quantitative technics

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INTRODUCTION

Managerial activities have become complex and it is necessary tomake right decisions to avoid heavy losses. Whether it is amanufacturing unit, or a service organization, the resources have to

be utilized to its maximum in an efficient manner. The future isclouded with uncertainty and fast changing, and decision-making acrucial activity cannot be made on a trial-and-error basis or byusing a thumb rule approach. In such situations, there is a greaterneed for applying scientific methods to decision-making to increasethe probability of coming up with good decisions. QuantitativeTechnique is a scientific approach to managerial decision-making.The successful use of Quantitative Technique for managementwould help the organization in solving complex problems on time,with greater accuracy and in the most economical way. Today,several scientific management techniques are available to solvemanagerial problems and use of these techniques helps managersbecome explicit about their objectives and provides additionalinformation to select an optimal decision. This study material ispresented with variety of these techniques with real life problemareas.

ABOUT QUANTITATIVE TECHNIQUE

Quantitative Techniques adopt a scientific approach to decision-

making. In this approach, past data is used in determining decisionsthat would prove most valuable in the future. The use of past data ina systematic manner and constructing it into a suitable model forfuture use comprises a major part of scientific management. Forexample, consider a person investing in fixed deposit in a bank, or inshares of a company, or mutual funds, or in Life InsuranceCorporation. The expected return on investments will varydepending upon the interest and time period. We can use thescientific management analysis to find out how much theinvestments made will be worth in the future. There are many

scientific method software packages that have been developed todetermine and analyze the problems.

Quantitative Technique is the scientific way to managerial decision-making, while emotion and guess work are not part of the scientificmanagement approach. This approach starts with data. Like rawmaterial for a factory, this data is manipulated or processed intoinformation that is valuable to people making decision. Thisprocessing and manipulating of raw data into meaningfulinformation is the heart of scientific management analysis.


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METHODOLOGY OF QUANTITATIVE TECHNIQUES

The methodology adopted in solving problems is as follows:

Figure: Methods of Quantitative techniques


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SCOPE OF QUANTITATIVE TECHNIQUE

The scope and areas of application of scientific management arevery wide in engineering and management studies. Today, there area number at quantitative software packages available to solve the

problems using computers. This helps the analysts and researchersto take accurate and timely decisions. This book is brought out withcomputer based problem solving. A few specific areas arementioned below.

Finance and Accounting: Cash flow analysis, Capital budgeting,Dividend and Portfolio management, Financial planning.

Marketing Management: Selection of product mix, Salesresources allocation and Assignments.

Production Management: Facilities planning, Manufacturing,Aggregate planning, Inventory control, Quality control, Workscheduling, Job sequencing, Maintenance and Project planning andscheduling.

Personnel Management: Manpower planning, Resource

allocation, Staffing, Scheduling of training programmes. General Management: Decision Support System andManagement of Information Systems, MIS, Organizational designand control, Software Process Management and KnowledgeManagement.

LINEAR PROGRAMMING

INTRODUCTION

Linear programming is a widely used mathematical modelingtechnique to determine the optimum allocation of scarce resourcesamong competing demands. Resources typically include rawmaterials, manpower, machinery, time, money and space. Thetechnique is very powerful and found especially useful because of itsapplication to many different types of real business problems inareas like finance, production, sales and distribution, personnel,marketing and many more areas of management. As its name


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implies, the linear programming model consists of linear objectivesand linear constraints, which means that the variables in a modelhave a proportionate relationship. For example, an increase inmanpower resource will result in an increase in work output.

DEFINITION

Linear programming is a mathematical technique for determiningthe optimum allocation of resources and obtaining a particularobjective when they are alternative uses of resources : money ,manpower, machine and other facilities. The objecttive in resourceallocation may be cost minimization or inversely profit maximization.The technique of linear programming is applicable to problems inwhich the total effectiveness can be expressed as a linear functionof individual allocations and the limitations on resources give rise to

linear equalities of the individual allocations.

APPLICATIONS AREAS OF LINEAR PROGRAMMING

The use of LP is made in regard to the problems of allocation,assignment, transportation etc. But the most important of these isthat of allocation of scare resources on which we shall concentrate.Some allocation problems are as follows:

Devising of a production schedule that could satisfy future demandsfor a firm product and at the same minimize production costs.

Choice of investment from a variety of shares and debentures so asto maximize return on investment.

Allocation of a limited publicity budget on various heads in order tomaximize its electiveness.

Selection of the product-mix to make the best use of availablerecourses like machines man-hours, etc, with a view to maximizeprofits.

Selecting the advertising mix that will maximize the benefit subjectto the total advertising budget, Linear programming can beeffectively applied.


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Determination of the distribution system that will minimizetransportation costs from several warehouses to various markets.

Designing, routing and assignment problems.

Manufacturing problems. To find number of items of each type thatshould be manufactured so as to maximize the profit subject toproduction restrictions imposed by limitations on the use ofmachinery and labor.

Transportation Problems . To find the least expensive way oftransporting shipments from the warehouse to customers.

Diet Problems . To determine the minimum requirement of nutrientssubject to availability of foods and their prices.

Blending Problems . To determine the optimum amount of severalconstituents to be used in producing a set of products whiledetermining the optimum quantity of each product to produce.

Assembling Problems. To have the best combination of basiccomponents to produce goods according to certain specifications.

Production Problems. To decide the production schedule to satisfydemand and minimize cost in face of fluctuating rates and storageexpenses.

Job assigning Problems. To assign job o workers for maximumeffectiveness and optimum results subject to restrictions of wagesand other costs.

Trim-Loss Problems. To determine the best way to obtain a varietyof smaller rolls of paper from a standard width of roll that is kept instock and at the same time, minimize wastage.

ESSENTIALS OF LINEAR PROGRAMMING MODEL

For a given problem situation, there are certain essential conditionsthat need to be

solved by using linear programming.

1. Limited resources : limited number of labour, material equipmentand finance

2. Objective : refers to the aim to optimize (maximize the profits orminimize the costs).

3. Linearity : increase in labour input will have a proportionateincrease in output.

4. Homogeneity : the products, workers' efficiency, and machinesare assumed to be identical.


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5. Divisibility : it is assumed that resources and products can bedivided into fractions. (in case the fractions are not possible, likeproduction of one-third of a computer, a modification of linearprogramming called integer programming can be used).

PROPERTIES OF LINEAR PROGRAMMING MODEL

The following properties form the linear programming model:

1. Relationship among decision variables must be linear in nature.

2. A model must have an objective function.

3. Resource constraints are essential.

4. A model must have a non-negativity constraint.GENERAL LINEAR PROGRAMMING MODEL

A general representation of LP model is given as follows:

Maximize or Minimize, Z = p1 x1 + p2 x2 pn xn

Subject to constraints,

w11 x1 + w12 x2 + w1n xn or = or w1 (i)

w21 x1 + w22 x2 w2n xn or = or w2 (ii)

. . . .

