SK-MB0048 (SET-2)

Sikkim Manipal University - MBA - MB0048 – Operation Research Semester: 2 - Assignment Set: 2

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ASSIGNMENT-02

Name : Sanjeev Kaushik

Registration No. : 571015350

Learning Center : Sidvin College, BSK, Bangalore

Learning Center Code: 2744

Course : Master of Business Administration

Subject : Operation Research

Semester : II

Module No. : MB0048

Date of submission : 18th Sep 2011

Marks awarded : ____________

Directorate of Distance EducationSikkim Manipal UniversityII Floor, Syndicate House

Manipal – 576 104

Signature of Coordinator Signature of Center Signature of Evaluator


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MBA SEMESTER IIMB0048 –Operation Research

Assignment Set- 2

Q1. State the different types of models used in OR. Explain briefly the general methods for solving these OR models?

Answer:

Most operations research studies involve the construction of a mathematical model. The model is a collection of logical and mathematical relationships that represents aspects of the situation under study. Models describe important relationships between variables; include an objective function with which alternative solutions are evaluated, and constraints that restrict solutions to feasible values.

Although the analyst would hope to study the broad implications of the problem using a systems approach, a model cannot include every aspect of a situation. Elements that are irrelevant or unimportant to the problem are to be ignored; hopefully leaving sufficient detail so that the solution obtained with the model has value with regard to the original problem.

Models must be both tractable, capable of being solved, and valid, representative of the original situation. These dual goals are often contradictory and are not always attainable. It is generally true that the most powerful solution methods can be applied to the simplest or most abstract model.

We provide in this section a description of the various types of models used by operations research analyst. The division is based on the mathematical form of the model. All the models described here are solved with Excel add-ins described in the Computation section of this site. In some cases the methods used to solve a model are described in the Methods sections. Student exercises for creating models are in the Problems section. Additional models related to problems arising in Operations management and Industrial Engineering are in the OM/IE section.

An operation research (OR) is a discipline explicitly devoted to aiding decision makers.

This section reviews the terminology of OR, a process for addressing practical decision problems and the relation between Excel models and OR.

o Linear Programming

A typical mathematical program consist of a single objective function, representing either a profit to be maximized or a cost to be minimized, and a set of constraints that circumscribe

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the decision variables. In the case of a linear program (LP) the objective function and constraints are all linear functions of the decision variables. At first glance these restrictions would seem to limit the scope of the LP Model, but this is hardly the case. Because of the simplicity, software has been developed that is capable of solving problems containing millions of variables and tens of thousands of constraints. Countless real-world applications have been successfully modeled and solved using linear programming techniques.

o Network Flow Programming

The term network flow program describes a type of model that is a special case of the more general linear program. The class of network flow programs includes such problems as the transportation problem, the assignment problem, the shortest path problem, the maximum flow problem, the pure minimum cost flow problem, and the generalized minimum cost flow problem. It is an important class because many aspects of actual situations are readily recognized as networks and the representation of the mode is much more compact than the general linear program. When a situation can be entirely modeled as a network, very efficient algorithms exist for the solution of the optimization problem, many times more efficient than linear programming in the utilization of computer time and space resources.

o Integer Programming

Integer programming is concerned with optimization problems in which some of the variables are required to take on discrete values. Rather than allow a variable to assume all real values in a given range, only predetermined discrete values within range are permitted. In most cases these values are the integers, giving rise to the name of the class of the models.Models with integer variables are very useful. Situation that cannot be modeled by linear programming are easily handled by integer programming. Primary among these involve binary decisions such as yes-no, build-no build or invest-not invest. Although one can model a binary decision in linear programming with a variable that ranges between 0 and 1, there is nothing that keeps the solution from obtaining a fractional value such as 0.5, hardly acceptable to a decision maker. Integer programming requires such a variable to be either 0 or 1, but not in between. Unfortunately integer programming models of practical size are often very difficult or impossible to solve. Linear programming methods can solve problems orders of magnitude larger than integer programming methods. Still, many interesting problems are solvable, and the growing power of computers makes this an active area of interest in Operations Research.

