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Quantitative Techniques In Management 4th Edition N D VOHRA - Solutions

14. Two companies, A and P, competing for the same product, are sharing a market. Each of these attempts to raise its share of the market. An advertising campaign by A can increase its share of the market by 7 per cent provided that P does nothing. Company P contemplates invoking a price cut if it
15. Solve the rectangular game whose paY-offmatrix for player A is: B B B3 A 8 4 -2 A 9 17 16 A3 10 20 -5
16. Two competitors are competing for the similar product. The paY-off matrix in terms of their advertising plan is shown below:Find the optimal strategies and the value of the game. Competitor B Large advt. Med. advt. Small advt. Large advt. 70 80 50 Competitor A Med. advt. 90 60 95 Small advt.
17. The management of a corporation is in the process of deciding whether to agree to negotiate with the striking union now or delay. The decision is difficult because the management does not know the union leadership's position. The union leaders may be adamant and insist on their original
18. Two firms, F1 and F2, make colour, and black and white television sets. F 1 can make either 300 per colour sets in a month or an equal number of black and white sets, and make a profit of Rs 200 per colour set and Rs 150 per black and white set. F2 can, on the other hand, make either 600 colour
19. Determine the optimal strategies and the value of the game from the following 4 x 4 paY-off matrix game Y X Y X -5 X 20 -5 X3 -5 X4 -20 -2 Y3 Y4 16 13 15 60 12 -70 10 50 -80
20. The paY-off table for a zero-sum game between A and B is shown below, where A is playing to maximise his paY-off. Solve the game for the optimal strategies of the two players and for the value of the game. B B B3 B4 A 150 -18 78 90 A 6 102 54 A3 130 -30 78 888. 70 80
21. The following matrix represents player A's paY-offin a two-person, zero-sum game:Find the optimal strategies for the two players and value of the game. Player B 30 40 -80 Player A 0 15 -20 90 20 50
22. From the following two-person, zero-sum game, fmd the optimal strategy for each player and the value of the game: Player B B B B3 B4 A 0 -1 3 5 Player A A A3 -5 2 4 5 -2 -3 4 -2
34. Formulate the following two-person, zero-sum game as an LPP and solve for the optimal strategies for both players and the value of the game. Player B Player A B B B3 A A2 A3 55 5 4 7 8 4 8 5 6
30. The theory of games is very restrictive in real world applications because of the nature of assumptions made. Mark the statement as T (True) or F (False).
Two leading firms, Nirmala Textiles Ltd. and Swati Rayons Ltd., for years have been selling shirting, which is but a small part of both firms' total sales. The Marketing Director of Nirmala Textiles raised the question, "What should the firm's strategies be in terms of advertising for the product
(a) Given the following pay-off matrix for a 2-person zero-sum game, the second row second column entry will be a saddle point only if:(i) p ~ 6 and q ~ 6(ii) p~ 6 and q~ 6(iii) p~ 6 and q~6(iv) None of the choices given above(b) The pay-off matrix for a 2-person zero-sum game is given below:From
Reduce the following two-person zero-sum game to 2 x 2 order, and obtain the optimal strategies for each player and the value of the game: Player B A B 3 B B3 B4 A2 3 24 42 0 2 4 Player A A3 4 2 4 0 A4 0 4 0 8
A company is currently involved in negotiations with its union on the upcoming wage contract.With the aid of an outside mediator, the table below was constructed by the management group. The pluses are to be interpreted as proposed wage increases while a minus figure indicates that a wage reduction
For the following pay-off matrix, find the value of the game and the strategies of players A and B using linear programming: Player A 1 Player B 2 3 1 3 -1 4 2 6 7 -2
Consider the problem given in Example 15.17 and re-solve by algebraic method.The pay-off matrix is reproduced here: Player A 1 Player B 2 3 1 3 -1 4 2. 6 7 -2
1. In a two-person game, both the players must have an equal number of strategies. Mark the statement as T (True) or F (False).
2. The zero-sum game implies that any gain of one player is exactly matched by a loss to the other so that their sum is equal to zero. Mark the statement as T (True) or F (False).
