. 8. Select the most negative value of Z

_{j}– C_{j}. That column is called key column. The variable corresponding to the column will become basic variable for the next table. 9. Divide the quantities by the corresponding values of the key column to get ratios; select the minimum ratio. This becomes the key row. The basic variable corresponding to this row will be replaced by the variable found in step 6.

10. The element that lies both on key column and key row is called Pivotal element.

11. Ratios with negative and “a” value are not considered for determining key row.

12. Once an artificial variable is removed as basic variable, its column will be deleted from next iteration.

13. For maximisation problems, decision variables coefficient will be same as in the objective function. For minimisation problems, decision variables coefficients will have opposite signs as compared to objective function.

14. Values of artificial variables will always is – M for both maximisation and minimisation problems.

15. The process is continued till all Z

_{j}– C_{j}³ 0.**Q.5 What do you mean by the two-phase method for solving a given LPP? Why is it used?**

Every linear programming problem (LPP) is associated with another linear programming problem involving the same data and optimal solutions. Such two problems are said to be duals of each other. One problem is called the primal, while the other problem is called the dual. The dual formulation is derived from the same data and solved in a manner similar to the original 'primal' formulation. In other words, you can say that dual is the 'inverse' of the primal formulation because of the following reasons.

- If the primal objective function is 'maximisation' function, then the dual objective function is 'minimisation' function and vice-versa.

- The column co-efficient in the primal constraint is the row co-efficient in the dual constraint.

- The co-efficients in the primal objective function are the RHS constraint in the dual constraint.

- The RHS column of constants of the primal constraints becomes the row of co-efficient of the dual objective function.

The concept of duality is useful to obtain additional information about the variation in the optimal solution. These changes could be effected in the constraint co-efficient, in resource availabilities and/or objective function co-efficient. This effect is termed as post optimality or sensitivity analysis.

**Characteristics of dual solutions**

If the primal problem possesses a unique non-degenerate, optimal solution, then the optimal solution to the dual is unique. However, dual solutions arise under a number of other conditions. Several of the cases which can arise are:

- When the primal problem has a degenerate optimal solution, the dual has multiple optimal solutions.

- When the primal problem has multiple optimal solutions, the optimal dual solution is degenerate.

- When the primal problem is unbounded, the dual is infeasible.

- When the primal problem is infeasible, the dual is unbounded or infeasible.

**Formulation of Dual Concepts**

Consider the following LPP

Maximise

*Z = c1x1 +c2x2 + . . .+ cnxn* Subject to the constraints

*a11 x1 + a12 x2 + . . . + a1n xn ≤ b1*

*a21 x*

*1 + a22 x2 + . . .*

*+ a2n xn ≤ b2*

*am1 x1 + am2 x2 + . . . + amn xn ≤ bm*

*x1, x2, . . ., xn ≥ 0*

To construct a dual problem, you must adopt the following guidelines:

i. The maximisation problem in the primal becomes a minimisation problem in the dual and vice versa

ii. (≤) type of constraints in the primal become (≥) type of constraints in the dual and vice versa.

iii. The coefficients c1, c2, . . .,cn in the objective function of the primal become b1, b2,…,bm in the objective function of the dual.

iv. The constants b1, b2,…,bm in the constraints of the primal become c1, c2, . . .,cn in the constraints of the dual

v. If the primal has n variables and m constraints the dual will have m variables and n constraints

vi. The variables in both the primal and dual are non-negative

Thus the dual problem will be

Minimise

*W = b1 y1 + b2 y2 + . . .**+bm ym* Subject to the constraints

*a11 y1 + a21 y2 + . . . + am1 ym*

*≥ c1*

*a12 y1 + a22 y2 + . . . + am2 ym ≥ c2*

*a1n y1 + a2n y2 + . . . + amn ym ≥ cn*

*y1, y2, . . ., ym ≥ 0*

Formation of dual LPP is easier when the standard form of LPP for maximisation problem must contain “≤” type of constraints, while for minimisation problem, it must contain “

*≥”*type of constraints. Two Phase Method:

Two-phase method for solving a given LPP can be divided in the two phses as mentioned below:

Phase I: Formulate the new problem. Start by eliminating the original objective function by the sum of the artificial variables for a minimisation problem and the negative of the sum of the artificial variables for a maximisation problem. The Simplex method optimizes the ensuing objective with the constraints of the original problem. If a feasible solution is arrived, the optimal value of the new objective function is zero (suggestive of all artificial variables being zero). Subsequently proceed to phase -II. If the optimal value of the new objective function is non-zero, it means there is no solution to the problem and the method terminates.

Phase II: Start phase II using the optimum solution of phase I as the base. Then take the objective function without the artificial variables and solve the problem using the Simplex method.

Why is it used?

The drawback of the penalty cost method is the possible computational error resulting from assigning a very large value to the constant M. To overcome this difficulty, Two - Phase Simplex method is considered where the use of M is eliminated by solving the problem in two phases.

**Q. 6**

**Indicate any four shortcomings of taking a simulation approach to solve an O.R. problem.**

Shortcomings of taking a simulation approach to solve an O.R. problem

The range of application of simulation in business is extremely wide. Unlike other mathematical models, simulation can be easily understood by the users and thereby facilitates their active involvement. This makes the results more reliable and also ensures easy acceptance for implementation. The degree to which a simulation model can be made close to reality is dependent upon the ingenuity of the OR team who identifies the relevant variables as well as their behavior.

In case of other OR models, simulation helps the manager to strike a balance between opposing costs of providing facilities (usually meaning long term commitment of funds) and the opportunity and costs of not providing them.

The simulation approach is recognised as a powerful tool for management decision-making. Shortcoming of taking a simulation approach to solve an O. R. problems are as follows;

- It does not produce optimal results. Solutions are approximate, and it is some less than formal but ‘satisfactory’ approach to problem-solving only.
- To be able to simulate systems, a fairly good knowledge of the parts or components of the system and their characteristics is required. The desire is to understand, explain and predict the dynamic behavior of the system or the sum total of these parts. Adequate knowledge of the system behavior.
- Each simulation run like a single experiment conducted under a given set of conditions as defined by a set of values for the input solution. A number of simulation runs will be necessary and thus can be time consuming. As the number of variables increases in terms of input, the difficulty in finding the optimum values increases considerably.
- Since simulation involves repetitions of the experiment, it is a time consuming task when manually done.
- As a number of parameters, increase, the difficulty in finding the optimum values increases to a considerable extent.
- Because of the simplicity in adoption of simulation process, one may develop to rely on this technique too often, although mathematical model is more suitable to the situation.
- One should not ignore the cost associated with a simulation study for data collection, formation of the model. A good simulation model may be very expensive. Often it takes years to develop a usable corporate planning model.
- The computer time as it is fairly significant.
- A simulation application is based on the premise that the behaviour pattern of relevant variables is known, and this very premise sometimes becomes questionable.
- Not always can the probabilities be estimated with ease or desired reliability. The results of simulation should always be compared with solutions obtained by other methods wherever possible, and “tempered” with managerial judgment