. . . .

wm1 x1 + wm2 x2 +wmn xn or = wm (iii)

Non-negativity constraint,

xi o (where i = 1,2,3 ..n)

GRAPHICAL METHOD

Linear programming problems with two variables can berepresented and solved graphically with ease. Though in real-life,the two variable problems are practiced very little, the interpretationof this method will help to understand the simplex method.

The graphic method of solving linear programming problems

consists of the following steps:


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Step:1 Graph the constraints

Step:2 Identify the feasible region

Step:3 Locate the solution points

solution method of solving the problem through graphical method isdiscussed with an example given below.

Example : A company manufactures two types of boxes, corrugatedand ordinary cartons. The boxes undergo two major processes:cutting and pinning operations. The profits per unit are Rs. 6 and Rs.4 respectively. Each corrugated box requires 2 minutes for cuttingand 3 minutes for pinning operation, whereas each carton boxrequires 2 minutes for cutting and 1 minute for pinning. Theavailable operating time is 120 minutes and 60 minutes for cutting

and pinning machines. Determine the optimum quantities of the twoboxes to maximize the profits.

The objective function is,

Zmax = 6x1 + 4x2

Constraints:The available machine-hours for each machine and thetime consumed by

each product are given.

Therefore, the constraints are,

2x1 + 3x2 120 ..........................(i)

2x1+ x2 60 ..........................(ii)

where x1, x2 0

Graphical Solution: As a first step, the inequality constraints areremoved by replacing equal to sign to give the following equations:

2x1 + 3x2 = 120 .......................(1)

2x1 + x2 = 60 .......................(2)

Find the co-ordinates of the lines by substituting x1 = 0 and x2 = 0in each equation. In equation (1), put x1 = 0 to get x2 and viceversa

2x1 + 3x2 = 120

2(0) + 3x2 = 120, x2 = 40

Similarly, put x2 = 0,


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2x1 + 3x2 = 120

2x1 + 3(0) = 120, x1 = 60

The line 2x1 + 3x2 = 120 passes through co-ordinates (0, 40) (60,

0).The line 2x1 + x2 = 60 passes through co-ordinates (0,60)(30,0).

The lines are drawn on a graph with horizontal and vertical axisrepresenting boxes x1 and x2 respectively.

Figure: Graphical presentation of a problem

SIMPLEX METHOD


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AIMS AND OBJECTIVES

In the previous lesson we have learnt linear programming with the

help of graphical now we will learn the linear programming with thehelp of Simplex Method using minimization and maximizationproblems and the degeneracy in LP problems and also the Dualityand Sensitivity Analysis.

INTRODUCTION

In practice, most problems contain more than two variables and areconsequently too large to be tackled by conventional means.Therefore, an algebraic technique is used to solve large problemsusing Simplex Method. This method is carried out through iterative

process systematically step by step, and finally the maximum orminimum values of the objective function are attained. The basicconcepts of simplex method are explained using the Example 1.8 ofthe packaging product mix problem illustrated in the previouschapter. The simplex method solves the linear programmingproblem in iterations to improve the value of the objective function.The simplex approach not only yields the optimal solution but alsoother valuable information to perform economic and 'what if'analysis.

DEFINITION

The simplex method is a method which goes from one Basic FeasibleSolution (BFS) or extreme point of the feasible region of an Linearprogramming problem expressed in tableau form to another BFS, insuch a way as to continually increase or decrease the value of theobjective function until optimality is reached. The simplex methodmoves from one extreme point to one of its neighboring extremepoint.

steps of the simplex method for minimization problem:To choose the variable with a negative Cj-Zj .

To determine the row to be replaced by selecting the one with thesmallest quantity to pivot column substitute rate ratio.

To calculate new values for the pivot row.

To compute the new values for errw.

Finally, to compute the Zj, Cj-Zj values for this tableau. If there areany Cj-Zj numbers less then 0, return to step 1.

ADDITIONAL VARIABLES USED IN SOLVING LPP


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Three types of additional variables are used in simplex method suchas,

(a) Slack variables (S1, S2, S3..Sn): Slack variables refer to theamount of unused resources like raw materials, labour and money.

(b) Surplus variables (-S1, -S2, -S3..-Sn): Surplus variable is theamount of resources by which the left hand side of the equationexceeds the minimum limit.

(c) Artificial Variables (a1, a2, a3.. an): Artificial variables aretemporary slack variables which are used for purposes ofcalculation, and are removed later.

Types of Additional variables

DUALITY AND SENSITIVITY

INTRODUCTION

For every Linear Programming Problem, whether maximization orminimization, there is another linear programming problemassociated with it, which is the mirror image of the orginal problemand is on the same data. The orginal problem is called the primalproblem and the other is called the dual problem. If the primalproblem deals with maximization then the dual problem will be ofminimization. Conversely, if the primal problem is of minimizationthen its dual will be dealing with maximization. One problem can bederived from the other and vice verse because both are formulatedfrom the same set of data, although arranged differently in theirmathematical presentation. Either one of the primal, then the otherwould automatically become its dual or the dual of the dual problemis the primal.

The selection of the primal problem depends on the case with whicha problem can be solved, i.e, the one that is easier to solve is


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designated as primal. However, under certain situations where allthe assumptions and requirements of linear programming problemsare met, the dual instead of the primal may offer the advantage offewer iteration, thus shortening the computation time is reachingthe optimal solution. The dual has an additional advantage ofchecking on the correctness of the primal solution because both theprimal and dual approach must arrive at the same optimal solution.

ADVANTAGE OF DUALITY

The dual solution has some powerful advantages over the primalsolution in certain situation. If a primal problem has a large numberof constraints and a small number of variables then the primalsimplex solution becomes more complex. For example if there arethree products all of which have to be processed sequentially on

eight constraints requiring at least eight slack variables and thiswould result in the initial simplex tableau having at the minimumeight rows to handle. However, if we convert it into a dual problemwhere the columns become rows and rows become column, we willhave the initial simplex tableau with only three rows to deal with,which obviously is easier.

Another advantage is that, since the optimal solution to theobjective function is the same for both primal and dual, a dualsolution can be used to check the accuracy of the primal solution.

Moreover, the management can draw economic interpretation fromduality which are very helpful to making decisions.

Construction of a dual problem:

The rules for constructing the dual from the primal ( or primal fromthe dual ) are:

a. If the objectives of one problem is to be maximized theobjective of the other is to be minimized.

b. The maximization problem should have all constrains andthe minimization problem has all constraints.

c. All primal and dual variables must be non-negative( 0 )

d. The element of right hand side of the constraints in oneproblem are the respective coefficients of the objectivefunction in the other problem.

e. The matrix of constraints coefficients for one problem is thetranspose of the matrix of constraint coefficients for the other

problem.