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o Nonlinear Programming

When expressions defining the objective function or constraints of an optimization model are not linear, one has a nonlinear programming model. Again, the class of situations appropriate for nonlinear programming is much larger than the class of linear programming. Indeed it can be argued that all linear expressions are really approximations for nonlinear ones.Since nonlinear functions can assume such a wide variety of functional forms there are many different classes of nonlinear programming models. The specific form has much to do with how easily the problem is solve, but in general a nonlinear programming model is much more difficult to solve than a similarly sized linear programming model.

o Dynamic Programming

Dynamic programming (DP) models are represented in a different way than other mathematical programming models. Rather than an objective function and constraints, a DP model describes a process in terms of states, decisions, transitions and returns. The process begins in some initial state where a decision is made. The decision causes a transition to a new state. Based on the starting state, ending state and decision a return is realized. The process continues through a sequence of states until finally a final state is reached. The problem is to find the sequence that maximizes the total return.The models considered here are for discrete decision problems. Although traditional integer programming problems can be solved with DP, the models and methods are most appropriate for situations that are not easily modeled using the constructs of mathematical programming. Objectives with very general functional forms may be handled and a global optimal solution is always obtained. The price of this generality is computational effort. Solutions to practical problems are often stymied by the “curse of dimensionally” where the number of stats grows exponentially with the number of dimensions of the problem.

o Stochastic Programming

The mathematical programming models, such as linear programming, network flow programming and integer programming generally neglect the effects of uncertainty and assume that the results of decisions are predictable and deterministic. This abstraction of reality allows large and complex decision problems to be modeled and solved using powerful computational methods.Stochastic programming explicitly recognizes uncertainty by using random variables for some aspects of the problem. With probability distributions assigned to the random variables, an expression can be written for the expected value of the objective to be optimized. Then a variety of computational methods can be used to maximize or minimize the expected value. This page provides a brief introduction to the modeling process.

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o Combinatorial Optimization

The most general type of optimization problem and one that is applicable to most spreadsheet models is the combinatorial optimization problem. Many spreadsheet models contain variables and compute measures of effectiveness. The spreadsheet user often changes the variables in an unstructured way to look for the solution that obtains the greatest or least of the measure. In the words of OR, the analyst is searching for the solution that optimizes and objective function, the measure of effectiveness. Combinatorial optimization provides tools for automating the search for good solutions and can be of great value for spreadsheet applications.

o Stochastic Process

In many practical situations the attributes of a system randomly change over time. Examples include the number of customers in a heckout line, congestion on a highway, the number of items in a warehouse, and the prices of financial security, t name a few. When aspects of the process are governed by probability theory, we have a stochastic process.The model is described in part by enumerating the states in which the system can be found. The state is like a snapshot of the system at a point in time that describes the attributes of the system. The example for this section is an Automated Teller Machine (ATM) system and the state is the number of customers at or waiting for the machine. Time is the linear measure through which the system moves. Events occur that change the state of the system. For the ATM example the events are arrivals and departures. In this section we describe the basic ideas associated with modeling to a stochastic process that are useful for both Discrete and Continuous Time Markov Chains.

o Discreet Time Markov Chains

Say a system is observed at regular intervals such as every day or every week. Then the stochastic process can be described by a matrix which gives the probabilities of moving to each state from every other state in one time interval. Assuming this matrix is unchanging with time, the process is called Discrete Time Markov Chain (DTMC). Computational techniques are available to compute a variety of system measures that can be used to analyze and evaluate a DTMC model. This section illustrates how to construct a model of this type and the measures that are available.

o Continuous Time Markov Chains

Here we consider a continuous time stochastic process in which the duration of all state changing activities are exponentially distributed. Time is a continuous parameter. The

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process satisfies the Markovian property and is called a Continuous Time Markov Chain (CTMC). The process is entirely described by a matrix showing the rate of transition from each state to every other state. The rates are the parameters of the associated exponential distributions. The analytical results are very similar to those of a DTMC. The ATM example is continued with the illustrations of the elements of the model and statistical measures that can be obtained from it.

o Simulation

When a situation is affected by random variables it is often difficult to obtain closed from equations that can be used for evaluation. Simulation is a very general technique for estimating statistical measures of complex systems. A system is modeled as if the random variables were known. Then values for the variables are drawn randomly from their known probability distributions. Each replication gives on observation of the system response. By simulating a system in this fashion for many replications and recording the responses, on can compute statistics concerning the results. The statistics are used for evaluation and design.

