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understanding management
Quantitative Techniques In Management 4th Edition N D VOHRA - Solutions
5. Solve the following LPP using dynamic programming. Maximise Subject to Z=30x +20x2 2x + 4x2 12 15x + 5x240 10
4. Solve the following LPP using dynamic programming. Minimise Subject to Z=40x +24x2 20x + 50x2 4800 80x + 50x2 7200 XX
3. (a) Solve the following network problem for shortest route between the origin and destination. The numbers on the arrows connecting different pairs of nodes represent distance in kms. Use dynamic programming.(b) Reconsider the network given in (a). If the road between nodes 6 and 9 is blown up
2. Obtain the shortest route (i) from A to E, and (ii) A to any point on D, for the following network, using dynamic programming. 8 B A 14 9 D 7 10 D; 5 10 12 C E 12 6 11 B 6 10. 10 C3 19
1. You are given the following network for a project, for which it is desired to obtain the longest path from start to finish. Required:(i) IdentifY the stages and states for the dynamic programming formulation of this problem.(ii) Use dynamic programming to solve this problem. Start 8 6 3 6 2 2 5
Solve the following LPP using dynamic programming Maximise Subject to Z= 8x + 16x2 X + x2 200 X2 125 3x + 6x2 900
You are given the following network wherein the numbers indicate the distance between various pairs of nodes. Use dynamic programming to determine the shortest path between the origin and destination. What is the second shortest path and the what is the distance involved? 6 A 10 9 B 5 12 10 F 5 6
A company has trained five salespersons in the past three months, whom it wants to assign to three sales regions: North, West and South. It has been estimated that increases in sales (in lakhs of rupees) over the next year in each region, if each were allocated different number of salesmen, are as
Four types of items are under consideration for loading on a plane, with an unlimited supply of each type. The weights and values of various types of items are given below:Determine as to which items should be loaded on the plane, and in what quantities, if the maximum capacity of the plane is 13
Solve this LPP using dynamic programming. Maximise subject to Z = 40x + 35x2 2x + 3x2 60 4x + 3x2 96
A company has Rs 6,00,000 to invest, and a total of three investments are available to it.If xi rupees (in lakhs) are invested in investment j, then a net present value (in lakhs of rupees) of ~{xj) is obtained, where ~{xj)'s are given below:The amount placed in each investment has to be in
10. A small project consists of seven activities for which the relevant data are given below:(i) Draw the network and find the project completion time.(ii) Calculate total float for each of the activities.(iii) Draw the time scaled diagram. Activity Preceding activities Activity duration (Days) A 4
Given the following information:Determine ES, EF, LS, LF, IF, and FF for each of the activities. Activity Immediate Predecessors Time (Days) ABCDEFG none 9 Activity H Immediate Predecessors Time (Days) D, G 4 none 4 I E, G 3 15 J H, I 3 none 10 K D A 7 L B 5 M J, K L 1961 10 C, F 4
8. (a) The following list of activities must be accomplished in order to complete a construction project:Construct a network for this project.Find the critical path and the duration the project.(b) Assume that you are the project manager of the project mentioned above. The project has pro- gressed
7. Activities involved in Generator Modernisation at BHEL Haridwar with their durations and imme diate predecessors are as under:(i) Identify and remove redundant immediate predecessor(s) if any.(ii) Develop an arrow diagram and find the project completion time.(iii) Calculate total float for the
Given that a project involves activities A, B, C, ... , I; each requiring completion time, in days, as per the following table:Given that activity A precedes activities D and E; activities B and D precede activity F; activity C precedes activity G; activities B and G precede H, and activities F and
5. For a small project of 12 activities, the details are given below. Draw the network and compute earliest occurrence time, latest occurrence time, critical activities and project completion time: Activity : A B C D E F G H I J K L Dependence : - B, C A C Duration (Days) : 9 4 7 00 8 7 5 29 E E 10
4. Bring out the dependency relationships from the network given in Figure 12.50. K L M N P 8 Q 4 6 R S 10 9
3. The information about various activities of a project and the precedence relationships are given below.Draw a network using as few dummy activities as possible. Activity Immediate predecessors Activity Immediate predecessors I F, G, H A BCDEFGH B, C, D A, B, C, D A, B, C, D A, B, C, D O, E, N K