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Shadow price:

The shadow price of a resource is the unit price that is equal to theincrease in profit to be realized by one additional unit of theresource. Thus the dual variable is also referred to as the shadow

price or computed price of a resource. This is the highest price themanufacturer would be willing to pay for the resource.

Economic interpretation of the dual:

The economic interpretation of the dual problem can be explainedby an example. A furniture manufacturing company plans to makechairs and tables form mahogany wood that is available in limitedquantities. A chair requires five feet of mahogany board and a table15 feet . it takes 10man-hours to make a chair and 20 man-hours to

produce a table. There are 450 feet of wood and 500 hours of labouravailable. Per unit profit is $40 and $90, respectively, for chairs andtables. The objective of the firm, of course, is to maximize profit withthe giver resources.

There are two issues that can be raised with the problem on hand.The first obviously is the optimal production plan with the given setof resources, which can be obtained through the simplex procedure.The second question has to do with few much should be paid foradditional unit of resources, which is our case are mahogany woodand labour man-hours. The answer to the second question lies inexamine the co-efficient of dual variable. In order to see this point,let us formulate the primal of the problem set forth above.

Define x1= number of chairs produced

X2= number of tables produced.

Objective function is to maximize profit,

That means

Maximize z(x)= 40x1 +90x2

Subject to 5x1 + 15x245010x1 + 20x2500

X1,x2 0

The dual of this primal problem is :

Minimize z(y)=450y1 + 500y2

Subject to 5y1 + 10y240

15y1 + 20y290

Y1,y20.


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SENSITIVITY ANALYSIS

Sensitivity analysis (SA) is the study of how the uncertainty in theoutput of a model (numerical or otherwise) can be apportioned todifferent sources of uncertainty in the model input. A relatedpractice is 'uncertainty analysis' which focuses rather on quantifyinguncertainty in model output. Ideally, uncertainty and sensitivityanalysis should be run in tandem.A technique used to determine how different values of anindependent variable will impact a particular dependent variableunder a given set of assumptions. This technique is used

within specific boundaries that will depend on one or more inputvariables, such as the effect that changes in interest rates will haveon a bond's price.Sensitivity analysis is a way to predict the outcome of a decision if asituation turns out to be different compared to the key prediction(s).Sensitivity analysis is very useful when attempting to determine theimpact the actual outcome of a particular variable will have ifit differs from what was previously assumed. By creating a given setof scenarios, the analyst can determine how changes in onevariable(s) will impact the target variable.

For example, an analyst might create a financial model that willvalue a company's equity (the dependent variable) given theamount of earnings per share (an independent variable) thecompany reports at the end of the year and the company's price-to-earnings multiple (another independent variable) at that time.

In more general terms uncertainty and sensitivity analysisinvestigate the robustness of a study when the study includes someform ofstatistical modelling. Sensitivity analysis can be useful tocomputer modelers for a range of purposes. including:

Support decision making or the development of recommendationsfor decision makers (e.g. testing the robustness of a result);

Enhancing communication from modellers to decision makers (e.g.by making recommendations more credible, understandable,compelling or persuasive);
http://en.wikipedia.org/wiki/Statistical_modellinghttp://en.wikipedia.org/wiki/Statistical_modelling


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Increased understanding or quantification of the system (e.g.understanding relationships between input and output variables);and

Model development (e.g. searching for errors in the model).

Let us give an example: in any budgeting process there are alwaysvariables that are uncertain. Future tax rates, interest rates, inflationrates, headcount, operating expenses and other variables may notbe known with great precision. Sensitivity analysis answers thequestion, "if these variables deviate from expectations, what will theeffect be (on the business, model, system, or whatever is beinganalyzed)?"

Applications

Sensitivity analysis can be usedTo simplify models

To investigate the robustness of the model predictions

To play what-if analysis exploring the impact of varying inputassumptions and scenarios

As an element of quality assurance (unexpected factors sensitivitiesmay be associated to coding errors or misspecifications).

It provides as well information on:

Factors that mostly contribute to the outputvariability

The region in the space of input factors for which the model outputis either maximum or minimum or within pre-defined bounds (seeMonte Carlo filtering above)

Optimal or instability regions within the space of factors for usein a subsequent calibration study

Interaction between factors

Sensitivity Analysis is common in physics andchemistry, in financial applications, risk analysis, signal

processing, neural networks and any area where models aredeveloped. Sensitivity analysis can also be used in model-based policy assessment studies. Sensitivity analysis can be used toassess the robustness ofcomposite indicators, also known asindices, such as the Environmental Performance Index.
http://en.wikipedia.org/wiki/Outputhttp://en.wikipedia.org/wiki/Optimization_(mathematics)http://en.wikipedia.org/wiki/Calibrationhttp://en.wikipedia.org/wiki/Interaction_(statistics)http://en.wikipedia.org/wiki/Financialhttp://en.wikipedia.org/wiki/Signal_processinghttp://en.wikipedia.org/wiki/Signal_processinghttp://en.wikipedia.org/wiki/Neural_networkshttp://en.wikipedia.org/w/index.php?title=Policy_assessment_studies&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Composite_indicators&action=edit&redlink=1http://en.wikipedia.org/wiki/Environmental_Performance_Indexhttp://en.wikipedia.org/wiki/Outputhttp://en.wikipedia.org/wiki/Optimization_(mathematics)http://en.wikipedia.org/wiki/Calibrationhttp://en.wikipedia.org/wiki/Interaction_(statistics)http://en.wikipedia.org/wiki/Financialhttp://en.wikipedia.org/wiki/Signal_processinghttp://en.wikipedia.org/wiki/Signal_processinghttp://en.wikipedia.org/wiki/Neural_networkshttp://en.wikipedia.org/w/index.php?title=Policy_assessment_studies&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Composite_indicators&action=edit&redlink=1http://en.wikipedia.org/wiki/Environmental_Performance_Index


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DECISION THEORY

Decision theory studies rational choices. It is used both to predictand explain actual choices and to improve actual decision making.The first purpose is called positive theory and the second is callednormative theory. While our primary aim is to predict and explainactual choices (positive theory), understanding how to improveactual decision making (normative theory) helps us to betterunderstand decision theory for our applications, as well.

Anatomy of Decision Theory:The steps of decision theory are:

1) Decision making environment

2) Objective of a decision maker

3) Alternative plans of actions

4) Decision pay off

Decision making environment

a) Deterministic situation or state of certaintyb) Risk Situation

c) Situation of Un certainty

Steps of Deterministic situation or state of certainty

a) Determine the alternative course of action

b) Calculate the pay offs, one for each course of action

c) Select the alternatives with largest profit or smallest costs

either by the method of complete enumeration or with the aidof appropriate mathematical models

Rules for decision making under risk:

There are several rules and techniques for decision making underrisk. Some important ones are:

1) Maximum likehood rule

2) EMV: formula: ( pay off 1st state of nature)*( Probability of the1st state of nature)+ ( pay off 2nd state of nature)*(Probabilityof the 2nd state of nature)+ ( pay off last state of


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nature)*(Probability of the last state of nature) and EOLcriterion

3) Decision trees

4) Utility functions

5) Bayestan Decision rule

6) Marginal Analysis

Decision tree analysis:

1)Define the problem

2)Structure or draw the decision tree

3)Assign probabilities to the states of nature

4)Estimate pay offs for each possible combination of alternativesand states of nature

5)Solve the problem by computing EMV for each state of naturenode. This is done by working backward.

