Course No.: BBM501, Course Title: OPERATIONS RESEARCHClass: B.B.M., Status of Course: MAJOR COURSE, Approved since session: 2012-13Total Credits:4, Periods(50 mts. each)/week:5(L-5+T-0+P/S-0),Min.pds./sem.:65

UNIT 1: INTRODUCTIONIntroduction to O.R. Origin and Historical Development, Nature and Characteristics of O.R., General Solution Methods of O.R., Phases of O.R. Study, Introduction to LP: The LP Model, Assumptions of LP, Formulation of LP.UNIT 2: SIMPLEX METHOD AND DUALITYGraphical Solution Method, Simplex Algorithm, Solution of Maximization and Minimization Problems, Big-M Method, Essence of Duality Theory, Primal Dual Relationships, Applications of LP in Management.UNIT 3: TRANSPORTATION PROBLEM AND ASSIGNMENT PROBLEMTransportation Problem: Mathematical Formulation, Methods for finding Optimal Solution, Degeneracy Assignment Problem: Mathematical Formulation, Hungarian Method.UNIT 4: GAME THEORY AND QUEUING THEORYGame Theory: Introduction, Two-Person Zero-Sum Game, Pure and Mixed Strategies, Applications Queuing Theory: Basic Structure of Queuing Model, Kendall’s Notation, Queuing Model: M/M/I infinite and finite queues.UNIT 5: NETWORK ANALYSISIntroduction, Components of a Network, Constructing a Project network diagram using Activities-on-Node (AoN) notation, Critical Path Method, PERT (Programme Evaluation and Review Technique), Application in Management.

SUGGESTED READINGS:Hiller FS and Lieberman GJ: OPERATION RESEARCHTaha HA: OPERATION RESEARCH AN INTRODUCTIONWagner HM: PRINCIPLES OF OPERATION RESEARCH WITH APPLICATION TO MANAGERIAL DECISIONS Budnick FS & Wojena R: PRINCIPLES OF OPERATIONS RESEARCH FOR MANAGEMENTBuffa ES & Dver JS: ESSENTIALS OF MANAGEMENT SCIENCE/OPERATION RESEARCHActoff RL & Sasieni MW: FUNDAMENTALS OF OPERATIONS RESEARCH

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DEI FACULTY OF SOCIAL SCIENCE, DAYALBAGH, AGRA BBM 501: OPERATIONS RESEARCH (2013-14 I Sem): QUESTION BANK

UNIT I: INTRODUCTION

CONCEPTS

Q 1: What is Operations Research? What are the areas of applications of Operations Research?Q 2: Give various definitions of O.R. Explain the nature and development of Operations Research.Q 3: Explain historical development of Operations Research.Q 4: What are the essential characteristics (or features) of Operations Research?Q 5: Mention different phases or steps in an Operation Research study.Q 6: Enumerate with brief description some of the techniques of operations research.Q 7: Discuss the use of scientific methodology in Operations Research. Q 8: Explain the basic formulation of the diet problemQ 9: Explain the basic formulation of the product mix problemQ 10: Give examples of heuristics in decision making.Q 11: What is Linear Programming? Discuss the scope and role of LP in solving management problems.Q 12: Discuss the limitations of LP.Q 13: Explain important features of a mathematical model.Q 14: Write short notes on the following: (i) Mathematical model, (ii) Applications of modeling, (iii) Use of scientific method in OR.

FORMULATION OF LINEAR PROGRAMMING PROBLEM

Q 15: The Whitt Window Company is a company with only three employees which makes two different kinds of hand-crafted windows; a wood-framed and an aluminum-framed window. They earn $60 profit for each wood-framed windows and $30 profit for each aluminum-framed window. Doug makes the wood frames, and can make 6 per day. Linda makes the aluminum frames, and can make 4 per day. Bob forms and cuts the glass, and can make 48 square feet of glass per day. Each wood-framed window uses 6 square feet of glass and each aluminum-framed window uses 8 square feet of glass. The company wishes to determine how many windows of each type to produce per day to maximize total profit. Formulate a linear programming model for this problem.