2. The Commerce Department of a university is considering to hold a faculty development programme.It has planned the following activities. Prepare a network diagram showing the interrelationships of the various activities: Activity Description A Design conference meetings and theme B Design front
1. Draw a network: ABCDERG F 1111ARE Activity Immediate predecessor(s) Activity Immediate predecessor(s) H I F C, D, G, H K L I I J, K B M J, K N M
A project consists of nine activities whose time estimates (in weeks) and other characteristics are given below:(i) Show the PERT network for the project.(ii) Identify the critical activities.(iii) What is the expected project completion time and its variance?(iv) What is the probability of
Following table gives the list of various activities involved in the launch of a new CREDIT CARD service by a company, their immediate predecessors and their expected durations (in days):where T0 = most optimistic time, Tm = most likely time, and TP = most pessimistic time (a) Draw an arrow diagram
project consists of 10 activities, each of which requires either, or both, of the two types of resources R1 and R2 for its performance. The duration of the activities and their resource requirements are as follows:Resource availability: R1 : 7 units, R2 : 5 units Determine the duration of the
Following are the manpower requirements for each activity in a project:(a) Draw the network and find out total float and free float for each activity.(b) The contractor stipulates that during the first 26 days only 4 to 5 men and during remaining days 8 to 11 men only can be made available.
The data for a project are:(a) Draw a project network diagram and find the critical path.(b) If a deadline of 17 weeks is imposed for completion of the project, what activities will be crashed, what would be the additional costs, and what would be the critical activities of the network after
The following table shows, for each activity of a project, the normal and crash times as also the normal and crash costs. The contract includes a penalty clause of Rs 200 per day in excess of 19 days.The overhead cost is Rs 400 per day.(a) Draw the project network and determine the critical path.
A project has the following characteristics:(a) Draw a PERT network for this project.(b) Find the critical path and the project completion time(c) Prepare an activity schedule showing the ES, EF, LS, LF, and slack for each activity.(d) Will the critical path change if activity G takes 10 weeks
(a) Draw a network corresponding to the following information.(b) Find the earliest and latest scheduling times of various activities.(c) Obtain the total, interfering, free, and independent floats for each of the activities.(d) Can this project be completed within 65 days?(e) What would be the
A cabinet manufacturing company is planning to introduce a new model of cabinets which requires the following tasks:The wheels are mounted after they are prepared. The base cannot be attached until the sides are assembled and the wheels mounted. The top cannot be attached nor the brackets inserted
A work project consists of twelve activities labelled A through L. Upon being asked to specify the order in which the jobs had to be done, the manager answered as follows:A, B, and C are the first activities of the project and can start simultaneously and immediately; A and 8 precede D while 8
The owner of a chain of fast-food restaurants is considering a new computer system for accounting and inventory control. A computer company sent the following information about the system installation(a) Construct an arrow diagram for this problem, determine the critical path and state the expected
For a project consisting of several activities, the duration and required resources of carrying out each of the activities and their availabilities are given below.Resources Availability:No. of operators = 50 Equipment X = 1 Equipment Y = 1 Equipment Z = 1 (a) Draw the network, identify critical
You are given in Figure 12.18 the network corresponding to a project. The activities of the project require manpower of similar type and the requirements, in terms of the number of men, are shown in parentheses next to the activity identification and timing (in days) for each. Determine how the
The following table shows for each activity needed to complete the project the normal time, the shortest time in which the activity can be completed of a building contract and the cost per day for reducing the time of each activity. The contract includes a penalty clause of Rs 100 per day over 17