3. The solution to a game implies determining optimal strategies for both the players and value of the game. Mark the statement as T (True) or F (False).
4. The solution to a two-person game is based on the assumption that player A will always play his strategy first and the other player, B, would play his strategy thereafter. Mark the statement as T (True) or F (False).
In a small town, there are only two stores that handle sundry goods-ABC and XVZ. The total number of customers is equally divided between the two, because price and quality of goods sold are equal. Both stores have good reputation in the community, and they render equally good customer
Can there be a 'two-person zero-sum' game which has a value of zero? If so, write the payoff matrix of one such game and find out the optimal strategies of the players.Yes, a 'two-person zero-sum' game can have a value of zero. One such game is given below, where player A has 3 and player B has 4
The following is the pay-off matrix of a game being played by A and 8. Determine the optimal strategies for the players and the value of the game. A's Strategy a b 8 B's Strategy b -7 4 -6 a2
Find the optimal strategies for A and 8 in the following game. Also obtain the value of the gameThe pay-offs in relation to a1 are equal to or greater than the corresponding pay-offs from a3• Thus, a1 dominates a3• Deleting a3, we get,Now, we observe that strategy b1 is dominated by b2• Thus,
Solve the following game using graphical approach:Here A has two strategies a1 and a2 which, suppose, he plays with probabilitiesx and 1-x respectively. When B chooses to play b1, the expected payoff for A shall be 8x + (- 6) (1 -x) or 14x - 6. Similarly, the expected pay-off functions in respect
Solve the following game using the graphical method a b -7 B's Strategy b 6 -4 7 A's Strategy a -2 -4 a3 -6 8
Solve the game with the following pay-off matrix: A's Strategy b B's Strategy b 4 6 a b3 3 a2 2 4 8
Formulate the following game as an LPP and obtain its solution: a A's Strategy a a3 b B's Strategy b 6 8 24 5 b3 3 6 1 7
Solve the following game:Since some of the entries in the paY-offmatrix are negative, we shall add a constant to each of the values to convert them all into positive values. Adding 5 to each of them, we get the following revised paY-offmatrix: B B's Strategy B B3 A 3 4 -2 -3 0 1 A3 -1 -4 2 A's
Shruti Ltd. has developed a sales forecasting function for its products and the products of its competitors, Purnima Ltd. There are four strategies S1, S2, S3 and S4 available to Shruti Ltd. and three strategies P1, P2 and P3 to Purnima Ltd. The pay-offs corresponding to all the twelve combinations
There is a famous Italian game called Two-finger Morra. This game is played by two people each of whom shows one or two fingers and simultaneously guesses the number of fingers his opponent will show. If just one player guesses correctly, he wins an amount equal to the sum of the fingers shown by
5. In the two-person games, both the players are assumed to be in full knowledge of the strategies available to each one and the pay-offs resulting from each combination of strategies. Mark the statement as T (True) or F (False).
6. For a two-person game with, respectively, 6 and 5 strategies available to the two players, a total of30 conditional pay-offs would be involved. Mark the statement as T (True) or F (False).
7. Player A's strategy is determined on the basis ofmaximin criterion whereas player B's is on minimax criterion. Mark the statement as T (True) or F (False).
20. In case ofm by 2 games, the value of the game is given by the highest point in the lower envelope. Mark the statement as T (True) or F (False).
21. In a two-person zero-sum game, if the optimal solution requires one player to use a pure strategy, the other player must also do the same. Mark the statement as T (True) or F (False).
22. The dominance principle suggests that strategies of one player are dominating the strategies of the other. Mark the statement as T (True) or F (False).
23. When each value in a column is less than or equal to the corresponding value in another column, the strategy of the first column is said to dominate that of the second column. Mark the statement as T (True) or F (False).
24. It is possible that a convex linear combination of the two strategies dominates a third strategy. Mark the statement as T (True) or F (False).
25. In a 2 by norm by 2 game, each of the players can mix at most two strategies if the multiple optimal solutions do not exist. Mark the statement as T (True) or F (False).