Q 16: A firm manufactures two types of products, A and B, and sells them at a profit of Rs. 2 on Type A, and Rs. 3 on Type B. Each product is processed on two machines, G and H. Type A product required 1 minute of processing time on G and 2 mins. on H; Type B product requires 1 minute on G and 1 minute on H. Machine G is available for no more than 6 hours 40 minutes, while Machine H is available for no more than 10 hours during any working day. Formulate a linear programming model for this problem.

Q 17: The Primo Insurance Company is introducing two new product lines: special risk insurance and mortgages. The expected profit is $5 per unit on special risk insurance and $2 per unit on mortgages. Management wishes to establish sales quotas for the new product lines to maximize total expected profit. The work requirements are as follows. Formulate a linear programming model for this problem.

Work-Hours per Unit

Department Special Risk MortgageWork-Hour Available

Underwriting 3 2 2400

2

Administration 0 1 800Claims 2 0 1200

Q 18: The owner of Metro sports wishes to determine how many advertisements to place in three selected monthly magazine A, B, C. His objective is to advertise in such a way that total exposure to principal buyers of expensive sports goods is maximized. Percentage of readers for each magazine is known. Exposure in any particular magazine is the number of advertisements placed multiplied by the number of principal buyers. The following data may be used:

MagazinesExposurecategory A B C

Readers 1 lakh 0.6 lakh 0.4 lakhPrincipal buyers 10% 15% 7%

Cost per advertisement (Rs)

5000 4500 4250

The budgeted amount is Rs 1 lakh for the advertisements. The owner has already decided that magazine A should have no more than six advertisements and that B and C each have at least two advertisements. Formulate a linear programming model for the problem.

Q 19: B&K grocery store sells two types of soft drinks: the brand name A1 Cola and the cheaper store brand B&K Cola. The margin of profit on the A1 Cola is about 5 cents per can, whereas on B&K Cola, it is 7 cents per can. On the average, the store sells no more than 500 cans of both colas a day. Although A1 is a better recognized name, customers tend to buy more of the B&K brand because it is considerably cheaper. It is estimated that the B&K brand outsells the A1 brand by a ratio of at least 2:1. However, B&K sells at least 100 cans of A1 a day. How many cans of each brand should the store carry daily to maximize profit?

Q 20: Electra produces two types of electric motors, each on a separate assembly line. The respective daily capacities of the two lines are 600 and 750 motors. Type 1 motor uses 10 units of a certain electronic component, and type 2 motor uses only 8 units. The supplier of the component can provide 8000 pieces a day. The profits per motor for types 1 and 2 are $60 and $40, respectively. Determine the optimum daily production mix.

Q 21: A company produces two products, A and B. The sales volume for A is at least 80% of the total sales of both A and B. However, the company cannot sell more than 100 unit of A per day. Both products use one raw material whose maximum daily availability is limited to 240 lb a day. The usage rates of the raw material are 2 lb per unit of A and 4 lb per unit of B. The unit prices for A and B are $20 and $50, respectively. Determine the optimal product mix for the company.

Q 22: A person wants to decide the constituents of a diet which will fulfill his daily requirements of protein, fats and carbohydrates at the minimum cost. The choice is to be made from four different types of foods. The yield per unit of the foods is given in the table below:

Food Type Proteins Fats Carbohydrates Cost per unit (Rs.)

1 3 2 6 452 4 2 4 403 8 7 7 854 6 5 4 65

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Min. requirement

800 200 700

Formulate the linear programming model for the problem.

Q 23: A manufacturing firm has three products on their production line. The production capacity for each product is 50, 30 and 45, respectively. The total availability in the production shop is of 300 man-hours. The manufacturing time required for each product is 0.5, 1.5 and 2.0 man-hours, respectively. The per unit profits of products are Rs. 10, 15 and 20, respectively. If the company has a minimum daily demand to be met of 25, 20, and 35 units for the respective products, formulate the problem as LP model to maximize the total profit.