Q2. What are the meaning and role of lower bound and upper bound in the branch and bound method?

Answer:

In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set (P,s) is an element of P which is greater than or equal to every element of S. the term lower bound is defined dually as an element of P which is lesser than or equal to every element of S. A set with an upper bound is said to be bounded from above by that bound, a set with a lower bound is said to be bounded from below by that bound.

A subset S of a partially ordered set P may fail to have any bounds or may have different upper and lower bounds. By transitivity, and element greater than or equal to an upper bound of S is again and upper bound of S, and any element lesser than or equal to any lower bound of S is again a lower bound of S. This leads to the consideration of least upper bounds: (or suprema) and greatest lower bounds (or infima).

The bounds of a subset S of a partially ordered set P may or may not be elements of S itself. If S contains an upper bound then that upper bound is unique and is called the greatest element of S. The greatest element of S (if it exists) is also the least upper bound of S. A special situation does occur when a subset is equal to the set of lower bounds of its own set of upper bounds. This observation leads to the definition of Dedekind cuts.

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The empty subset of a partially ordered set P is conventionally considered to be both bounded from above and bounded from below with every element of P being both an upper bound and lower bound of

And 5 are both lower bounds for the set {5, 10, 34, 13934}, but 8 is not. 42 is both upper and lower bound for that set.

Every subset of the natural numbers has a lower bound, since the natural numbers have a least element (0, or 1 depending on the exact definition of natural numbers). Every finite subset of a totally ordered set has both upper and lower bounds.

An infinite subset of the natural numbers cannot be bounded from above. An infinite subset of the integers may be bounded from below or bounded from above, but not both. An infinite subset of the rational numbers may or may not be bounded from below and may or may not be bounded from above.

Q3. (A) Give and algorithm to solve an assignment problem.

Answer:

Hungarian method algorithm is based on the concept of opportunity cost and is more efficient in solving assignment problems.

Adopt the following steps mentioned below to solve an AP using the Hungarian method algorithm.

Step1: Prepare row ruled matrix by selecting the minimum values for each row and subtract it from other elements of the row.

Step2: Prepare column reduced matrix by subtracting minimum value of the column from other values of that column.

Step3: Assign zero row-wise if there is only one zero in the row and cross (cancel) (X) other zeros in that column.

Step4: Assign column wise if there is only one zero in that column and cross other zeros in that row.

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Step5: Repeat steps 3 and 4 till all zeros are either assigned or crossed. If the number of assignments is equal to number of rows present, you have arrived at an optimal solution if not, then proceed to step6.

Step6: Mark ([]) the unassigned rows. Look for crossed zero in that row. Mark the column containing the crossed zero. Look for assigned zero in that column. Mark the row containing assigned zero. Repeat this process till all makings are over.

Step7: Draw straight line through unmarked rows and marked column. The number of straight line drawn will be equal t o number of assignments made.

Step8: Examine the uncovered elements. Select the minimum.

- Subtract it from uncovered elements- Add it tat the point of intersection of lines.- Leave the rest as it is.- Prepare a new table.

Step9: Repeat steps 3 to 7 till optimum assignment is obtained.

Step10: Repeat steps 5 to 7 till number of allocations = number of rows.

The assignment algorithm applies the concept of opportunity costs. The cost of nay kind of action or decision consists of the opportunities that are sacrificed in taking that action. If we do one thing, we cannot do another.

(B) Show that an assignment problem is a special case of transportation problem?

Answer:

The assignment problem is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research in mathematics. It consists of finding a maximum weight matching in a weighted bipartite graph.

In it most general form, the problem is as follows:

There are a number of agents and a number of tasks. Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent-task assignment. It is required to perform all tasks by assigning exactly one agent to each task in such a way that the total cost of the assignment is minimized.