Applications of Decision theory:

1) It is used both to predict and explain actual choices and toimprove actual decision making.

2) Decision theory may also be used to predict equilibriumachieved by some evolutionary process. In particular, thisevolutionary process may be the biological evolutionaryprocess or a cultural evolutionary process.

3) It is used to consider profit-maximizing decisions thatfirms make in perfectly competitive markets

4) the decision theory toolbox considers the additional toolsnecessary to deal with a world with uncertainty. In somedetail, basic rules that an individual or organization shouldfollow if he were rational in the face of uncertainty inbusiness.

TRANSPORTATION MODEL

INTRODUCTION

Transportation is a special class of linear programming problemwidely applied in business area. Transportation problem deals with


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the distribution of goods from several points of supply (sources) to anumber of points of demand (destination) at minimum cost.Usually, we have a given capacity goods at each sources and agiven requirement for the goods at destination. The objective ofsuch a problem is to schedule shipments from sources to destinationso that total transportation and production cost are minimized.Transportation models can also be used when a firm is trying todecide where to locate a new facility. Before going a newwarehouse, factory, or sales office, it is good practice to consider.

APPLICATION OF TRANSPORTATION MODEL IN BUSINESSAREA

1. When total supply equals total demand, transportation applied inbalancing the demand and supply for minimizing total cost.

2. Transportation model applied to optimizing minimum total costwhen supply is not balanced with demand.

3. Transportation model applied to allocation of total finished goodscosts.

4. Transportation model applied very accurately in such kind ofsituation, when it is not possible to transport goods from certainsources to certain destinations, due to unfavorable weatherconditions, road hazards, etc.

5. Transportation model applied to reducing total shipment cost by

satisfying total supply and demand condition.

METHODS FOR BASIC FEASBLE SOLUTION:

There are several methods to obtain an initial solution. Discussbelow-

1 .NORTH-WEST CORNER METHOD:

A systematic procedure for establishing an initial feasible solution totransportation problem. The steps are-

Step 1.Select the north-west (upper left hand) corner for a shipment.

Step 2.Make as large a shipment as possible in the north-westcorner or cell. This operation will completely exhaust either thesupply available at one source or the demand at one destination.

Step 3.Adjust the supply and demand values. If all supply anddemand values are exhausted then stop, otherwise move one cell tothe right or one cell down depending on the supply and demandvalues, go to step 2.


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2. VOGELS APPROXIMATION METHOD (VAM):

An algorithm used to find a relatively efficient initial feasible solutionto a transportation problem. This method gives better initial solutionin terms of less transportation cost and do not the cost. The steps in

VAM are as follows:Step 1.Given a balanced transportation table calculate penaltynumbers for each row and column by taking the difference betweenthe lowest and next lowest unit transportation costs. This differenceindicates the penalty or extra cost which has to be paid for notassigning an allocation to the cell with the minimum transportationcost.

Step 2.Select the row or column having the largest penalty number.If there is a tie then it can be broken by selecting the cell where

maximum allocation can be made.Step 3.Assigning the maximum number of units possible to thelowest cost in the row or column selected in step 2. Adjust theavailabilities and requirements after allocation at sources anddestinations. Eliminate from further consideration a row or columnfor which availability and requirement conditions are met.

Step 4.Repeat steps 1 to 3 until the entire available capacity atvarious sources and requirements at various destinations have beenmet.

3. INSPECTION METHOD:

It is another method used in transportation problem for initialsolution. Here at first consider lowest cost cell and then completethis cell with supply and demand. This process is going on it untilcompletion of allocated cell.

METHOD FOR OPTIMUM SOLUTION:

After the computation of an initial solution the next step is to checkto optimality. An optimum solution is one in which there is no otherof transportation routes (allocation) that will reduce the totaltransportation costs.

1 .STEPPING STONE METHOD:

An iterative technique for moving from an initial feasible solution toan optimal solution in transportation problem. Starting from the

initial feasible solution the following steps are followed-


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Step 1.Compute improvement index for each of the empty cells. Thisis computed by calculating the opportunity costs of an empty cell.This means that if we shift one unit from a cell containing positiveshipment to the empty cell that will be the net cost. If all empty cellshave positive improvement index, then the given solution is anoptimum solution.

Step 2.If there are several empty cells with negative improvementsindices, then we select the cell having the largest negativeimprovement index and shift the maximum possible units to that cellwithout violating the supply and demand constraints. After it, againgo to step 1.

2. THE MODIFIED DISTRIBUTION METHOD (MODI):

A technique that is used to evaluate the empty cells in a

transportation problem. The methods involves the following stepStep 1.For an initial basic solution with m + n n occupied cells,calculate the Ui and Vj for rows and columns. To start with any oneof uis or vjs is given the value zero. It is better to assign zero for aparticular ui or vj where there are maximum number of allocations ina row or column respectively as it will considerably reduce thearithmetic work. By using the relation cij=ui + vj for all occupiedcells (i ,j ) the computation of uis and vjs for others rows andcolumns can be completed.

Step 2.For unoccupied cells, calculate opportunity cost by using therelation:

ij = cij-(ui+vj) for all i and j

Step 3.Examine sign of each ij.

a) If 0, then current basic feasible solution is optimum.

b) If 0, then current basic feasible solution will remainunaffected but an alterntive solution exist.

c) If one or more 0, then an optimum solution can be obtainedby entering unoccupied cell (i, j) in the basis. An unoccupied cellhaving the largest negative value of is chosen for entering intothe new transportation schedule.

Step 4.Construct a closed path (loop) for the unoccupied cell withlargest negative opportunity cost. Start the close loop with theselected unoccupied cell make a plus sign (+) in this cell, trace apath along the rows (or column) to an occupied cell, mark the cornerwith minus sign (-) and continue down the column (or row ) to anoccupied cell and mark the corner with alternatively plus sign (+)


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and minus sign (-). Close the path back to the selected unoccupiedcell.

Step 5.Locate the smallest quantity allocated to a cell marked with aminus sign. Allocated this value to the selected unoccupied cell and

add it to other occupied cells marked with plus signs and subtract itfrom the occupied cells marked with minus signs.

Step 6.Obtain a new solution after allocating units to the unoccupiedcell according to step 5 and calculate the new total transportationcost.

Step 7.Test the revised solution for optimality.