Draw a network corresponding to the following information. Obtain the early and late start and completion times, and determine the critical activities. Activity : 1-2 Duration : 1-3 2-6 3-4 3-5 4-6 5-6 5-7 6-7 4 6 87 4 6 5 19 10
Information on the activities required for a project is as follows:Draw the network and calculate the earliest start (ES), earliest finish (EF), latest start (LS), and latest finish (LF) times of each of the activities. Name Activities Node : < A : 1-2 1-3 ? B C DE F G H J K 1-4 2-5 3-5 3-6 3-7 4-6
Draw an arrow diagram showing the following relationships: Activity : A B C D Immediate prede.: - E A, B B, C A, B C F G H I J D, E, F D G K L M N GH, JKI, L
The following precedence relationships are given for a project consisting of eight activitiesIdentify redundant immediate predecessor(s) if any. Activity Immediate Predecessor(s) ABCDEFGH A A A, C B, D B, D, E B, D E, F, G
27. A production manager faces the problem of job allocation between two assembly lines. The production rate of assembly line 1 is 10 units per hour, and the production rate of assembly line 2 is 12 units per hour. The normal working period for both lines is 8 hours per day. The production manager
26. The Soft-Ad Company is planning for the next month an advertising campaign for one of its clients, Amco. Soft-Ad is considering advertising during sports programmes and during soap operas on TV. Amco has specified the following goals:Further, the company has budgeted Rs 12 lakhs for these
25. The Super-Star Company produces three models of microcomputers-SS 100, SS 150 and SS250. Most of the components are imported from the Far East, and the company only assembles the microcomputers. The operation hours required to produce one unit of the microcomputers are 2 hours for SSlO0, 3
24. Consider the following Goal Programming model:All variables to be non-negative.The modified simplex tableau giving IBFS is shown below. Carry out the simplex iterations until an optimal solution is reached. State clearly the optimal solution and to what extent the goals have been achieved.
23. The PQR manufacturing Company produces four different airplane parts from fabricated sheet metal for several major aircraft companies. The manufacturing process consists of four operations-stamping, assembly, finishing and packaging. The processing times per unit for each operation and total
22. Amro Mutual Fund has Rs 20 lakh available for investment in Government bonds, blue chips, speculative stocks, and gold. The expected annual return on these investments is given below:The investment policy of Amro is:(a) The amount of investment can in no case be more than Rs 20 lakh.(b) Never
21. A farmer has 1,000 acres of land on which he intends to plant corn, wheat and soyabeans. Each acre of corn costs Rs 10,000 for preparation, requires 7 worker days of labour, and yields a profit of Rs 3,000. An acre of wheat costs Rs 12,000 to prepare, requires 10 worker days of labour, and
20. A rural clinic hires its staff from nearby cities and towns on a part-time basis. The clinic attempts to have a general practitioner (GP), a nurse and an internist on duty during at least a portion of each week. The clinic has a weekly budget of Rs 1,200. A GP charges the clinic Rs 40 per hour,
19. You have been retained as a consultant to formulate a model for job evaluation in an organisation.The data collected for 5 representative benchmark jobs across the hierarchy of all blue collar jobs evaluated on a 5-point rating scale for four factors viz. Skill, Effort, Working Condition, and
18. The ABC Ice Cream Company has a distribution depot in Greater Kailash Part I for distributing ice cream in South Delhi. There are four vendors located in different parts of South Delhi (call them A, B, C and D) who have to be supplied ice cream everyday. The following matrix displays the
1 7. The expected times required to be taken by a salesman in travelling from one city to another are as follows:How should the salesman plan his trip so that he covers each of these cities no more than once, and completes his trip in minimum possible time required for travelling? Time (in hours)
16. A machine operator processes four types of items on his machine and he must choose a sequence for them. The set-up cost per change depends on the items currently on machine and the set-up made according to the following table:If he processes each of the items once each week, then how should he
A salesman has to visit five cities A, B, C, D, and E. The distances (in hundred km) between the five cities are as follows:If the salesman starts from city A and has to come back to city A, which route should he select so that the total distance travelled by him is minimum. A ABC C 17 16 18 E 14 D
