MB0048-OPERATIONS RESEARCH

1Q. Discuss the various stages involved in the methodology of Operations Research. Brieflyexplain the techniques and tools of Operations Research.Ans: The basic dominant characteristic feature of operations research is that itemploys mathematical representations or models to analyse problems. Thisdistinct approach represents an adaptation of the scientific methodologyused by the physical sciences. The scientific method translates a givenproblem into a mathematical representation which is solved andretransformed into the original context.Steps in the OR methodology

DefinitionThe first and the most important step in the OR approach of problem solvingis to define the problem. One needs to ensure that the problem is identifiedproperly because this problem statement will indicate the following threemajor aspects:Description of the goal or the objective of the studyIdentification of the decision alternative to the systemRecognition of the limitations, restrictions, and requirements of theSystemConstructionBased on the problem definition, you need to identify and select the mostappropriate model to represent the system. While selecting a model, youneed to ensure that the model specifies quantitative expressions for theobjective and the constraints of the problem in terms of its decisionvariables. A model gives a perspective picture of the whole problem andhelps in tackling it in a well-organised manner. Therefore, if the resultingmodel fits into one of the common mathematical models, you can obtain aconvenient solution by using mathematical techniques.SolutionAfter deciding on an appropriate model, you need to develop a solution forthe model and interpret the solution in the context of the given problem. Asolution to a model implies determination of a specific set of decisionvariables that would yield an optimum solution. An optimum solution is onewhich maximises or minimises the performance of any measure in a modelsubject to the conditions and constraints imposed on the model.ValidationA model is a good representation of a system. However, the optimal solution must work towards improving the systems performance. You can test thevalidity of a model by comparing its performance with some past dataavailable from the actual system. If under similar conditions of inputs, your model can reproduce the past performance of the system, then you can be sure that your model is valid.ImplementationWe need to apply the optimal solution obtained from the model to thesystem and note the improvement in the performance of the system. Youneed to validate this performance check under changing conditions.Operations Research Techniques and ToolsThe different techniques and tools used in OR are as follows:Linear programming You can use linear programming to find asolution for optimising a given objective. The objective may be tomaximise profit or to minimise cost. You need to ensure that both theobjective function and the constraints can be expressed as linearexpressions of decision variables. You will learn about the various usesof linear programming in Unit 2.Inventory control methods The production, purchasing, and materialmanagers are always confronted with questions, such as when to buy,how much to buy, and how much to keep in stock. The inventory modelaims at optimising these inventory levels.Goal programming In linear programming, you take a single objectivefunction and consider all other factors as constraints. However, in reallife there may be a number of important objective functions. Goalprogramming has several objective functions, each having a targetvalue. Programming models are developed to minimise deviations fromthese targets.Queuing model The queuing theory is based on the concept ofprobability. It indicates the capability of a given system and the changespossible in the system when you modify the system. In formulating aqueuing model, you need not take into account all the constraints. Thereis no maximisation or minimisation of an objective function. Therefore,the application of queuing theory cannot be viewed as an optimisationprocess. You can use the queuing theory to estimate the requiredbalance between customer waiting time and the service capability of the system.Transportation model The transportation model is an important classof linear programs. The model studies the minimisation of the cost oftransporting a commodity from a number of sources to severaldestinations.

2Q. a. Explain the steps involved in linear programming problem formulation. Discuss inbrief the advantages of linear programming.b. Alpha Limited produces & sells two different products under the brand names black& white. The profit per unit on these products in Rs. 50 & Rs. 40 respectively. Both theproducts employ the same manufacturing process which has a fixed total capacity of50,000 man-hours. As per the estimates of the marketing research department ofAlpha Limited, there is a market demand for maximum 8,000 units of Black & 10,000units of white. Subject to the overall demand, the products can be sold in any possiblecombination. If it takes 3 hours to produce one unit of black & 2 hours to produce oneunit of white, formulate the model of linear programming.2Ans:a. Linear Programming (LP) is a mathematical technique designed to helpmanagers in their planning and decision-making. It is usually used in anorganisation that is trying to make the most effective use of its resources.A few examples of problems in which LP has been successfully applied are:Developments of a production schedule that will satisfy future demandsfor a firms product and at the same time minimise total production andinventory costs.Establishment of an investment portfolio from a variety of stocks orbonds that will maximise a companys return on investment.Mathematical Formulation of LPPThe procedure for mathematical formulation of a linear programmingproblem consists of the following major steps:Procedure of mathematical formulation of a linearprogramming problem

Advantages of LPPThe advantages of linear programming techniques may be outlined asfollows:Linear programming technique helps in making the optimum utilisation ofproductive resources. It also indicates how decision makers can employproductive factors most effectively by choosing and allocating theseresources.The quality of decisions may also be improved by linear programmingtechniques. The user of this technique becomes more objective and lesssubjective.Linear programming technique provides practically applicable solutionsbecause there might be other constraints operating outside the problem.These constraints must also be taken into consideration. Just becauseso many units must be produced does not mean that all those can besold. So the necessary modification of its mathematical solution isrequired for the sake of convenience to the decision maker.In production processes, high lighting of bottlenecks is the mostsignificant advantage of this technique. For example, when bottlenecksoccur, some machines cannot meet the demand while others remain idlefor some time.2b.