NETWORK ANALYSISPERT & CPM

CONCEPT AND MEANING

Network Analysis (N.A.) is an important concept that deals withproject management. A project is a well-defined set of jobs, tasks, oractivities, which must be completed in a specified time andsequence order. The jobs, tasks or activities are interrelated andrequire resources such as money, material, personnel and otherfacilities. Therefore, the major focus of project management is toschedule jobs,tasks or activities in an efficient way so as to reduceor have minimum cost of its completion within the specified period.Network analysis, sometimes called network planning andscheduling techniques, is a set of operations research techniquesneeded and useful for planning, scheduling and controlling large andcomplex projects. The diagrammatical representation of thesetechniques, which consists of arrows (activities) and nodes (events)is known as network diagram.

Possible areas of application of network analysis include:

(a) Construction projects such as building highways, house, bridge,e.t.c;

(b) Maintenance or planning of oil refinery, ship repair and otherlarge operations;

(c) Preparation of bids and proposals for large projects;

(d) Development of new weapon/system or manufacturing system;

(e) Manufacturing and assemblage of large items such as airplanes,ships and computers; and


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(f) Simple projects such as home remodeling, moving to a newhouse, house cleaning and painting.

Limitations of the method

Differences may occur about the activities that precede eachother.

Duration of activities are estimates it is unlikely that thedurations of all activities are known. They are generally basedon experience on similar projects, the availability and quality ofstaff and the level of resources available.

It is often necessary to make decisions on employing extra

staff to accelerate progress of the project or specific elementsof it. This increase in cost has to be off-set against potentialsavings which may accrue. This may arise due to unforeseencircumstances.

Objectives of network analysis

1. Minimization of total cost

2. Minimization of total time

3. Minimization of time for a given cost4. Minimization of a given total time

5. Minimization of idle resources

6. Network analysis can also be employed to minimize production

delays , interruptions and conflicts

NETWORK DIAGRAM

The pictorial representation of the interrelation of the variousactivities and events concerning a project is known as networkdiagram.

Network models:

there are four network models that can be used to solve a verity ofproblem.

Minimum-spanning tree technique:

This minimal-spanning tree technique connects nodes at a

minimum distance.


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Steps for the minimal-spanning tree:

There are five steps for the minimal-spanning tree problem.

1. Select any node in the network.

2. Connect this node to the nearest node that minimizes the total

distance.3. Considering all of the nodes that are now connected

4. Find and connect the nearest mode that is not connected.

5. Repeat the third step until all nodes are connected

Maximal-flow technique :

The maximal-flow technique finds the most that can flow througha network.

Four steps of the Maximal-flow technique:

1. Pick any path from the start (source) to the finish (sink) withsome flow.

2. Find the arc on this with the smallest flow capacity available.

3. For each node on this path, decrease the flow capacity in thedirection of flow by the amount c .

4. Repeat these steps until an increase in flow is no longerpossible.

Shortest-route technique:

minimizes the distance though a network.

Steps of the shortest-Route technique

1. Find the nearest node to the origin (plant).

2. Find the next-nearest node to the origin (plant),.

3. Repeat this process until you have gone through the entirenetwork.

All-Node-Pairs Shortest path:

The floyd-Warshall algorithm is useful for finding the shortestpath between all pairs of nodes in a network.


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Three steps of the All-Node-pairs shortest path Algorithm :

1. Initialize distance and node adjacency matrices.

2. Check the distance matrix for shorter paths between nodesusing node I as an intermediate node.

3. Repeat the second step using the other nodes in sequence asthe intermediate node

RULES IN CONSTRUCTING A NETWORK

1. No single activity can be represented more than once in anetwork. The length of an arrow has no significance.

2. The event numbered 1 is the start event and an event withhighest number is the end event. Before an activity can beundertaken, all activities preceding it must be completed. That

is, the activities must follow a logical sequence (or interrelationship) between activities.

3. In assigning numbers to events, there should not be anyduplication of event numbers in a network.

4. Dummy activities must be used only if it is necessary to reducethe complexity of a network.

5. A network should have only one start event and one endevent.

Some conventions of network diagram are shown in

Figures below:Some Conventions followed in making Network Diagrams


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PROCEDURE FOR NUMBERING THE EVENTS USING FULKERSON'SRULE

Step1: Number the start or initial event as 1.

Step2: From event 1, strike off all outgoing activities. This wouldhave made one or more events as initial events (event which do

not have incoming activities).

Number that event as 2.

Step3: Repeat step 2 for event 2, event 3 and till the end event.

The end event must have the highest number.

Example : Draw a network for a house construction project. The

sequence of activities with their predecessors is given in thefollowing table, below.

METHODS OF NETWORK ANALYSIS

The two well known methods of network analysis are:

(a) Critical path method (CPM); and

(b) Program evaluation review technique (PERT).Both methods (CPM and PERT) focus their procedures on identifyingthe critical path. The critical path takes the longest or maximumtime duration of the project; while activities on the critical path arecalled the critical activities.

Some major features of both methods are discussed as follows:

(a) CPM

(i) CPM is used for activities which are deterministic;

(ii) It is an activity-oriented technique;


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(iii) It is good in establishing a trade-off for optimum balancingbetween scheduled time and cost of the project; and

(iv) CPM was developed mostly for maintenance and constructionprojects.

(b) PERT

(i) It is an event-oriented technique;

(ii) It handles probablististic type of data; and

(iii) It has activities of non-receptive nature and therefore used forone time project.

PERT:

This was developed by the US Navy for the planning and control

of the Polaris missile program and the emphasis was on

completing the program in the shortest possible time. In addition

PERT had the ability to cope with uncertain activity completion

times.

In PERT activities are shown as a network of precedence

relationships using activity-on-arrow network construction

1. Multiple time estimates2. Probabilistic activity times

PERT is used in project management for non-repetitive jobs

(research and development work), where the time and cost

estimates tend to be quite uncertain. This technique uses

probabilistic time estimates.

CPM:

CPM (Critical Path Method) was developed by Du Pont and theemphasis was on the trade-off between the cost of the project

and its overall completion time (e.g. for certain activities it may

be possible to decrease their completion times by spending more

money - how does this affect the overall completion time of the

project.

In CPM activities are shown as a network of precedence

relationships using activity-on-node network construction

1. Single estimate of activity time

2. Deterministic activity times


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CPM is used in production management for the jobs which are

repetitive in nature where the activity time estimates can be

predicted with considerable certainty due to the existence of past

experience.

Project management generally consists of three phases.

Planning: Planning involves setting the objectives of the project.

Identifying various activities to be performed and determining the

requirement of resources such as men, materials, machines, etc.

The cost and time for all the activities are estimated, and a

network diagram is developed showing sequential

interrelationships (predecessor and successor) between various

activities during the planning stage.

Scheduling: Based on the time estimates, the start and finish

times for each activity are worked out by applying forward and

backward pass techniques, critical path is identified, along with

the slack and float for the non-critical paths.

