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understanding management
Quantitative Techniques In Management 4th Edition N D VOHRA - Solutions
13. Three products, A, Band Care produced at three machine centres, X, Yand Z. Each product involves operation at each of the machine centres. The time required for each operation on various products is indicated in the following table:(a) Formulate a linear programming problem on the basis of
14. A pharmaceutical company has 100 kg of material A, 180 kg of material Band 120 kg of material C available per month. They can use these materials to make three basic pharmaceutical products, namely 5-10-5, 5-5-10 and 20-5-10, where the numbers in each case represent the percentage by weight of
15. Noah's Boats makes three different types of boats. All boats can be made profitably in this company, but the company's monthly production is constrained by the limited amount of labour, wood and screws available each month. The director will choose the combination of boats that maximises his
16. A trucking company with Rs 40,00,000 to spend on new equipment is contemplating three types of vehicles. Vehicle A has a IO-tonne paY-load and is expected to average 35 km per hour. It costs Rs 80,000. Vehicle B has a 20-tonne paY-load and is expected to average 30 km per hour. It costs Rs
17. MaximiseSubject to Z=7x + 2x2 + 3x3 + 4x4,
18. A furniture company manufactures four models of desks. Each desk is first constructed in the carpentry shop and is next sent to the finishing shop, where it is varnished, waxed and polished. The number of man-hours oflabour required in each shop is as follows:Because oflimitations in capacity
19. Minimise Subject to Z=6x+4x2
20. Solve the LPP by two-phase method:MinimiseSubject to Z = x1 + x2
21. Using two-phase method, solve the following problem:MinimiseSubject to 150x150x2100x3
22. MinimiseSubject to Z=60x + 80x2
23. A small metal press shop makes Ash Trays and Tea Trays. The process consists ofbuying sheet metal from the market, cutting out appropriate pieces for ash tray /tea tray on stamping machines, forming the stamped pieces into ash tray/tea tray and finally painting in the Spray Painting Booth. The
24. A company manufactures two products, A and B. Product A Yields a contribution of Rs 30 per unit and product B Rs 40 per unit towards profits and fixed costs. It is estimated that the sales of product A for the coming month will not exceed 20. Sales of product B have not been estimated but the
25. A company produces two products, A and B. The sales volume of product A is at least 60 percent of the total sales of the two products. Both the products use the same raw material of which the daily availability is limited to 100 tonnes. Products A and B use this material at the rate of 2 tonnes
26. A firm makes two products, X and Y and has total production capacity of 9 tonnes per day, X and Y requiring the same production capacity. The firm has a permanent contract to supply at least 2 tonnes of X and at least 3 tonnes of Y per day to another company. Each tonne of X requires 20 machine
27. A factory is engaged in producing two items, / 1 and / 2, each of which has to pass through three machinesM1, M2 andM3. One unit of/1 requires one hour of machining onM1, 2 hours onM2 and 1 hour on M3, while one unit of /2 requires machining of an hour each on M1 and M2, and 3 hours on M3• In
28. An advertising company wants to cover two types of audiences: Type A, consisting of customers with annual income greater than Rs 150,000 and Type B, consisting of customers with annual income less than Rs 150,000. The total advertising budget of the company is Rs 2,10,000. One programme of TV
29. A company is considering three magazines for releasing its advertisements. The relevant data are given here:The company has allocated Rs 1,00,000 for magazine advertising. It has also decided that there should be at least two insertions each in magazines A and C, and that the number of
30. Solve the following LPP:What would happen if the objective function of this problem was of the 'maximisation' type? Minimise Subject to Z=120x1 + 60x2 20x + 30x2 900 40x + 30x2 1200 XX
31. Two products A and B are processed on three machines M1, M2 and M3. The processing times per unit, machine availability and profit per unit are:Any unutilized time on machine M3 can be given on rental basis to another firm at an hourly rate of Rs 1.50. Formulate the mathematical model, solve it
32. A manufacturer produces three commodities, C1, C2 and C3, which require, respectively, 2, 2 and 3 units of material and, respectively, 2, 2 and 1 hours oflabour. The unit profit on C1, C2 and C3 is Rs 6, Rs 3 and Rs 2 in that order.In a given week, if 300 units of material and 120 hours of
33. Given below are the objective function, the constraints, and the final simplex tableau for a linear programming product-mix problem:Objective function:(a) Write the optimal product-mix and the profit contribution shown by the above solution.(b) Is this solution feasible? Why? Give briefreason
34. MaximiseSubject to Z=10x + 20x2,
35. Solve the following problem by Simplex method:MaximiseSubject to 22X+30Y+25Z
36. Solve the following LPP:MaximiseSubject to Z=40x + 35x2,
3 7. A company, during the festival season, combines two factors A and B to form a gift pack which must weigh 5 kg. At least 2 kg of A and no more than 4 kg of B should be used. The net profit contribution to the company is Rs 5 per kg for A and Rs 6 per kg for B. Determine the optimal product mix.