14. The following table is the cost matrix for travel among a particular set of locations. Determine a minimum cost, travelling salesman itinerary: States 1 2 3 4 5 1 2345 2 15 15 15 22 22 17 18 22 16 17 15 17 19 15 16 19 20 19 5 19 17 21 20
13. The dietician at the local hospital is planning the breakfast menu for the maternity ward patients.She is planning a special nonfattening diet, and has chosen cottage cheese and scrambled eggs for breakfast. She is primarily concerned with Vitamin E and iron requirements in planning the
12. A company manufactures two products P1 and P2, which require time on three different machines M1, M2 andM3• Product P1 requires 2 hours on machineM1 and 1 hour onM1• Product P2 requires 4 hours on machine M2 and 2 hours on machine M3• There are 50 hours available on machine M1, 16 hours
11. A manufacturer of toys makes two types of toys, A and B. Processing of these toys is done on two machines X and Y. Toy A requires 2 hours on machine X and 6 hours on machine Y. Toy B requires 4 hours on machine X and 5 hours on machine Y. There are 18 hours of time per day available on machine
10. MaximiseSubject to Z=2x+4x2 + 3x3
9. A manufacturer of baby dolls makes two types of dolls. One is sold under the brand name Molina and the other Suzie. Processing of these two dolls is done on two machines A and B. The processing time for each Molina is 3 hours and 4 hours on machines A and B respectively, and that for each Suzie
8. Find the optimal solution to the following integer programming problem, using Gomory's cutting plane method.Given below is the optimal tableau of the associated Linear Programme, with no restriction on the integrality of x' s. Maximise Subject to Z=7x + 9x2 -x+3x2 LE 18 7x1 x2 LE 105 x and x2 to
7. An exporter of ready-made garments makes two types of shirts: X and Y. He makes a profit of Rs 10 and Rs 40 per shirt on X and Y, respectively. He has two tailors, A and B, at his disposal to stitch these shirts. Tailors A and B can devote at the most 7 hours and 15 hours per day, respectively.
6. A company is considering four investment opportunities. The following table presents information about the investments (in Rs thousands):In addition, it is given that projects 1 and 2 are mutually exclusive, while project 4 is contingent upon the prior acceptance of project 3. Formulate an
5. A manufacturing company faces the following problem. Should they make or buy each of several products? The company policies specify that they will either make or buy the whole lot of each product in a complete lot. The company has four products to make or buy with six machines involved in making
4. In a project selection model, eight projects, numbered 1, 2, ... 8, are in consideration. The following conditions are given. You are required to write constraints to satisfy each of them.(a) Only one of the projects 3 and 8 has to be selected.(b) If only project 4 is selected that the project 7
3. An air-conditioning and refrigeration company has been awarded a contract for the airconditioning of a new computer installation. The company has to make a choice between two alternatives: (a) hire one or more refrigeration technicians for 6 hours a day, or (b) hire one or more part time
2. Three machines are available to make 5,000 units which are required for the internal use of a company as components to another product. If the production is to be made by any one of these machines, a set-up cost is incurred apart from the cost of making each unit on different machines. The cost
1. ABC Construction Company is faced with the problem of determining which projects it should undertake over the next 4 years. The following table gives information about each project:Formulate a zero-one programming model to maximise the estimated present value. Estimated Capital requirements
9. "Goal programming appears to be the most appropriate, flexible and powerful quantitative technique for complex decision problems involving conflicting objectives." Discuss this statement.
8. "In goal programming, we attempt to 'satisfy' or come as close as possible to satisfying, the various goals." Discuss.
7. Differentiate between non-preemptive and preemptive goal programming problems. How is the solution method different in the two cases?
6. Give a detailed account of the solution of 0-1 integer programming problems by the branchand-bound method.
5. Discuss the cutting-plane algorithm for solving integer programming problems.
4. State the Travelling Salesman problem. Discuss the branch and bound method of solving it.State the linear programming formulation of an assignment problem.
3. What is integer linear programming? Explain the merits and demerits of 'rounding off' a continuous optimal solution to an LP problem to obtain an integer solution.