Controlling: Controlling refers to analyzing and evaluating the

actual progress against the plan. Reallocation of resources,

crashing and review of projects with periodical reports are carried

out.

Far more than the technical benefits, it was found that PERT/CPM

provided a focus around which managers could brain-storm and

put their ideas together. It proved to be a great communication

medium by which thinkers and planners at one level could

communicate their ideas, their doubts and fears to another level.

Most important, it became a useful tool for evaluating the

performance of individuals and teams.

There are many variations of CPM/PERT which have been usefulin planning costs, scheduling manpower and machine time.

CPM/PERT can answer the following important questions:

How long will the entire project take to be completed?

What are the risks involved?

Which are the critical activities or tasks in the project whichcould delay the entire project if they were not completed ontime?


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Is the project on schedule, behind schedule or ahead ofschedule?

If the project has to be finished earlier than planned, what isthe best way to do this at the least cost?

PERT/CPM NETWORK COMPONENTS

PERT / CPM networks contain two major components

i. Activities, and

ii. Events

Activity: An activity represents an action and consumption ofresources (time, money, energy) required to complete a portion

of a project. Activity is represented by an arrow.

An Activity

Event: An event (or node) will always occur at the beginning and

end of an activity. The event has no resources and is represented

by a circle. The ith event and jth event are the tail event andhead event respectively.

An Event

Merge and Burst Events

One or more activities can start and end simultaneously at an

event.

Preceding and Succeeding Activities

Activities performed before given events are known as preceding

activities, and activities performed after a given event are known

as succeeding activities.


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Preceding and Succeeding Activities

Activities A and B precede activities C and D respectively.

Dummy Activity

An imaginary activity which does not consume any resource and

time is called a dummy activity. Dummy activities are simply

used to represent a connection between events in order to

maintain a logic in the network. It is represented by a dotted line

in a network, see Figure.

Dummy Activity

CRITICAL PATH ANALYSIS

The critical path for any network is the longest path through the

entire network. Since all activities must be completed to

complete the entire project, the length of the critical path is also

the shortest time allowable for completion of the project. Thus if

the project is to be completed in that shortest time, all activities

on the critical path must be started as soon as possible.These activities are called critical activities. If the project has to

be completed ahead of the schedule, then the time required for

at least one of the critical activity must be reduced. Further, any

delay in completing the critical activities will increase the project

duration.

The activity, which does not lie on the critical path, is called non-

critical activity. These non-critical activities may have some slack

time. The slack is the amount of time by which the start of anactivity may be delayed without affecting the overall completion


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time of the project. But a critical activity has no slack. To reduce

the overall project time, it would require more resources (at extra

cost) to reduce the time taken by the critical activities to

complete.

Scheduling of Activities: Earliest Time and Latest Time

Before the critical path in a network is determined, it is necessary

to find the earliest and latest time of each event to know the

earliest expected time (TE) at which the activities originating

from the event can be started and to know the latest allowable

time (TL) at which activities terminating at the event can be

completed.

Forward Pass Computations (to calculate Earliest, TimeTE)

Procedure

Step 1: Begin from the start event and move towards the end

event.

Step 2: Put TE = 0 for the start event.

Step 3: Go to the next event (i.e node 2) if there is an incoming

activity for event 2, add calculate TE of previous event (i.e event

1) and activity time.Note: If there are more than one incoming activities, calculate TE

for all incoming activities and take the maximum value. This

value is the TE for event 2.

Step 4: Repeat the same procedure from step 3 till the end event.

Backward Pass Computations (to calculate Latest Time

T Network Model L)

Procedure

Step 1: Begin from end event and move towards the start event.Assume that the direction of arrows is reversed.

Step 2: Latest Time TL for the last event is the earliest time. TE of

the last event.

Step 3: Go to the next event, if there is an incoming activity,

subtract the value of TL of previous event from the activity

duration time. The arrived value is TL for that event. If there are

more than one incoming activities, take the minimum TE value.

Step 4: Repeat the same procedure from step 2 till the start

event.


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DETERMINATION OF FLOAT AND SLACK TIMES

Float : As discussed earlier, the non critical activities have

some slack or float. The float of an activity is the amount of time

available by which it is possible to delay its completion timewithout extending the overall project completion time.

For an activity i = j, let

tij = duration of activity

TE = earliest expected time

TL = latest allowable time

ESij = earliest start time of the activity

EFij = earliest finish time of the activity

LSij = latest start time of the activity

LFij = latest finish time of the activity

Total Float TFij: The total float of an activity is the difference

between the latest start

time and the earliest start time of that activity.

TFij = LSij ESij ....................(1)

or

TFij = (TL TE) tij ....................(2)

Free Float FFij: The time by which the completion of an activity

can be delayed from its earliest finish time without affecting the

earliest start time of the succeeding activity is called free float.

FFij = (Ej Ei) tij ....................(3)

FFij = Total float Head event slack

Independent Float IFij: The amount of time by which the start

of an activity can be delayed without affecting the earliest starttime of any immediately following activities, assuming that the

preceding activity has finished at its latest finish time.

IFij = (Ej Li) tij ....................(4)

IFij = Free float Tail event slack

Where tail event slack = Li Ei

The negative value of independent float is considered to be zero.

Critical Path: After determining the earliest and the latest

scheduled times for various activities, the minimum time requiredto complete the project is calculated. In a network, among


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various paths, the longest path which determines the total time

duration of the project is called the critical path. The following

conditions must be satisfied in locating the critical path of a

network.

An activity is said to be critical only if both the conditions aresatisfied.

1. TL TE = 0

2. TLj tij TEj = 0

Example : A project schedule has the following characteristics as

shown in the table

Project Schedule

i. Construct PERT network.

ii. Compute TE and TL for each activity.

iii. Find the critical path.

Solution:

(i) From the data given in the problem, the activity network is

constructed as shown in the following figure.

Activity Network Diagram

(ii) To determine the critical path, compute the earliest, time T

Network Model E and latest time TL for each of the activity of the

project. The calculations of TE and TL are as follows:

To calculate TEfor all activities,

TE1 = 0

TE2 = TE1 + t1, 2 = 0 + 4 = 4

TE3 = TE1 + t1, 3 = 0 + 1 =1TE4 = max (TE2 + t2, 4 and TE3 + t3, 4)


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= max (4 + 1 and 1 + 1) = max (5, 2)

= 5 days

TE5 = TE3 + t 3, 6 = 1 + 6 = 7

TE6 = TE5 + t 5, 6 = 7 + 4 = 11

TE7 = TE5 + t5, 7 = 7 + 8 = 15TE8 = max (TE6 + t 6, 8 and TE7 + t7, 8)

= max (11 + 1 and 15 + 2) = max (12, 17)

= 17 days

TE9 = TE4 + t4, 9 = 5 + 5 = 10

TE10 = max (TE9 + t9, 10 and TE8 + t8, 10)