3Q. a. What is degeneracy in transportation problem? How it can be resolved?b. Solve the following transportation problem using Vogels approximation method.Distribution Centers

FactoriesC1C2C3C4Supply

F1327650

F2752360

F3254525

Requirements60402015

3.a.Ans: Degeneracy in transportation problemA basic solution to an m-origin, n destination transportation problem canhave at the most m+n-1 positive basic variables (non-zero), otherwise thebasic solution degenerates. It follows that whenever the number of basiccells is less than m + n 1, the transportation problem is a degenerate one.The degeneracy can develop in two ways:Case 1 - The degeneracy develops while determining an initial assignmentvia any one of the initial assignment methods discussed earlier.To resolve degeneracy, you must augment the positive variables by asmany zero-valued variables as is necessary to complete the requiredm + n 1 basic variable. These zero-valued variables are selected in such amanner that the resulting m + n 1 variable constitutes a basic solution.The selected zero valued variables are designated by allocating anextremely small positive value to each one of them. The cells containingthese extremely small allocations are then treated like any other basic cells.The s are kept in the transportation table until temporary degeneracy isremoved or until the optimum solution is attained, whichever occurs first. Atthat point, we set each = 0.Case 2 - The degeneracy develops at the iteration stage. This happenswhen the selection of the entering variable results in the simultaneous driveto zero of two or more current (pre-iteration) basic variables.

To resolve degeneracy, the positive variables are augmented by as manyzero-valued variables as it is necessary to complete m+n-1 basic variables.These zero-valued variables are selected from among those current basicvariables, which are simultaneously driven to zero. The rest of theprocedure is exactly the same as discussed in case 1.Note - The extremely small value is infinitely small and it never affects thevalue it is added to or subtracted from. Introduce in unallocated minimumcost cell to avoid forming a loop.3.b.F1327650 10

F2752360

F3254525

Req6040 02015

37610 0

72360

24525

60 502015

72360

24525 0

502015

72360 40

2520 015

7340 25

2515 0

725

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=2*40+10*3+25*2+20*2+15*3+25*7=4204Q. a. Explain the steps in Hungarian method. Differentiate between Transportation andAssignment problem.b. Find the optimal assignment of four jobs and four machines when the cost ofassignment is given by the following table:J1J2J3J4

M110987

M23456

M32112

M44356

4.Ans.a. Hungarian method algorithm is based on the concept of opportunity cost and is more efficient in solving assignment problems. The following steps are adopted to solve an AP using the Hungarian method.

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

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

Step3: Assign zero row-wise if there is only one zero in the row and cross (X) or cancel other zeros in that columnStep4: Assign column wise if there is only one zero in that column and cross other zeros in that row.Step5: Repeat steps 3 and 4 till all zero are either assigned or crossed. If the number of assignments is equal to number of rows present, you have arrived at an optimal solution, if not, proceed to step 6Step6: Mark () the unassigned rows. Look for crossed zero in that row. Mark the column containing the crossed zero. Look for assigned zero in that column. Mark the row containing assigned zero. Repeat this process till all the makings are done.Step7: Draw a straight line through unmarked rows and marked column. The number of straight line drawn will be equal to the number of assignments made.Step8: Examine the uncovered elements. Select the minimum. Subtract it from the uncovered elements. Add it all the point of intersection of lines. Leave the rest as is. Prepare a new table.

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

Step 10: Repeat steps 5 to 7 till number of allocations = number of rows.

Difference between transportation and assignment The transportation problem is a special type of linear programmingproblem in which the objective is to transport a homogeneous productmanufactured at several plants (origins) to a number of differentdestinations at a minimum total cost. The assignment problem is aspecial case of transportation problem, where the objective is to minimisethe cost or time of completing a number of jobs by a number of persons, andto maximise revenue and sales efficiently.4b.5Q. Define Simulation. Explain the Simulation procedure. Discuss the use of Simulationwith an example.5.Ans:A definition of simulation as given by Shannon:Simulation is the process of defining a model of a real system andconducting experiments with this model for the purpose of understandingthe behaviour (within the limits imposed by a criterion or a set of criteria) forthe operation of a system.Simulation procedure:In any simulation problem, the variables to be studied will be given withassociated probabilities. The initial conditions will also be specified. You canchoose random numbers from table. However, to get uniform results, therandom numbers will be specified. The first step involves coding the datathat is, you assign random numbers to the variable. Then you identify therelationship between the variables and run the simulation to get the results

simple example of a queuing process.A sample of 100 arrivals of customers at a retail sales depot is according tothe following distribution:Samples of 100 Arrivals of Customers

