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OPERATIONS RESEARCHINTRODUCTION TO OPERATIONS RESEARCH Operations Research is a new branch of Mathematics dealing in theoptimization problems in real-life situations. It is also a quantitative technique todeal many management problems, In this discipline ,we study cost minimization ofvarious inventory problems, the minimization of transportation costs of sendinggoods from various warehouses to different centers, the profit maximization or costminimization in linear programming models, the assignment of different person todifferent jobs so that total time taken to perform the jobs is minimized, thecongestion problem in traffic places, airline counters, supermarket, to find out ofthe waiting time of customers in the queue, the project completion time withlimited resources and many other similar problems.ORIGIN OF OPERATIONS RESEARCH The germination of the concept Operations Research occurred duringWorld War 1. In England in the year 1915, F.W. Lanchester attempted to treatmilitary operations quantitatively. He derived equations relating the outcome of abattle to both the relative numerical strength of the combatants and their relativemanpower. He modeled a situation involving strategic choices and then tested thata model against a known situation. During the same period, Thomas Alva Edison in America was studyingthe process of Anti-submarine warfare. He devised a war game to be used forsimulating problems of naval manoeuvre. In 1917, A.K. Erlang, a DanishMathematician has developed solutions for some waiting line problems. In 1915,F.W.Harris had developed the first model on an inventory problem for economic 1
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lot size In 1930 W.Leontieff developed a linear programming model representingthe entire United status economy. Active research works were done during WorldWar II in Great Britain and United States of America.DEFINITION OF OPERATIONS RESEARCH Operations Research was initially a subject dealing with militaryoperations during the World War II. Later several techniques were developed tosuit much of humanities progress in science, technologies, business administration,etc. A vast variety of fields is brought in the purview of this branch of science.There are served definitions for O.R.They only specify the applications of the discipline. None of the definitions is welldefined. We mention some of the definitions here.DEFINITION BY D’CLARKE Operations Research is defined as the art of winning wars withoutactually fighting.DEFINITION BY ACOFF, ANNOFF AND CHURCHMANN O.R. is the application of scientific methods, techniques and tools toproblem involving the operations of system so as to provide those in control of theoperations with optimum solution to the problem.DEFINITION BY T.L.SASTRY O.R. is the art of giving bad answers to problems where otherwiseworse answers are given. 2
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DEFINITION BY JAMES LUNTRY O.R. is the sophisticated name given to multidisciplinary problem-oriented approach to the top management problem. It involves the applications ofscientific methods in situations where executives require description, prediction,and comparison for the purpose of decision making.DEFINITION BY AMERICAN SOCIETY OF O.R. O.R. is an experimental and applied science devoted to observe,understanding and predicting the behaviors of purposeful man-machine system,many of these definitions only broad line the applications of O.R. to war, industry,management and humanity progress.APPLICATIONS OF OPERATIONS RESEARCH Here we mention only some of the areas where O.R. techniques can beapplied. This science is applied widely in areas of accounting facilities, planning,finance, manufacturing, marketing, purchasing and in organizations andgovernment and quasi government activities. We mention some of the applicationsof O.R. in the above areas. Cash flow planning, credit policy planning of delinquent accountstrategy are some of the areas in accounting where O.R. techniques are used. Warehouses locations, Transportation loading and unloading, factorysize and location, Hospitals planning are some of the areas of facilities planning . In finance, it is a applied to qualitative study of investment analysis,portfolio management, dividend policy, etc. In marketing, O.R. is applied to studythe selection of product-mix, prediction scheduling time, advertising allocation,etc. 3
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In order to arrive at an optimal solution to the problems in O.R., first wehave to construct a model. Once a project is selected, we have to describe theproblem as a model. The model should describe all the features of the problem.We have to express the description of the problem in a mathematical formulation.This formulation has not be done satisfying all the assumptions of the problem.MODELS AND MODELLING Modelling a real life situation helps us to study the different behavior ofthe problem corresponding to the description of the problem. Great efforts havebeen taken by experts to model business situations, military operations, motion ofplanets and stars, congestion in traffic places and so on. A model is an abstraction on an idealized representation of a real lifeproblem. The object of a model is to provide means for analyzing the behavior ofthe system for future improvement. A map of multiple activity chart, a projectnetwork, the representation of the behavior of a queuing system, a model toforecast the future, based on the past and the present factors of a time series, etc..are all examples of models, A model can be a picture, map, a curve or an equation.The reliability of the decision drawn from the model may depend upon the validityof the model on the basic assumption on which the model is built. Modelling is the essence of operations research building a model helpsus to convert the complexities and uncertainties of a decision making problem intoa concrete logical structure which is amenable to formed analysis. A model is avehicle for arriving a well structured problem of reality. A commentator of acricket match describes the play as a model to enable us to predict the futurecourse of events of the play. It is a descriptive model available for further analysis.It is not always possible to analyse a situation only with the description of the 4