= max (10 + 7 and 17 + 5) = max (17, 22)

= 22 days

To calculate TL for all activities

TL10 = TE10 = 22

TL9 = TE10 t9,10 = 22 7 = 15

TL8 = TE10 t 8, 10 = 22 5 = 17

TL7 = TE8 t 7, 8 = 17 2 = 15

TL6 = TE8 t 6, 8 = 17 1 = 16

TL5 = min (TE6 t5, 6 and TE7 t5, 7)

= min (16 4 and 15 8) = min (12, 7)

= 7 days

TL4 = TL9 t 4, 9 = 15 5 =10TL3 = min (TL4 t3, 4 and TL55 t 3, 5)

= min (10 1 and 7 6) = min (9, 1)

= 1 day

TL2 = TL4 t2, 4 = 10 1 = 9

TL1 = Min (TL2 t1, 2 and TL3 t1, 3)

= Min (9 4 and 1 1) = 0

Various Activities and their Floats


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(iii) From the table, we observe that the activities 1 3, 3 5, 5

7,7 8 and 8 10 are critical activities as their floats are zero.

Critical Path of the Project

The critical path is 1-3-5-7-8-10 (shown in double line in the

above figure) with the project duration of 22 days.

PROJECT EVALUATION REVIEW TECHNIQUE, PERT

In the critical path method, the time estimates are assumed to be

known with certainty. In certain projects like research and

development, new product introductions, it is difficult to estimate

the time of various activities. Hence PERT is used in such projects

with a probabilistic method using three time estimates for an

activity, rather than a single estimate, as shown in Figure.

PERT Using Probabilistic Method with 3 Time Estimates


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Optimistic time tO:

It is the shortest time taken to complete the activity. It means

that if everything goes well then there is more chance of

completing the activity within this time.

Most likely time tm:

It is the normal time taken to complete an activity, if the activitywere frequently repeated under the same conditions.

Pessimistic time tp:

It is the longest time that an activity would t ake to complete. It is

the worst time estimate that an activity would take if unexpected

problems are faced.

Taking all these time estimates into consideration, the expected

time of an activity is arrived at.

The average or mean (ta) value of the activity duration is givenby,

Ta= t0+4tm+tp/6 .....................(5)

The variance of the activity time is calculated using the formula,

Ta= t0+4tm+tp/6 ...................(6)

Probability for Project Duration

The probability of completing the project within the scheduled

time (Ts) or contracted time may be obtained by using the

standard normal deviate where Te is the expectedtime of project completion.


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...............(7)

Probability of completing the project within the scheduled time is,

P (T Ts) = P ( Z Z0 ) (from normal tables).................(8)

Example : An R & D project has a list of tasks to be performed

whose time estimates

are given in the table, as follows.

Time expected for each activity is calculated using the formula

(5):

Ta= t0+4tm+tp/6

= 4+4(6)+8/6 = 36/6 = 6 days for activity ASimilarly, the expected time is calculated for all the activities. The

variance of activity time is calculated using the formula (6).

Similarly, variances of all the activities are calculated. Construct a

network diagram and calculate the time earliest, TE and time

Latest TL for all the activities.

Network Diagram


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Time Estimates for R & D Project

a. Draw the project network.

b. Find the critical path.

c. Find the probability that the project is completed in 19 days. If

the probability is less that 20%, find the probability ofcompleting it in 24 days.

Solution:

Calculate the time average ta and variances of each activity as

shown in the following table.

Te & s2 Calculated

From the network diagram Figure, the critical path is identified as

1-4, 4-6, 6-7, with project duration of 22 days. The probability of

completing the project within 19 days is given by,


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Thus, the probability of completing the R & D project in 19 days is

9.01%. Since the probability of completing the project in 19 days

is less than 20%, we find the probability of completing it in 24

days.

GAME THEORY

Game theory is a probabilistic model which is used in analyzing, anddriving rules for making decisions when two or more people arecompeting for some objectives. Game theory attempts to look at therelationships between participants in a particular model and predicttheir optimal decisions.

There are two primary use of game theory: descriptive andprescriptive.


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In the descriptive use, game theory has been used to study a widevariety of human and animal behaviors

In the prescriptive (normative) use, game theory has also beenused to attempt to develop theories of ethical or normative

behavior.

Strengths and Weaknesses of Game Theory

Strengths Game Theory

(1) Game theory is a model type, which are generally explicit andunambiguous in nature.

For this reason, solution to game theory are easy to criticize orsubscribe to.

(2) Game theory as model if applied it provides a limitedrepresentative of reality and for this

reason a solution to game theory is a solution to a surrogate forthe real problem.

(3) A solution to game theory is objective. For this reason thesolution can easily be manipulated, augmented and eliminated.

(4) There are always consistencies in solution to game theory

(reliability).(5) Game theory can give a decision marker an opportunity ofknowledge to acceptance or rejection of a hypothesis.

(6) Game theory is applicable to a close system because the payoffsof the all participants are added up to be zero (winning = (+) andloses = (-) then the game is called zero- sum game. Otherwise, it isknown as non-zero-sum game

Weaknesses of Game Theory

(1) For practical considerations, game theory always imposedconstraints because it is the

only correct way to formulate the problem.

(2) Game theory is based on the assumption that the parties arerational and few in numbers.

(3) It is based on the assumption that each player knows theobjectives of his opponent.


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Applications of Game Theory in Business:

1) predicts outcomes of a group of interacting agents where an

action of a single agent directly affects the payoff of other

participating agents2)studies of multi person decision problems.

3)a bag of analytical tools designed to help to understand thephenomena that are to be observed when decision-makers interact.

4) studies the mathematical models of conflict and cooperationbetween intelligent rational

decision-makers.

5) To focus on Price and quantity competition

Types of games

There are several types of game:

Cooperative or non-cooperative

A game is cooperative if the players are able to form bindingcommitments. For instance the legal system requires them toadhere to their promises. In noncooperative games this is not

possible.

Symmetric and asymmetric

A symmetric game is a game where the payoffs for playing aparticular strategy depend only on the other strategies employed,not on who is playing them. If the identities of the players can bechanged without changing the payoff to the strategies, then a gameis symmetric

Most commonly studied asymmetric games are games where there

are not identical strategy sets for both players. For instance, theultimatum game and similarly the dictator game have differentstrategies for each player.

Zero-sum and non-zero-sum

In zero-sum games the total benefit to all players in the game, forevery combination of strategies, always adds to zero (moreinformally, a player benefits only at the equal expense of others).

Many games studied by game theorists (including the infamousprisoner's dilemma) are non-zero-sum games, because the outcome


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has net results greater or less than zero. Informally, in non-zero-sumgames, a gain by one player does not necessarily correspond with aloss by another.