26. Ifa constant is added to all the elements ofa paY-offmatrix, it would affect only the value of the game and not the optimal strategies of the players. Mark the statement as T (True) or F (False).
27. Every two-person zero-sum game can be represented by a pair of linear programming problems with primal-dual relationship. Mark the statement as T (True) or F (False).
28. Ifsome elements are negative in a paY-offmatrix ofa two-person, zero-sum game, a constant should be added to each of the elements of the matrix such that all of them become positive. Mark the statement as T (True) or F (False).
19. Every game is solvable irrespective of whether the players adopt pure or mixed strategies. Mark the statement as T (True) or F (False).
18. The graphic approach to the solution of games can be applied when one of the players has two strategies available. Mark the statement as T (True) or F (False).
9. It is possible that multiple minimax/maximin strategies might exist in a two-player game. Mark the statement as T (True) or F (False).
10. If the largest element in the paY-offmatrix is negative, the game should favour player B irrespective of what the optimal strategies of the players are. Mark the statement as T (True) or F (False).
11. If the value of the game is a negative value, it implies that the game is favouring player B. Mark the statement as T (True) or F (False).
12. A game is said to be fair when both the players know about the optimal strategy of each other and the losing player pays off the amount involved to the gaining player. Mark the statement as T (True) or F (False).
13. Mixed strategy for a player can involve no more than two strategies. Mark the statement as T (True) or F (False).
14. Mixed strategies for each of the players are determined such that same paY-offwould be expected irrespective of the strategy adopted by the opponent. Mark the statement as T (True) or F (False).
15. Mixed strategies are adopted by the players when they are doubtful about the optimal strategies for them. Mark the statement as T (True) or F (False).
16. Strategies involved in mixed strategies of the players are played in a random manner in the ratio determined in the optimal mix. Mark the statement as T (True) or F (False).
17. An optimal strategy (0, 2/9, 7/9) implies that in every nine plays, the first strategy should never be played, the second strategy twice and the third strategy be adopted in the remaining seven times. Mark the statement as T (True) or F (False).
29. The constant value, if added to each element of the pay-off matrix to formulate the given problem as an LPP, should be subtracted from the value of the game, determined from the solution to the LPP. Mark the statement as T (True) or F (False).
10. A state i is said to be transient ifthere is a way to leave it but it never returns to it. Mark the statement as T (True) or F (False).
12. Following are the various activities involved in a project. The cost and time information for these activities is given below:(a) Draw the PERT network.(b) Find out the critical path and the expected project completion time.(c) What is the minimum possible project completion time after crashing
The data for a PERT network is given in the following table:(i) Draw a network and estimate the earliest and latest event times for all nodes and hence derive critical path.(ii) Estimate the expected duration of the project and corresponding variance.(iii) What is the probability that the project
30. (a) For an activity in a PERT network, the following information is available:Expected time= 9.5 days Optimistic time = 5 days Variance = 6.25 days2 Calculate most likely and pessimistic times of this activity.(b) The critical activities of a project, their expected durations and standard
A small project consisting of eight activities has the following characteristics:(a) Draw the PERT network.(b) Find out the critical path and the expected project completion time.c) If a 30-week deadline is imposed, what is the probability that the project will be finished within the time limit?
28. A firm has 50 workers and one unit of each of machines M1, M2 and M3 available for carrying out a project. The activities comprising the project, their durations and resource requirements are as follows:Allocate the resources to the various activities and determine the duration of the project.