Time between Arrivals (min)Frequency

0.52

16

1.510

225

2.520

314

3.510

47

4.54

52

Study of the time required to service customers by adding up the bills,receiving payment, making change and placing packages in hand trucks,yields the distribution depicted below:

Service time (min) Frequency

0.512

121

1.536

219

2.57

35

Estimate the average percentage customer waiting time and averagepercentage idle time of the server by simulation for the next 10 arrivals.Solution:Step 1: Convert the frequency distributions of time between arrivals andservice time to cumulative probability distributions.Step 2: Allocate random numbers 00 to 99 for each of the values of timebetween arrivals and service time. The range allocated to each valuecorresponds to the value of cumulative probability.Step 3: Using random numbers from table, sample the random time ofarrival and service time for ten sets of random numbers.Step 4: Tabulate waiting time of arrivals and idle time of servers.Step 5: Estimate the percentage waiting time of arrivals and percentage idletime of servers corresponding to the ten samples.The service facility is made available at clock time zero and the server hasto be idle for 3.5 minutes, when the service for first arrival starts. The serviceis completed at 5.0 minutes and again the server is idle for 2 minutes till thesecond arrival joins the system. The first three arrivals get immediateservice and they dont have to wait, as the server is idle when they arrive.The fourth arrival that joins at 9.0 minutes has to wait for 0.5 minute, whilethe service to the third is completed. Similarly the waiting time and idle timecan be computed for further arrivals.Total elapsed time = 29 minutesWaiting time of arrival = 1 minutePercentage of waiting time = ( 1 x 100) / 29 = 3.4Idle time for server = 14.5 minutesPercentage of idle time = ( 14.5 x 100) / 29 =50%6Q. Explain the following:a. Integer programming modelb. PERT and CPMc. Operating Characteristics of a Queuing SystemAns.a. The IPP is a special case ofLinear Programming Problem (LPP), where all or some variables areconstrained to assume non-negative integer values. In LPP, the decisionvariables as well as slack or surplus variables were allowed to take any realor fractional value. However, there are certain real life problems in which thefractional value of the decision variables has no significance. For example, itdoes not make sense saying 1.5 men working on a project or 1.6 machinesin a workshop. The integer solution to a problem can, however, be obtainedby rounding off the optimum value of the variables to the nearest integervalue. This approach can be easy in terms of economy of effort, time andcost that might be required to derive an integer solution but this solution maynot satisfy all the given constraints. Secondly, the value of the objectivefunction so obtained may not be an optimal value. Integer programmingtechniques come to our rescue during such scenarios.Integer LP problems are those in which some or all of the variables arerestricted to integer (or discrete) values. An integer LP problem hasimportant applications. Capital budgeting, construction scheduling, plantlocation and size, routing and shipping schedule, batch size, capacityexpansion, fixed charge, etc are few problems which demonstrate the areasof application of integer programming.b. PERT and CPMPERTSome key points of PERT are as follows:PERT was developed in connection with an Research and Development(R&D) work. Therefore, it had to cope with the uncertainties that areassociated with R&D activities. In PERT, the total project duration isregarded as a random variable. Therefore, associated probabilities arecalculated in order to characterise it.It is an event-oriented network as in the analysis of a network, emphasisis given on the important stages of completion of a task rather than theactivities required to be performed to reach a particular event or task.PERT is normally used for projects involving activities of non-repetitivenature in which time estimates are uncertain. It helps in pinpointing critical areas in a project, so that necessaryadjustment can be made to meet the scheduled completion date of theproject. CPMCPM was developed in connection with a construction project, whichconsisted of routine tasks whose resource requirements and durationwere known with certainty. Therefore, it is basically deterministic.CPM is suitable for establishing a trade-off for optimum balancingbetween schedule time and cost of the project.CPM is used for projects involving activities of repetitive nature.c. Operating Characteristics of a Queuing SystemA queuing model has thefollowing operating characteristics which enables us to understand andefficiently manage a queue:Queue length: The number of customers in the waiting line reflects oneof the two conditions. Short queues could mean either good customerservice or too much capacity. Similarly, long queues could indicateeither low server efficiency or the need to increase capacityNumber of customers in system: The number of customers in queueand also those being served in the queue relates to the serviceefficiency and capacity. Large values imply congestion, potentialcustomer dissatisfaction and a need for more capacity.Waiting time in queue: Long lines do not reflect long waiting times ifthe service rate is fast. However, when waiting time seems long tocustomers, they perceive that the quality of service is poor. Long waitingtimes may indicate a need to adjust the service rate of the system orchange the arrival rate of customers.Waiting time in system: The total elapsed time from entry into thesystem until exit from the system may indicate problems with customers,server efficiency or capacity. If some customers are spending too muchtime in the service system, there may be a need to change the prioritydiscipline, increase productivity or adjust capacity in some way.Service facility utilisation: The collective utilisation of the servicefacilities reflects the percentage of time the facilities are busy.Management is interested in maintaining high utilisation but thisobjective may adversely impact the other operating characteristic.