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situation we have to formulate the problem into concrete mathematicalrepresentation in the form of a curve, graph or equations. Models, could beclassified as iconic model, analogue models and symbolic model.ADVANTAGES A iconic model is concrete. It is easy to construct the model. It is easy study the model then the system itself.DISADVANTAGES This model is not suited for further manipulation. It cannot be used to study the changes in the operation of the system. It is not possible to make any modification of the model. Adjustment with changing situations cannot be done in this model.ANALOGUE MODEL In an analogue model, one set of properties is used to represent anotherset of properties. After analysing the model for decision making the results of theanalysis can be re-interpreted in terms of the original system. For example,Contour lines on a map are analogues of elevation as they represent the size andfall of heights, Graphs are analogues as distance is used to represent a wide varietyof variables such as time, percentage, weight, etc. It is easier to manipulate theanalogue model. But it is less specific and less concrete. 5
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SYMBOLIC MODEL Symbolic models employ a set of mathematical symbols and functionsto represent the decision variables and their functions to describe the behavior ofthe system. Almost all the models in O.R. are symbolic model.ADVANTAGES These models are most abstract and most general. These models are amicable for experimental manipulation. They yield reasonably good results to the real life problem. A good model should have the following characteristics : It should be capable of taking into account new formulations with having any significant change in its frame. The assumptions should be well defined and the number of assumptions should be as small as possible. The assumptions should be simple and coherent. Only a limited number of variables should be used. It should be acceptable to parametric treatment. 6
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ADVANTAGE OF A MODEL It is a description of a physical problem. It gives a systematic approach to a problem and is subject to logical treatment. It is easy to make decisions based on a model. If a model is built on a broad based assumption, it is easy to modify it according to new situations. Model help us finding avenues for new research and improvement in a system.LIMITATION OF A MODEL Models are only an attempt in describing a system and should be taken to be as absolute representation of a system. Model constructed is valid only if all the assumptions of the model are true in the system for which the model is constructed. Validity of the model is subject to experimental testing. 7
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PERT PROGRAM EVALUATION AND REVIEW TECHNIQUEINTRODUCTION Network scheduling is a technique used for planning and schedulinglarge projects in the fields of construction, maintenance, fabrication purchasing,computer system installation, research and development designs etc. The techniqueis a method of minimizing trouble spots, such as, production bottlenecks, delaysand interruptions by determining critical factors and coordinating various parts ofoverall job. There are two basic planning and control techniques that utilize anetwork that to complete a pre determined project or scheduling. These are :Program Evaluation and Review Technique(PERT) and the Critical PathMethod(CPM) several variations of these have also been developed. One suchimportant variation being the Review Analysis of Multiple Projects(RAMP) whichis useful for guiding the „activities‟ of several projects at one time.NETWORK ANALYSIS CPM was developed in 1957 by J.E. Kelly of Remington and M.RWalker of Dupont to aid in the scheduling of routine plant overhaul, maintenanceand construction work. This method differentiates between planning andscheduling. Planning refers to the determination of activities that must beaccomplished and the order in which such activities should be performed toachieve the objectives of the project. 8
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PERT was developed in the late 1950‟s by the US Navy SpecialProjects Office in operation with the management consulting firm of Booz , Allenand Hamilton. The technique received substantial favourable publicity for its use inthe engineering and development program of the Polaris missile, a complicatedproject that had 250 prime characters and over 9000 sub characters. But now thistechnique is very popular in the hands of project planner and controller of variousdepartments in government and in industry. In PERT, we usually assume that thetime to perform the activity is uncertain and as such three time estimates are used.METHODOLOGY OF PERT/CPM NETWORKS The methodology involved in applying PERT for any project may besplit into the following steps: PROJECT PLANNING The purpose of this is to identify all important events/activitieswhich are essential for completion as well as making up of the project and theirdependence upon one another is shown explicitly in the form of a network. TIME ESTIMATION Estimates of the time required perform each of network activitiesare made, the estimates are based upon manpower and equipment availabilityand certain assumptions that may have been made in planning the project. Byincorporating the time required for completing each of the activities in thenetwork, the project duration as well as the criticality of the activities are found. Atthis stage it is also possible to compute the probability of completing the project ora part of the project by a specified time. 9