Q 24: An investor has a sum of Rs. 1,00,000/= which he wishes to invest in income–producing securities and government bonds so as to maximize his annual return. The five possible investments he has available are given below:

Organisation Type of Scrip Paying Periodicity of Income

Oil Company A Stock 11% Annual dividend

Oil Company B Stock 7.5% Annual dividend

Utility Company C Stock 8% Annual dividend

Utility Company D Stock 6% Annual dividend

Government Bond 5% Annual interest

The investor has made the following decision regarding his portfolio planning:

The Total Investment in May not exceedOil companies Rs. 30,000/=

Utilities Rs. 50,000/=Oil Company A Rs. 20,000/=

Utility C Rs. 30,000/=Oil Stocks The total

investment inUtility Stocks

Oil Company A and Utility C combined The investmentin Government Bonds

Formulate a linear programming model for the problem.

Q 25. Glass Discount Store is opening a new department with a storage area of 10,000 sq.ft. Management considers four products for display. Prod. A costs $55, sells for $130, and needs 24 sq.ft./unit for storage Prod. B costs $100, sells for $120, and needs 20 sq.ft./unit for storage Prod. C costs $200, sells for $295, and needs 36 sq.ft./unit for storage Prod. D costs $300, sells for $399, and needs 50 sq.ft./unit for storageIt is required that at least 10 units of each product are on display. The company's goal is profit maximization. How many units of each product should be on display if the company has $600,000 available for purchasing the products?

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UNIT II: SIMPLEX METHOD AND DUALITY

CONCEPTS

Q 1: Define the following terms: (a) Objective function, (b) Decision variables, (c) Optimal solution, (d) Feasible solution, (e) Unrestricted variables, (f) Slack, (g) Surplus.Q 2: Give some example of linear programming problems.Q 3: Establish difference between a) Feasible solution, b) Basic feasible solution, c) Degenerate basic feasible solution, d) Optimal basic feasible solution, e) adjacent solution, and f) Corner-point feasible solution.Q 4: Explain the assumptions of LP model.Q 5: Provide a schematic description of simplex method.Q 6: Write the difference in the simplex solution procedure for a maximization problem and a minimization problem of an LP.Q 7: Explain the concept of duality in LP.Q 8: Describe the rules for writing the dual of a LP.Q 9: Describe a linear programming application in management.

GRAPHICAL SOLUTION

Q 10: Consider the following constraints:(a) –3x1 + x2 7. (b) 2x1 – 3x2 8. Determine the feasible space for each individual constraint, given that x1, x2 0.

Q 11: Solve the following problem by graphical method:

Min 2x1 + 3x2

subject tox1 > = 125

x1 + x2 >= 350 2x1 + x2 <= 600 x1, x2 >= 0

Q 12: Solve the following LP by graphical method.

Maximize Z = x1 +2x2

subject to x1 + x2 1 4x1 + x2 3, and x1, x2 0.

Q 13: Consider the following problem.

Maximize Z = 3x1 + 2x2,subject to 2x1 + x2 6

x1 + 2x2 6and x1 0 , x2 0.

(a) Use the graphical method to solve this problem. Circle all the corner points on the graph.

(b) For each CPF solution, identify the pair of constraint boundary equations it satisfies.(c) For each CPF solution, identify its adjacent CPF solutions.(d) Calculate Z for each CPF solution. Use this information to identify an optimal solution.

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Q 14: A dietitian wants to design a breakfast menu for certain hospital patients. The menu is to include two items A and B. Suppose that each ounce of A provides 2 units of vitamin C and 2 units of iron and each ounce of B provides 1 unit of vitamin C and 2 units of iron. Suppose the cost of A is 4¢/ounce and the cost of B is 3¢/ounce. If the breakfast menu must provide at least 8 units of vitamin C and 10 units of iron, how many ounces of each item should be provided in order to meet the iron and vitamin C requirements for the least cost? What will this breakfast cost? Solve by graphical method of LP.

SIMPLEX METHOD

Q 15: Use simplex method to solve the LP

Maximize Z = 4x1 + 10x2

subject to 2x1 + x2 50 2x1 + 5x2 100 2x1 + 3x2 90 x1, x2 0.

Q 16: Solve the following problem by simplex method

Max Z = 3 x1 + 8 x2S.t. 2 x1 + 4 x2 <= 1600

6 x1 + 2 x2 <= 1800 x2 <= 350 x1, x2 >= 0

Q 17: Solve the following problem by simplex method

Maximize Z = 3x1 + 2x2,subject to 2x1 + x2 6

x1 + 2x2 6and x1 0 , x2 0.