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If the number of agents and tasks are equal and the total cost of the assignment for all tasks is equal to the sum of the costs for each agent (or the sum of the costs for each task, which is the same thing in this case), then the problem is called the Linear assignment problem. Commonly, when speaking of the Assignment problem without any additional qualification, then the Linear Assignment problem is meant.

The Hungarian algorithm is one of many algorithms that have been devised that solve the linear assignment problem within time bounded by a polynomial expression of the number of agents.

The assignment problem is a special case of the transportation problem, which is a special case of the minimum cost flow problem, which in turn is a special case of a linear program. While it still possible to solve any of these problems using the simplex algorithm each specialization has more efficient algorithms designed to take advantage of its special structure. If the cost function involves quadratic inequalities it is called the quadratic assignment problem.

Suppose that a taxi firm has three taxis (the agents) available, and three customer (the tasks) wishing to be picked up as soon as possible. The firm prides itself on speedy pickups, so far each taxi the cost of pickup a particular customer will depend on the time taken for the taxi to reach the pickup points. The solution to the assignment problem will be whichever combination of taxis and customers results in the least cost.

However, the assignment problem can be made rather more flexible than it first appears. In the above example, suppose that there are four taxi available, but still only three customers. Then a fourth dummy task can be invented, perhaps called “sitting still doing nothing”, with a cost of 0 for the taxi assigned to it. The assignment problem can then be solved in the usual ay and still give the best solution to the problem.

Similar tricks can be played in order to allow more tasks than agents, tasks to which multiple agents must be assigned (for instance, a group of more customers than will fit in one taxi), or maximizing profit rather than minimizing cost.

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Q4. What is the game in game theory? What are the properties of a game? Explain the ‘best strategy’ on the basis of minimax criterion of optimality.

Game theory studies strategic interaction between individuals in situations called games. Classes of these games have been given names. This is a list of the most commonly studied games. Games can have several features; a few of the most common are listed here.

Number of players: Each person who makes a choice in a game or who receives a payoff from the outcome of those choices is a player.

Strategies per player: In a game each player chooses from a set of possible actions, known as strategies.

If the number is the same for all players, it is listed here.

- Number of pure strategy Nash equilibrium: Nash equilibrium is a set of strategies which represents mutual best responses to the other strategies. In other words, if every player is playing their part of Nash equilibrium, no player has an incentive to unilaterally change his or her strategy. Considering only situations where players play a single strategy without randomizing ( a pure strategy) a game can have any number of Nash equilibrium.

- Sequential Game: A game is sequential if one player performs his/her actions after another; otherwise the game is a simultaneous move game.

- Perfect information: A game has perfect information if it is a sequential game and every player knows the strategies chosen by the players who preceded them.

- Constant sum: A game is constant sum if the sum of the payoffs to every player are the same for every set of strategies. In these games on player gains if and only if another player loses.

the properties of a game :Competitive SituationsCompetitive situations occur when two or more parties with conflicting interests operate.

The situations may occur as follows.

1 Marketing different brands of a commodity.Two (or more) brands of detergents (soaps) try to capture the market by adopting various methods (courses) such as ‘advertising through electronic media’, ‘providing cash discounts to consumers’ or ‘offering larger sales commission to dealers’.

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2 Campaigning for elections.Two (or more) candidates who contest an elections try to capture more votes by adopting various methods (courses) such as ‘campaigning through T.V.’, ‘ door to door campaigning’ or ‘campaigning through public meetings’.

3 Fighting military battles.Two forces fighting a war try to gain supremacy over one another by adopting various courses of action such as ‘direct ground attack on enemy camp’, ‘ground attack supported by aerial attack’ or ‘playing defensive by not attacking’.We consider each of the above situations to be a competitive game where the parties (players) adopt a course of action (play the game).

The “best strategy” is mixed strategy While playing a game, mixed strategy of a player is his predecision to choose his course of action according to certain pre-assigned probabilities.

Thus, if player A decides to adopt courses of action 1 2 A and A with perspective probabilities 0.4 and 0.6, it is mixed strategy.