27. The time durations and labour requirements of different activities of a project are:Additional information:1. Labour rate is Rs 8 per day.2. Overhead charge for the project is Rs 50 per day.3. No activity can be started unless the needed labour force is available.4. The firm has the option of
The following activities are related to a given project.A certain type of crane is needed in carrying out activities 2-8, 3-6, 4-6 and 5-7. Only one crane of this type is available.Do you think the project completion would be delayed due to this? If yes, by how much? Activity Time Activity Time 1-2
32. A new version of a popular spreadsheet software is due to be released in a year. However, the preliminary project team has not been able to come up with an expected completion date. But, as the table that follows shows, they have been able to estimate duration for each activity (in weeks)
(a) Given that an activity takes generally 20 minutes to perform while the optimistic and pessimistic estimates of its performance are, respectively, one-half and thrice this time, answer the following: (i) What is the expected duration of this activity? (ii) What is its variance? (iii) In
38. A civil engineering firm has to bid for the construction of a darn. The activities and estimates are given as follow:The policy of the firm with respect to submitting bids is to bid the minimum amount that will provide a 95% of probability of at best breaking even. The fixed costs for the
37. The data for a small PERT project is as given below, where a represents optimistic time, m most likely time and b pessimistic time estimates (in days) of the activities A, B, ... , J, K.Precedence relationships: A, B, C can start simultaneously. A Draw the arrow network of the project.
36. A construction company is preparing a PERT network for laying the foundation of a new art museum.Given the following set of activities, their predecessor requirements and three time estimates of completion time:(i) Draw the PERT network.(ii) Compute the slack for each activity and determine the
35. (a) Using PERT, a manager was able to determine that the expected project completion time for the construction of a pleasure yatch is 21 months and the variance is 4 months2?(i) What is the probability that the project will be completed in over 22 months?(ii) What is the probability that the
A project consisting of twelve distinct activities is to be analysed by using PERT. The following information is given (time estimates are in days):Draw the PERT network. Indicate the expected total slack for each activity and hence indicate the average critical path. Within how many days would you
25. The following table lists the activities of a maintenance project.(i) Draw the project network.(ii) Find the critical path and duration of the project.(iii) Suppose we are required to employ a special piece of equipment on activities 1-3, 3-6, 2-5, 5-8 and 8-9, one at a time. Will it affect the
Given the following information about a project:It is further given that the normal cost of doing all activities of the project in normal time is Rs 18,400 and the overhead cost is Rs 500 per day. Further, a penalty of Rs 2,000 per day in excess of 21 days would be levied. You are required to
16. The time and cost estimates of different activities of a project and their precedence relationships are given below:Overhead costs amount to Rs 1,000 per week.It is stipulated that the contractor will have to pay a penalty of Rs 2,000 per week for completing the project beyond 12 weeks.Crash
15. A small project is having seven activities. The relevant data about these activities is given below:(i) Find out the normal duration and the minimum duration.(ii) What is the percentage increase in cost to complete the project in 21 days?(CA, November, 1997) Activity Dependence Normal Duration
14. In planning a project to introduce a new product in the market, a company lists the various activities, their normal times and costs, and their crash times and costs. They are:If the indirect cost is Rs 4,000 per week, find the optimal project duration. Activity Immediate predecessor Normal
The following table gives data on normal time and cost, and crash time and cost for a project.(i) Draw the network and find out the critical path and the normal project duration.(ii) Find out the total float associated with each activity.(iii) If the indirect costs are Rs 100 per week, find out the
The following data relate to a project:(a) Draw the network, find the critical path and the duration of the project (the minimum as well as the regular).(b) If the project manager wants to reduce the completion time to 25 days, which activities would be required to be crashed and by how much? What
19. The following are the various activities involved in a small project. The cost and time information for these activities is given here:(a) Draw the project network.(b) Find out the critical path and the expected project completion time.(c) If the project is to be completed within 21 days with
24. Consider the following project. Assume that each activity requires manpower only and that the manpower requirements for the various activities are as follows:Further, assume that idle workers are never kept on the payroll, that excess manpower is laid off immediately, and that inadequacies are
For a project, the following information is given:The firm loses Rs 500 for every week beyond a total of 34 weeks. Consider each of the proposals given here individually and state whether it should be accepted by the firm.(a) Activity J can be reduced to 9 weeks by taking resources from activity K,
22. The Madras Construction Company is bidding on a contract to install a line of microwave towers. It has identified the activities, along with their expected times, predecessor restrictions, and worker requirements:The contract specifies that the project must be completed in 14 weeks. The company
21. The activities of a project and the labour requirements in respect thereof areHow should the activities be scheduled if it is desired to smoothen the labour employment on the project, as far as possible? What is the labour requirement? Activity Time Workers Activity Time Workers 1-2 6 6 3-6 1-3
20. The following table gives the activities in a small project and other relevant information:(a) Draw the Network, compute Earliest Start Time (E), Latest Finish Time (F) for each of the activities and find out the project completion time and identify the Critical Path.(b) Draw the Time Scaled
The Oil India Corporation is considering whether to go for an offshore drilling contract to be awarded in Bombay High. If they bid, value would be Rs 600 million with 65% chance of gaining the contract.The Corporation may set up a new drilling operation or move already existing operation, which has
An ice cream manufacturer sells soft scoop ice cream in special pressurised containers and is planning production for the summer, which is the peak period. The company wishes to ensure that it has the best quantity of containers on hand: too few and sales will be lost; too many and the surplus will
A company is considering the introduction of a new product to its existing product range. It has defined two levels of sales as 'high' and 'low' on which to base its decision and has estimated the chances that each market level will occur, together with their costs and consequent profits or losses.