Simultaneous and sequential

Simultaneous games are games where both players movesimultaneously, or if they do not move simultaneously, the laterplayers are unaware of the earlier players' actions (making themeffectively simultaneous). Sequential games (or dynamic games)are games where later players have some knowledge about earlieractions

Perfect information and imperfect information

An important subset of sequential games consists of games of

perfect information. A game is one of perfect information if allplayers know the moves previously made by all other players. Thus,only sequential games can be games of perfect information becauseplayers in simultaneous games do not know the actions of the otherplayers. Most games studied in game theory are imperfect-information games. Interesting examples of perfect-informationgames include the ultimatum game and centipede game.Recreational games of perfect information games include chess, go,and mancala. Many card games are games of imperfect information,for instance poker or contract bridge.

Many-player and population games

Games with an arbitrary, but finite, number of players are oftencalled n-person games. Evolutionary game theory considers gamesinvolving a population of decision makers, where the frequency withwhich a particular decision is made can change over time inresponse to the decisions made by all individuals in the population.

Differential games

Differential games such as the continuous pursuit and evasion gameare continuous games where the evolution of the players' statevariables is governed by differential equations.

Metagames

These are games the play of which is the development of the rulesfor another game, the target or subject game. Metagames seek tomaximize the utility value of the rule set developed. The theory ofmetagames is related to mechanism design theory.


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Usefulness of Game theory in Business

Game theory was once hailed as a revolutionary

interdisciplinary phenomenon bringing together psychology,mathematics, philosophy and an extensive mix of otheracademic areas. Eight Noble Prizes have been awarded tothose who have progressed the discipline; but beyond theacademic level, is game theory actually applicable in today'sworld? Yes!

The classical example of game theory in the business worldarises when analyzing an economic environment characterizedby an oligopoly. Competitive firms are faced with a decision

matrix similar to that of a Prisoner's Dilemma. Each firm hasthe option to accept the basic pricing structure agreed upon bythe other companies or to introduce a lower price schedule.Despite that it is in the common interest to cooperate with thecompetitors, following a logical thought process causes thefirms to default. As a result everyone is worse off. Althoughthis is a fairly basic scenario, decision analysis has influencedthe general business environment and is a prime factor in theuse of compliance contracts.

Game theory has branched out to encompass many otherbusiness disciplines. From optimal marketing campaignstrategies, to waging war decisions, ideal auction tactics andvoting styles, game theory provides a hypothetical frameworkwith material implications. For example, pharmaceuticalcompanies consistently face decisions regarding whether tomarket a product immediately and gain a competitive edgeover rival firms, or prolong the testing period of the drug; if abankrupt company is being liquidated and its assets auctionedoff, what is the ideal approach for the auction; what is the best

way to structure proxy voting schedules? Since these decisionsinvolve numerous parties, game theory provides the base forrational decision making.

Another important concept, zero-sum games, also stemmedfrom the original ideas presented in game theory and the NashEquilibrium. Essentially, any quantifiable gains by one partyare equal to the losses of another party. Swaps, forwards,options and other financial instruments are often described as"zero-sum" instruments, taking their roots from a concept that

now seems distant. (For an in depth explanation about gametheory, check out Game Theory: Beyond the Basics.)


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SIMULATION

INTRODUCTION

Simulation is one of the most widely used in quantities analysistools. It is a mathematical model which is used to find out somealternative course of action.

There are various it sector to using the Simulates model.ForExample Business students take courses that use management

games to stimulate realistic competitive business situations. Yourlocal civil defense organization may carry out rescue and evacuationpractices it simulates the natural disaster conditions of a hurricaneor tornado.

PROCESS OF SIMULATION

1. Define Problem

2. Introduce the variables

3. Construct a numerical model

4. Possible Courses of Action for testing

5. Run the experiment

6. Consider the result

7. Select best course of Action.

Advantages:

Simulation is a tools that has become widely accepted by managersfor several reasons

1) It is relatively straightforward and flexible

2) Recent advances in software make some simulation modelsvery easy to develop.


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3) It can be used to analyze large and complex real worldsituations than can not be solved by conventional quantitativeanalysis models.

4) Simulation allows us to study the interactive effect of individual

components or variables to determine which ones areimportant.

5) Time compression is possible with simulation. The effect ofordering, advertising or other policies over May months oryears can be obtained by computer simulation in a short time.

Disadvantages:

1) Good simulation models for complex situation can be veryexpensive. It is often a long, complicated process to develop

a model.

2) Managers must generate all of the conditions andconstraints for solutions that they want to examine.

3) Each simulation model is unique. Its solutions andinferences are not usually transferable to other problems.

Various uses in Profession of simulation:

Simulation is using in the following way

1) Civil defense organization.

2) Conditions of disasters.

3) Business Students.

4) Managers in organization.

5) Mathematical Term etc.

Applicability of Quantitative Techniques As a Tools of

Business Decision Making

(1)Business Mathematics


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This course is designed to improve the quantitative reasoning

skills of business students. It provides an introduction to two

important knowledge bases: linear functions and systems and

the fundamentals of the derivative and integration and theiruses in business decision making. The focus of the course will

be on the application of these mathematical concepts to

personal business, management, marketing, and finance

issues. Excel spreadsheet applications will be used extensively

throughout the course.

(2)Probability and Statistics

A course in elementary probability and statistical concepts with

emphasis on data analysis and presentation; frequency

distributions; probability theory; probability distributions,

sampling distributions, statistical inference, hypothesis testing.

(3)Quantitative Techniques in Management

An introduction to quantitative techniques in management.

Topics include linear programming, assignment problems,

transportation algorithms, network and inventory models, and

decision theory.

4)Management Information system

This course provides methodology of the design, analysis, and

evaluation of management information systems (MIS). Topics

include organizational implications of information technology,

planning and control systems, implementation of an integrated

system, technical treatment of MIS management, andapplication of computers via computer packages in business

environments.

(5)Quantitative Techniques

Advanced applications of quantitative techniques to the

solution of business problems. Topics include classical

optimization techniques, non-linear programming, topics in

mathematical programming, and graph theory.


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(6)Operations Management

Basic review of service and production systems designs and

performance evaluation. Topics include operations strategy,

staff and production scheduling, Just-in-Time and time-basedcompetition, project management, and the role of technology

in service and manufacturing operations.

(7)Forecasting for Decision Making

Review of different approaches to forecasting used by

management at different levels of decision making.

Techniques will include smoothing and decomposition, causal

and judgmental methods. Computer applications and modeling

will be emphasized.

(8)Project Management

Survey of management techniques applicable to a wide variety

of business-related project types. Emphasis on the project

management cycle, including selecting, scheduling, budgeting,

and controlling projects. Desired qualifications and roles of

project managers. Extensive use of project management

software will be required.

9)Cost Benefit Management

An introduction to an overview of the field of cost benefit

management. Fundamental theoretical evaluation of

cost/benefit of a project. Includes: the selection of the best

investment criteria, the external environment spillover effectsand the application of cost/benefit management decision

making under uncertainty.

quantitative technics

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