Q 18: Solve the following LP by simplex method.

Maximize Z = x1 +2x2

subject to x1 + x2 1 4x1 + x2 3, and x1, x2 0.

Q 19: Use Big – M method to solve the following problem:

Minimize Z = 60 x1 + 80 x2S.t. 20 x1 + 30 x2 >= 900 40 x1 +30 x2 >= 1200

x1, x2 >= 0

Q 20: Solve the following problem by Big-M method:Min 2x1 + 3x2

subject tox1 > = 125

x1 + x2 >= 350 2x1 + x2 <= 600 x1, x2 >= 0Q 21: Solve the following problem by Big-M method:

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Max 50x1 + 40x2

subject to 3x1 + 5x2 <= 150 x2 <= 20 8x1 + 5x2 <= 300 x1 + x2 >= 25 x1, x2 >= 0

DUALITY AND DUAL LINEAR PROGRAMMING

Q 22: Determine the dual of the following problem:

Max Z = 30 x1 + 20 x2S.t. 2 xl + 3 x2 <= 45

4 x1 + 5 x2 <= 85 x1, x2 >= 0

Q 23: Determine the dual of the following problem:

Max 50x1 + 40x2

subject to 3x1 + 5x2 <= 150 x2 <= 20 8x1 + 5x2 <= 300 x1, x2 >= 0

Q 24: Determine the dual of the following problem:

Min 2x1 - 3x2

subject to x1 + 2x2 <= 12 4 x1 - 2x2 >= 3 6x1 - x2 = 10 x1, x2 >= 0

Q 25: Determine the dual of the problem

Minimize Z = 5x1 + 2x2 + x3

subject to 2x1 + 3x2 +x3 20 6x1 + 8x2 + 5x3 30 7x1 + x2 + 3x3 40 x1 + 2x2 + 4x3 50 and, x1, x2, x3 0.

Q 26: Write the dual of the following LP

Maximize Z= 2x1 + 5x2 + 3x3

subject to 2x1 + 4x2 -3x3 8 -2x1 - 2x2 + 3x3 -7 x1 + 3x2 - 5x3 -2 4x1 + x2 + 3x3 4, and x1, x2, x3 0.

UNIT III: TRANSPORTATION AND ASSIGNMENT MODELS

CONCEPTS

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Q 1: Describe Transportation problem (TP).Q 2: Explain the main elements of transportation problem.Q 3: What do you understand by unbalanced TP? How is it converted to balanced TP?Q 4: Describe the main features of transportation simplex method. Q 5: Describe the following: (a) North-west corner rule, (b) Least cost method, and (c) Vogel’s approximation method.Q 6: Describe an industrial application of transportation problemQ 7: Describe the Assignment Problem (AP). How is it special case of TP? Explain.Q 8: Explain the Hungarian method for solving AP.Q 9: Describe an industrial application of assignment problem

TRANSPORTATION PROBLEM Q 10: Determine the basic feasible solution to the following transportation problem.

Distribution Centres

Sources D1 D2 D3 D4 SupplyS1 2 3 11 7 6S2 1 0 6 1 1S3 5 8 15 9 10

Requirements 7 5 3 2

Q 11: Find the basic feasible solution for the following transportation problem using (i) north west corner rule (ii) Vogel’s Approximation Method.

Destination1 2 3 4 Supply

1 10 20 5 7 10

2 13 9 12 8 20

Source 3 4 5 7 9 30

4 14 7 1 0 40

5 3 12 5 19 50

Demand 60 60 20 10

Q 12: Powerco has three electric power plants that supply the needs of four cities. Each power plant can supply the following KWH of electricity: Plant 1 – 35 million; Plant 2 – 50 million; Plant 3 – 40 million. The peak power demands (in KWH) in the four cities, which occur at the same time (2 p.m.), are as follows: City 1 – 45 million; City 2 – 20 million; City 3 – 30 million; City 4 – 30 million. The cost of sending 1 million KWH from a plant to a city depends on the distance the electricity must travel, and is given in the following table (in $). Solve this problem as a transportation problem.