Mr X is considering whether to make an investment in a project with the following likely returns:The utility function of Mr X is approximated as follows:Should the project be undertaken by him? Consider (a) Expected monetary value criterion, and (b) Expected utility criterion. Amount (Rs) 2,00,000
Re-solve Example 13.22 by preparing a pay-off table.For the given problem, the various acts and states of nature, the events, are given here: Courses of action: A : do nothing A2 : accept A and then stop A3 : accept B and then stop A4 : accept A and, if successful, then accept B As : Events: E :
The annual demand for a seasonal product follows the distribution shown here.The manufacturer of this item can produce it by one of the three methods:(a) Using the existing equipment at a cost of Rs 8 per unit.0.30 0.10 (b) Buy special equipment for Rs 22,000 whose salvage value at the end of the
An oil company has recently acquired rights in a certain area to conduct surveys and test drillings to lead to lifting oil if it is found in commercially exploitable quantities.The area is considered to have good potential for finding oil in commercial quantities. At the outset, the company has the
Suppose Delhi Developers Limited (DDL), a construction company, has recently acquired a piece of land in a city on which it plans to construct a shopping complex. The company has now to decide the size of the complex. It is considering three options: a small-sized complex with 40 condominiums and a
Technico Ltd. has installed a machine costing Rs 4 lacs and is in the process of deciding on an appropriate number of a certain spare parts required for repairs. The spare parts cost Rs 4,000 each but are available only if they are ordered now. In case the machine fails and no spares are available,
40. An engineering firm is tendering for a contract to supply a steel fabrication with a target duration of 60 days. The tasks have been analysed as follows:(a) You are required to calculate the expected project duration and identify the critical path.The firm is awarded the contract and starts
The manager of a firm has two alternatives to choose from, for the next quarter.(a) To take a contract to supply an item to a company which would result in a sure profit of Rs 20,000.(b) To make and introduce a new product in the market. The likely profit/loss possibilities along with the likely
An individual, whose current assets amount to Rs 80,000, has the following utility function over the relevant portion of his overall assets scale(a) He is offered a bet in which he has a sixty per cent chance of gaining Rs 20,000 and a forty per cent chance of losing Rs 20,000. Should he accept the
Leisure Hotels Ltd. is planning to build another 700-room complex. It has been suggested that because the existing hotels average only 70% occupancy on an annual basis, the new complex should have only 500 rooms. It has been estimated that the cost per room per annum is Rs 21,000.The following data
A stockist of a particular commodity makes a profit of Rs 30 on each sale made within the same week of purchase, otherwise he incurs a loss of Rs 30 on each item. The data on the past sales are given here:(a) Find out the optimum number of items the stockist should buy every week in order to
An investor is given the following investment alternatives and percentage rates of return.Over the past 300 days, 150 days have been medium market conditions and 60 days have had high market increases.On the basis of these data, state the optimal investment strategy for the investor. States of
The relevant data for three alternatives to invest Rs one lakh are given below:Which investment should be chosen and what would be the rate of return? Investment A (Dealership) B (Property) C (Fixed deposit) Relevant Data The probability of success in exporting a product is 0.7 and in that case the

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