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SCHEDULING The scheduling computations give the earliest and the latest allowable start and finish time for each activity, and as a by product, they identify the critical path through the network, and indicate the amount of “slack” time may associated with the non critical paths. TIME COST TRADE OFF’S If the scheduled time to complete the project as determined in step 3 satisfactory, the project planning and scheduling may be complete However, if one interested in determining the cost of reducing the project completion time. Then time cost trade-offs of activity performs time must be considered for those activities on the critical and near critical path(s). RESOURCE ALLOCATION The feasibility of each schedule must be checked with respective manpower and equipment requirements. Establishing complete feasibility of a specific schedule may require replanning and rescheduling or time-cost trade-offs. Hence a final solution may require the performance of a number of cycles of steps 3, 4 and 5. PROJECT CONTROL When the network plan and the schedule have been developed to a satisfactory extent, they are repaired to final form for use in the field. The project is controlled by checking progress against the schedule, assigning and scheduling manpower and equipment, and analysing the effects of delays. 10
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PROBABILTY CONSIDERATIONS IN PERT The network methods discussed so far may be termed as deterministic, since estimated activity times are assumed to be the expected values. But no recognition is given to the fact that expected activity time is the mean of a distribution of possible values which could occur. Under the conditions of uncertainty, the estimate time for each activity are PERT network is represented by a probability distribution. This probability distribution of activity time is based upon three different time estimates mode for each activity. These are as follows: to = the optimistic time, is the shortest possible time to complete the activity if all goes well. tp = the pessimistic time, is the longest time that an activity could take if everything goes wrong. tm = the most likely time, is the estimate of the normal time an activity would take. If only one time where available, this would be it. Otherwise it is mode the probability distribution.PROBABILITY OF MEETING THE SCHEDULE TIME With PERT, it is possible to determine the probability of completing acontract on schedule. The scheduled dates are expressed as number of time unitsfrom the present time. Initially they may be the latest time, T L, for each event, butafter a project is started we shall know how far it has progressed at any given date,and the scheduled time will be the latest time if the project is to be completed on itsoriginal schedule. 11
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The probability distribution of times for completing an event can beapproximated by the normal distribution due to the control limit theorem. Thus theprobability of completing the project by schedule time(TS) is given by: Prob (z<(Ts-Te)/σ) the standard normal variate is given by, Z=(Ts-Te)/σe Where Te = Expected completion time of the project. σ e = Number of the standard deviations the scheduled time lies from the expected time. (i.e) the standard deviations of the scheduled time. Using the commutative normal distribution table, the corresponding valueof the standard normal variate is read off. This will give the require probability ofcompleting the project on schedule time.RULES OF NETWORK CONSTRUCTION For the construction of a network, generally, the following rules arefollowed Each activity is represented by one and only one arrow. Each activity must be identified by its starting and end node which implies that. i. Two activities should not be identified by the same completion events, and 12
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ii. Activities must be represented by their symbols are by the corresponding ordered pair of starting completion events. Notes are numbered to identify an activity uniquely Tail node should be lower than the head node, of an activity. Between any pair of nodes, there should be one and only one activity, however more than one activity may emanate from and terminate to a node. Arrows should be kept straight and not curved or bent.NUMBERING THE EVENTS After the network is drawn in a logical sequence, every event isassigned a number. The number sequenced must be such so as to reflect the flow ofthe network. In event numbering, the following rules should be observed: a. Event number should be unique. b. Event numbering should be carried out on a sequential basis from left to right. c. The initial event which has all outgoing arrows with no incoming arrow is numbered 0 or 1. d. The head of an arrow should always bear a number higher than the one assigned at the tail of the arrow. e. Gaps should be left in the sequence of event numbering to accommodate subsequent inclusion of activities, if necessary. 13
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BASIC CONCEPTS OF NETWORK ANALYSIS A fundamental ingredient in both PERT and CPM is the use ofnetwork system as a means of graphically depicting the current problems orproposed project. Because of its importance to a basic understanding of both PERTand CPM, the network concept will be examined. When a network is beingconstructed, certain conventions are followed to represent a project graphically, forit is essential that the relationship between activities and events are correctlydepicted. Before illustrating the network representation, it is necessary to definesome of the concepts.ACTIVITY All projects may be viewed as being composed of operations or taskscalled activities, which require the expenditure of time and resources for theaccomplishment. An activity as depicted by a single arrow ( ) on the projectnetwork. The activity arrows are called arcs. The activity arrow is not scaled, thelength of the activity time is only a matter of convenience and clarity, and does notrepresent important of time. The head of the arrow shows the sequence or flow ofactivities. An activity cannot begin until the completion of the preceding activities.It is important that activities be defined so that beginning and end of each activitycan be identified clearly.PREDECESSOR ACTIVITY Activities that must be completed immediately prior to the start ofanother activity are called predecessor activities. 14