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Plant \ City 1 2 3 41 8 6 10 92 9 12 13 73 14 9 16 5

Q 13: Consider a transportation problem with three warehouses and four markets with the following data:

Warehouse \ Market

M1 M2 M3 M4 Supply

W1 2 2 2 1 3W2 10 8 5 4 7W3 7 6 6 8 5

Demand 4 3 4 4

Q 14: Solve the following transportation problem

1 2 3 4 SupplyA 12 13 4 6 500B 6 4 10 11 700C 10 9 12 4 800

Demand 400 900 200 500

Q 15: Solve the following transportation problem

1 2 3 4 SupplyA 3 2 7 6 5000B 7 5 2 3 6000C 2 5 4 5 2500

Demand 6000 4000 2000 1500

Q 16: A company has two plants (A, B) producing a certain product that is to be shipped to three distribution centers (1,2,3). The unit production costs are the same at the two plants, and the shipping cost per unit is shown below.

Shipments are made once per week. During each week, each plant produces at most 60 units and each distribution center needs at least 40 units. How many units should be shipped from each distribution center to each distribution center, so as to minimize cost?

ASSIGNMENT PROBLEM

Q 17: A department has five employees with five jobs to be performed. The time (in hours) each man will take to perform each job is given in the following cost matrix.

EmployeesI II III IV V

Jobs A 10 5 13 15 16B 3 9 18 13 6C 10 7 2 2 2D 7 11 9 7 12

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E 7 9 10 4 12

How should the jobs be allocated, one per employee, so as to minimize the total man hours?

Q 18: In the modification of plant layout of a factory, four new machines M1, M2, M3, M4 are to be installed in a machine shop. There are five vacant places A, B, C, D and E that are available. Because of limited space, machine M2 can not be placed at C and M3 cannot be placed at A. The cost of placing of machine i at place j (in rupees) is shown below:

A B C D EMachine M1 9 11 15 10 11

M2 12 9 - 10 9M3 - 11 14 11 7M4 14 8 12 7 8

Find the optimal assignment schedule.

Q 19: Solve the following assignment problem

24 10 21 1114 22 10 1515 17 20 1911 19 14 13

Q 20: Solve the following assignment problem

10 15 99 18 56 14 3

UNIT IV: GAME THEORY AND QUEUING THEORY

CONCEPTS

Q 1: What is game theory? State assumptions underlying it. Discuss its importance in business decisions.Q 2: Explain the concept of dominance in game theory.Q 3: Explain the concept of minimax criterion in game theory.Q 4: Explain the concept of unstable solution in game theory.Q 5: Explain a) Saddle point b) Pure strategy c) Mixed strategyQ 6: What is queuing theory? Explain with examples the situations in which it can be applied.Describe various forms and structures of queuing problems.Q 7: Describe an industrial application of queuing problemQ 8: (a) Describe classifications of queuing models.

(b) What are the Kendall’s notations for representing queuing models?(c) Consider a barber shop. Demonstrate that it is a queuing system by describing its components.

Q 9: Derive various analytical expressions for M/M/1 queues.

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GAME THEORY

Q 10: Reduce the size of the game whose matrix is given by

Player BI II III

I - 4 6 3Player A II - 3 - 3 4

III 2 - 3 4

Q 11: Reduce the size of the game whose matrix is given by

Player BI II III

I 9 8 -7Player A II 3 -6 4

III 6 7 7

Q 12: For the following payoff table, determine the optimal strategy for each player by successively eliminating dominated strategies. (Indicate the order in which you eliminated strategies).

Player 2Strateg

y1 2 3

1 1 2 0Player 1 2 2 - 3 - 2

3 0 3 - 1

Q 13: Use minimax criterion to solve the game whose payoff matrix is given by

Player BI II III IV V

I - 2 0 0 5 3Player A II 3 2 1 2 2

III - 4 - 3 0 - 2 6IV 5 3 - 4 2 - 6

Q 14: Find the saddle point of the game having the following payoff table.

Player 2Strateg

y1 2 3

1 1 - 1 1Player 1 2 - 2 0 3

3 3 1 2

Use the minimax criterion to find the best strategy for each player. Does this game have a saddle point? Is it a stable game?