2. "It may not be possible to obtain an optimal integer solution by rounding off the continuous optimum of a linear programming problem." Comment.
1. Discuss integer linear programming. Give an example each of a pure and a mixed integer linearprogramming problem.
30. In solving a preemptive goal programming problem, once the /ij values are found to be all greater than or equal to zero, with respect to the first-priority goal, the optimal solution to the problem is reached. Mark the statement as T (True) or F (False).
29. Various goals in preemptive goal programming are given equal or differential weightage. Mark the statement as T (True) or F (False).
28. Preemptive goal programming deals with situations where goals have different priority levels, so that some of them are considered more important than others. Mark the statement as T (True) or F (False).
27. The weights used in non-preemptive goal programming are indicative of decision-maker's utility for the various objectives. Mark the statement as T (True) or F (False).
26. Penalty weights may be assigned to deviational variables related to different goals, in accordance with their relative importance to the decision-maker. Mark the statement as T (True) or F (False).
25. Non-preemptive goal programming deals with situations where all goals are of comparable importance and it assumes that the marginal rate of substitution between the objectives is linear. Mark the statement as T (True) or F (False).
24. One-sided goals set a lower or an upper limit which may not fall short of or exceeded, respectively. In two-sided goals, both lower and upper limits are set. Mark the statement as T (True) or F (False).
23. One of the deviational variables for a goal constraint must equal zero. Mark the statement as T (True) or F (False).
22. Both the deviational variables related to a goal-one of underachievement and the other of overachievement-can together assume positive values. Mark the statement as T (True) or F (False).
21. The inequalities related to structural constraints are rigid while those representing goals are flexible. Mark the statement as T (True) or F (False).
20. The deviational variables for any goal constraint measure the amount of discrepancy between the value of the objective and the goal specified in respect thereof. Mark the statement as T (True) or F (False).
19. Each of the goals in a goal programming problem is either underachieved or overachieved. Mark the statement as T (True) or F (False).
18. The main idea of goal programming is to determine the individual preferences of the decisionmakers in terms of goals stated and achieve them all. Mark the statement as T (True) or F (False).
17. The multiple goals specified in a goal programming problem are usually conflicting and incommensurate. Mark the statement as T (True) or F (False).
16. Goal programming is used where multiple criteria, instead of a single objective, are used in decision-making. Mark the statement as T (True) or F (False).
15. In branch and bound algorithm for solving a travelling salesman problem, we may initially consider a tour by considering travelling in the order 1-2-3-... -n. The distance covered in this tour would provide a lower bound on the total distance of the optimal tour. Mark the statement as T (True)
14. An assignment problem solved by branch and bound method may or may not have the same optimal solution as obtained by Hungarian assignment method. Mark the statement as T (True) or F (False).
13. Basically, branching and bounding keeps dividing up the feasible region into smaller and smaller parts until the solution that optimises is determined. Mark the statement as T (True) or F (False).
12. Branch and bound method is not a standardised method. It is applie Mark the statement as T (True) or F (False).d differently for different problems.
11. A cut may or may not eliminate any points that are feasible for the IPP. Mark the statement as T (True) or F (False).
10. When two or more constraints have fractional values in the RHS, it is desirable to generate the cut by using the constraint whose RHS has a fractional part closest to 1. Mark the statement as T (True) or F (False).
9. In using cutting-plane algorithm, a cut is introduced by using a constraint in the optimal solution, obtained by simplex method, which has a fractional b; value. Mark the statement as T (True) or F (False).
8. In a pure IPP, solution to all variables, decision and slack or surplus, would always be in integer values, provided that all the coefficients in the constraints and all the right hand side values of the constraints be integers. Mark the statement as T (True) or F (False).
7. If the solution obtained by simplex method to a given problem yields only integer values, it may or may not represent optimal solution to the corresponding integer programming problem. Mark the statement as T (True) or F (False).
6. The solution to the integer programming problems can always be obtained by rounding off the fractional values obtained in solution to the problems by using graphic/simplex method. Mark the statement as T (True) or F (False).
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