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SUCCESSOR ACTIVITY Activities that cannot be started until one or more of the otheractivities are completed, but immediately succeed them are called successoractivities.CONCURRENT ACTIVITY Activities which can be accomplished concurrently are known asconcurrent activities. It may be noted that an activity can be a predecessor orsuccessor to an event or it may be concurrent with one or more of the otherActivities.EVENT An Event represent a specific accomplishment in the project and takesplace at a particular instant of time, and does not, therefore, consume time orresources. An event in a network is a time oriented reference point that signifiesthe end of the activity and the beginning of another. Events are usually representedin the project network by circles (o). The event circles are called nodes. Therefore,a major difference between activities and events is that activities represent thepassage of time where as events are point in time. All activity arrows must beginand end with event nodes as shown below Start Finish event Activity event 15
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MERGE EVENT Where more than one activity comes and joins, the event is known asmerge eventBURST EVENT When more than one activity leaves an event, the event is known asburst event.MERGE AND BURST EVENT An activity may be a merge and burst event simultaneously as withrespect some activities it can be merge event and with respect to some otheractivities it may be burst event. Merge event Burst Event Merge & BurstDUMMY ACTIVITY In most projects many activities can be performed concurrently orsimultaneously. It is possible that two activities could be drawn by the samebeginning and end events, In situations where two are more activities can beperformed concurrently, the concept of dummy activity is introduced to solve thisproblem. Therefore there will be only one activity between two events. 16
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An activity which does not consume either any resource or time isknown as dummy activity. A dummy activity represented by dotted line in thenetwork diagram. Predecessor Successor Activity Activity Dummy activityPERT SYSTEM OF THREE TIME ESTIMATE The traditional single estimate of duration of any activity is replaced bythree time estimates in PERT system an optimistic, a pessimistic, and a most likelytime.OPTIMISTIC TIME (a or to) The time estimate of an activity when everything is assumed to go wellas per plan. In other words, it is the estimate of the minimum possible time, whichan activity takes in completion under ideal conditions. However no provisions aremade for breakdown, delays, etcMOST LIKELY TIME (m or tm) The time which the activity will take most frequently performed anumber of times the model value.PESSIMISTIC TIME (tp) The unlikely but possible performance time if whatever could gowrong, goes wrong in series. In other words it is the longest time the activity can 17
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conceivably take. This however does not include major catastrophies like labourstrikes, acts of God unrest, etc.EST- It means Earliest start time for an activity represent the time at which anactivity begins at the earliest.EFT- EFT means Earliest finish time of an activity is it earliest start time „+‟(plus) the required time to perform the activity.LFT- LFT means latest finish time. Latest finish time of an activity represent thelatest by which an activity must be completed without delaying the completion ofproject.LST- Latest start time for an activity is the Latest finish time „-‟(minus) theactivity duration methods.FORWARD PASS METHOD (For Earliest Event Time) Based on fixed occurrence time of the initial network event, theforward pass computation yields the earliest start and earliest finish times for eachactivity and indirectly the earliest expected occurrence time for each event.BACKWORD PASS METHOD (For latest allowable time) The latest occurrence event time (L) specifies the time by which allactivities entering into that event must completed, without delaying the total 18
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project. These are computed by reversing the method of calculation used forearliest event times.CRITICAL PATH METHOD The longest path is called the critical path. An activity is said be criticalif the delay in its start will delay the project completion time.PERT-ALGORITHM The various step involved in developing PERT network for analyzingany project are summarized below Make a list of activities that make up the project including immediate Predecessors. Making use of step1 sketch the required network. Denote the most likely time by tm, the optimistic time to and pessimistic time by tp. Using beta distribution for each activity duration the expected time t e for te = (to+tm+tp)/6 Tabulated various time (i.e) expected activity times, earliest and latest times and mark the EST and LFT on the arrow diagram. Determine the total float for each activity by taking the difference between EST & LFT. Identify the critical activities and connect them with the beginning node and the ending node in the network diagram by double line arrows. The critical path and expected date of completion of the project Using the values of tp and to to compute the variance (σ2) of each activity . This is done with the following formula, σ2 = [(tp-to)/6]2 19
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Compute the standard normal deviate Due date –Expected date of completion Zo = √project variance Use standard normal tables to find the probability p(z ≤ zo) of completing the project within the scheduled time, where Z ~ N(0,1). 20
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