Q 15: Solve the following game:

Strategy B1 B2 B3A1 12 - 8 - 2A2 6 7 3

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A3 - 10 - 6 2

Q 16: Use minimax criterion to solve the game whose payoff matrix is given by

Player BI II III

I 12 -8 -2Player A II 6 7 3

III -10 -6 2

QUEUING THEORY

Q 17: In each of following situations, identify the customer and the server: (i) Planes arriving at an airport.(ii) Taxi stand.(iii) Letters processed in a post office.(iv) Registration for classes in a university.(v) Check-out operation in a supermarket.(vi) Parking lot operation.

Q 18: Customers arrive at a booking office (single server) at a rate of 25 per hour. Time required to serve a customer has exponential distribution with a mean of 120 seconds. Find:

(a) mean waiting time of a customer in the queue,(b) average queue length,(c) average number of customers in the system

Q 19: A self–service store employs one cashier at its counter. An average of 9 customers arrives every 5 minutes while the cashier can serve 10 customers in 5 minutes. Assuming Poisson distribution for arrival rate and exponential distribution for service rate, find

(i) Average number of customers in the system.(ii) Average number of customers in queue or average queue length.(iii) Average time a customers spends in the system.(iv) Average time a customer waits before being served.

Q 20: Customers arrive at a booking office window, being manned by a single individual at a rate of 25 per hour. Time required to serve a customer has exponential distribution with a mean of 120 seconds. Find the mean waiting time of a customer in the queue.

Q 21: Trucks arrive at Schips Truck Dock at the rate of 3 per hour. Trucks are loaded at the rate of 4 per hour. Assuming Poisson distribution for arrival rate and exponential distribution for service rate, find

(i) Probability that the dock is idle.(ii) Average number of customers in the system.(iii) Average number of customers in queue or average queue length.(iv) Average time a customers spends in the system.(v) Average time a customer waits before being served.

UNIT V: NETWORK PLANNING

CONCEPTS

Q 1: What are the challenges involved in managing large-scale projects?What kinds of decision-making situations may be analyzed using PERT and CPM techniques?Q 2: What is a critical path? Can a project have multiple paths?Describe critical-path method in detail.

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Q 3: Describe various assumptions/approximations of PERT approach.Describe PERT method in detail.Q 4: Describe various applications of network planning techniques in management. Q 5: Describe an industrial application of project management

CRITICAL PATH METHOD

Q 6: Draw a network for the following project.(a) A is the start activity and K is the end activity.(b) J is the successor activity to F.(c) A precedes B.(d) C and D are successor activity to B.(e) D is the preceding activity to G.(f) E and F occur after activity C.(g) E precedes F.(h) G precedes H.(i) H precedes J.(j) F restrains the occurrence of H.(k) K succeeds activity J.

Q 7: Draw a network for an example project of buying a small business with the following activity list.

Activity Description Immediate Predecesso

rs A Develop a list of sources for financing -B Analyse the financial records of the

business-

C Develop a business plan BD Submit a proposal to lending institution A,C

Q 8: Draw a network for a project with the following activity list.

Activity Immediate Predecessors

A -B -C BD A,CE CF CG D, E, F

Q 9: Christine has done some planning for her sales management training project and has the following activity list. Construct the project network for this project.

Activity Activity Description Immediate Predecessors

Estimated Duration

A Select location - 2 weeksB Obtain speakers - 3 weeksC Make travel plans for

speakersA, B 2 weeks

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D Prepare and mail brochures A, B 2 weeksE Take reservations D 3 weeks

Q 10: Christine has done more detailed planning for her sales management training project and has the following activity list.

Activity Activity Description Immediate Predecessors

Estimated Duration

A Select location - 2 weeksB Obtain Keynote speaker - 1 weeksC Obtain other speaker B 2 weeksD Make travel plans for keynote

speakerA, B 2 weeks

E Make travel plans for other speakers

A, C 3 weeks

F Make food arrangements A 2 weeksG Negotiate hotel rates A 1 weeksH Prepare brochure C, G 1 weeksI Mail brochure H 1 weeksJ Take reservations I 3 weeksK Prepare handouts C, F 4 weeks

Construct the project network for this project.