Q5. What do you understand by

i) Queue discipline

Answer:

The pattern of selection for service from the pool of customers is of two types. The common pattern is to select in the order in which the customers arrive. “First come first served” is a common example. In issuing materials from a store’s inventory sometimes the storekeeper follows the “Last in First out” principle because of the convenience it offers for removal from stocks and handling.

There are queues according to priority to certain types of customers. This type of queuing can have two approaches; non pre-emptive and pre-emptive. In case of non pre-emptive priority, the customer getting service is allowed to continue with service till completion, even if a “priority customer” arrives midway during the service. This is a common form of priority. Pre-emptive priority involves stopping the service of the non priority customers as soon as the priority customer arrives.

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ii) Arrival Process

Answer:

Arrivals may originate from one or several sources referred to as the calling population. The calling population can be limited or unlimited. An example of a limited calling population may be that of a fixed number of machines that fail randomly. The arrival process consists of describing how customers arrive to the system. If Ai is the inter arrival time between the arrivals of the (i-1)th and ith customers we shall denote the mean (or expected) inter-arrival time by E(A) and call it () = 1/(E(A)) the arrival frequency.

iii) Service Process

Answer:

The service mechanism of a queuing system is specified by the number of servers (denoted by s), each server having its own queue or a common queue and the probability distribution of customer’s service time. Let Si be the service time of the ith customer, we shall denote the mean service time of a customer by E(S) and (E(S)) the service rate of server.

Q6. State two major reasons for using simulation. Explain the basic steps of Monte – Carlo simulation. Briefly describe the application in finance & accounting.

Answer:

Using simulation, an analyst can introduce the constants and variables related to the problem set-up the possible courses of action and establish criteria which act as measure of effectiveness.

The major reasons for applying simulation technique to OR problems may be listed as below:

It is an appropriate tool to use in solving a problem when experimenting on the real system.o Would be disruptive

o Would be too expensive

o Does not permit replication events

o Does not permit control over key variables

It is desirable tool for solving a business problem when a mathematical modelo Is too complex to solve

o Is beyond the capacity of available personnel

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o Is not detailed enough to provide information on all important decision variables

The major reasons for adopting simulation in place of other mathematical techniques are:o It may be the only method available, because it is difficult to observe the actual

reality.o Without appropriate assumption, it is impossible to develop a mathematical solution.

o It may be too expensive to actually observe the system.

o There may not sufficient time to allow the system to operate for a very long time.

It provides a trail-and-error movement towards the optical solution. The decision-maker selects an alternative, experiences the effect of the election, and then improves the selection. In this way, the selection is adjusted until it approximates the optical solution.

MONTE-CARLO Simulation

The Monte-Carlo method is a simulation technique in which statistical distribution functions are created by using a series of random numbers. This approach has the ability to develop many months or years of data in a matter of few minutes on a digital computer.

The method is generally used to solve the problems which cannot be adequately represented by mathematical models, or where solution of the mode, is not possible by analytical method.

The Monte-Carlo simulation procedure can be summarized in the following steps:

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Step 1: Define the problem:

Identify the objective of the problem, and Identify the main factors which have the greatest effect on the objectives of the

problem.

Step 2: Construct an appropriate model

Specify the variables and parameters of the model. Formulate the appropriate decision rules, i.e. state the conditions under which the

experiment is to be performed Identify the type of distribution that will be used – models use either theoretical

distributions or empirical distributions to state the patterns the occurrence associated with the variables.

Specify the manner in which time will change. Define the relationship between the variables and parameters.

Step 3: Prepare the model for experimentation

Define the starting conditions for the simulation

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Specify the number of runs of simulation to be made

Step 4: Using step 1 to 3, experiment with the model

Define a coding system that will correlate the factors defined in step 1 with the random number generator and create the random numbers to be used in the simulation.

Associate the generated random number with the factors defined in step 1 and coded in step 4.

Step 5: Summarize and examine the results obtained in step 4.

Step 6: Evaluate the results of the simulation

Step 7: formulate proposals for advice to management on the course of action ot be adopted and modify the model, if necessary.

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SK-MB0048 (SET-2)