Q 11: Construct the project network for a project with the following activity list.

Activity Immediate Predecessors

Estimated Duration

A - 1 monthB A 2 monthsC B 4 monthsD B 3 monthsE B 2 monthsF C 3 monthsG D, E 5 monthsH F 1 monthI G, H 4 monthsJ I 2 monthsK I 3 monthsL J 3 monthsM K 5 monthsN L 4 months

Q 12: Draw a network for production planning.

Activity Description Time (weeks) Preceded byA Market research 15 -B Make drawings 15 -C Decide production policy 3 AD Prepare sales programme 5 AE Prepare operation sheets 8 B, CF Buy materials 12 B, CG Plan labour force 1 EH Make tools 14 E

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I Schedule production 3 D, GJ Produce product 14 F, H, I

Q 13: Consider the following data for the activities of a project:

Activity : A B C D E FImmediate predecessors

: - A A B, C - E

Duration (days) : 2 3 4 6 2 8

Draw the network and find the critical path.

Q 14: You are given the following information about a project consisting of six activities:

Activity Immediate Predecessors

Estimated Duration

A - 5 monthsB - 1 monthC B 2 monthsD A, C 4 monthsE A 6 monthsF D, E 3 months

(a) Construct the project network for this project.(b) Find the earliest times, latest times, and slack for each activity. Which is a critical

path?

Q 15: Find the critical path for the following Shopping Center Expansion Project:

ACTIVITY DESCRIPTION IMMEDIATE PREDECESSOR

TIME ( WEEKS )

A Architectural Drawing - 5B Identify new tenants - 6C Prospectus for tenants A 4D Select Contractor A 3E Prepare Building Permits A 1

F Approval of Building Permits

E 4

G Construction D,F 14H Finalisation of Contract

with tenantsB,C 12

I Occupation by tenants G,H 2

Q 16: Draw the project network and find the critical path for the following New Product Development Project:

Activity

Description Required Predecessor

Duration

A Product design (None) 5 months

B Market research (None) 1C Production

analysisA 2

D Product model A 3

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E Sales brochure A 2F Cost analysis C 3G Product testing D 4H Sales training B, E 2I Pricing H 1J Project report F, G, I 1

P.E.R.T. METHOD

Q 17: Sharon Lowe, vice president for marketing for the Electronic Toys Company, is about to begin a project to design an advertising campaign for a new line of toys. She wants the project completed within 57 days in time to launch the advertising campaign at the beginning of the Christmas season. Sharon has identified the six activities (labeled A, B, …, F) needed to execute this project. Considering the order in which these activities need to occur, she also has constructed the following project network.

Using the PERT three-estimate approach, Sharon has obtained the following estimates of the duration of each activity.

Activity Optimistic Estimate

Most Likely Estimate

Pessimistic Estimate

A 12 days 12 days 12 daysB 15 days 21 days 39 daysC 12 days 15 days 18 daysD 18 days 27 days 36 daysE 12 days 18 days 24 daysF 2 days 5 days 14 days

(a) Find the estimate of the mean and variance of the duration of each activity.(b) Find the mean critical path.(c) Use the mean critical path to find the approximate probability that the advertising

campaign will be ready to launch within 57 days.(d) Now consider the other path through the project network. Find the approximate

probability that this path will be completed within 57 days.

Q 18: Consider the following building project to put an A/C unit in a factory.

Task ID Task DescriptionPrerequisit

es

Optimistic

Duration

Most Likely

Duration

Pessimistic

Duration

1 Build internal components none 1 2 3

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START FINISH

A

B

C F

D

E

2 Modify roof and floor none 2 3 4

3 Construct collection stack 1 1 2 3

4 Pour concrete and install frame

2 2 4 6

5 Build high-temperature burner

3 1 4 7

6 Install control system 3 1 2 9

7 Install air pollution device 4, 5 3 4 11

8 Inspection and testing 6, 7 1 2 3

(a) Draw the project network. (b) Find the estimate of the mean and variance of the duration of each activity.(c) Find the mean critical path.

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