38. Solve the following LPP: Maximise Subject to Z=2x+4x2 2x1 + x2 18, 3x+2x 30, x1 + 2x2 = 25, 20
39. A firm manufactures two products A and B, which pass through manual and machine processes. The time required by each unit and the total available time is as given here:The enthusiastic production manager requires that at least 300 units of A and 300 units of B should be produced. It is known
40. MaximiseSubject to Z 50x110x2 + 120x3
A firm is engaged in producing two products, A and B. Each unit of product A requires 2 kg of raw material and 4 labour hours for processing, whereas each unit of product B requires 3 kg of raw material and 3 hours of labour, of the same type. Every week, the firm has an availability of 60 kg of
The Agricultural Research Institute suggested to a farmer to spread out at least 4800 kg of a special phosphate fertiliser and not less than 7200 kg of a special nitrogen fertiliser to raise productivity of crops in his fields. There are two sources for obtaining these-mixtures A and B. Both of
Continuing with Example 2.1, suppose that each of the products are required to be packed.Every unit of product A requires 4 hours while every unit of product B needs 3.5 hours for packaging. Suppose that in the packaging department, 105 hours are available every week.Under these conditions, what
Solve graphically the following LPP:The constraints are shown plotted on the graph in Figure 2.6. Also, iso-profit lines have been graphed. Maximise Subject to Z = 8x + 16x2 X + X2 200 X2 125 3x + 6x2 900 X1, X220
Solve graphically: Minimise Subject to Z=6x + 14x2 5x + 4x2 60 3x + 7x2 84 X + 2x2 18 X1, X2 0
24-hour supermarket has the following minimal requirements for cashiers:Period 1 follows immediately after period 6. A cashier works eight consecutive hours, starting at the beginning of one of the six time periods. Determine a daily employee worksheet which satisfies the requirements with the
An agriculturist has a 125-acre farm. He produces radish, muttarand potato. Whatever he raises is sold fully in the market. He gets Rs 5 per kg for radish, Rs 4 per kg for muttar and Rs 5 per kg for potato. The average per acre yield is 1500 kg of radish, 1800 kg of muttar and 1200 kg of potato. To
A manufacturing firm needs 5 component parts. Due to inadequate resources, the firm is unable to manufacture all its requirements. So the management is interested in determining as to how many, if any, units of each component should be purchased from outside and how many should be produced
A firm produces three products A, Band C. It uses two types of raw materials I and II of which 5,000 and 7,500 units, respectively, are available. The raw material requirements per unit of the products are given below:The labour time for each unit of product A is twice as that of product Band three
The Marketing Department of Everest Company has collected information on the problem of advertising for its products. This relates to the advertising media available, the number of families expected to be reached with each alternative, cost per advertisement, the maximum availability of each medium
A multinational company has two factories that ship to three regional warehouses. The costs of transportation per unit are:Factory F2 is old and has a variable manufacturing cost of Rs 20 per unit. Factory F1 is modern and produces for Rs 10 per unit. Factory F2 has a monthly capacity of 250 units,
A leading Chartered Accountant is attempting to determine a "best" investment portfolio and is considering six alternative investment proposals. The following table indicates point estimates for the price per share, the annual growth rate in the price per share, the annual dividend per share and a
A company engaged in producing tinned food delicacies has 300 trained employees on its rolls each of whom can produce one can of food in a week. Due to the developing taste of the public for this kind offood, the company plans to add the existing labour force by employing 150 people in a phased
Solve graphically: Maximise Subject to Z = 10x + 15x2 2x1 + x2 26 2x + 4x2 56 X-5 X1, X2 0
Solve graphically the following linear programming problem: Minimise Subject to Z = 3x + 5x2 -3x + 4x2 12; 2x + 3x 12; 2x - X2 -2; x4; x2 2, and X1, X20
A retired person wants to invest upto an amount of Rs 30,000 in fixed income securities. His broker recommends investing in two bonds: Bond A yielding 7% and Bond B yielding 10%. After some consideration, he decides to invest at most Rs 12,000 in Bond Band at least Rs 6,000 in Bond A. He also wants
A local travel agent is planning a charter trip to a major sea port. The eight day and seven night package includes the fare for round trip, surface transportation, board and lodging and selected tour options. The charter trip is restricted to 200 persons and past experience indicates that there
Let us assume that you have inherited Rs 100,000 from your father-in-law that can be invested in a combination of only two stock portfolios, with the maximum investment allowed in either portfolio set at Rs 75,000. The first portfolio has an average return of 10%, whereas the second has 20%. In
A manufacturer employs three inputs: man hours, machine-hours and cloth material to manufacture two types of dresses. Type A dress fetches him a profit of Rs 160 per piece, while type B, that of Rs 180 per piece. The manufacturer has enough man-hours to manufacture 50 pieces of type A or 20 pieces
1. Basically, linear programming is a resource allocation problem that deals with the best allocation of limited resources to a number of competing activities. Mark the statement as T (True) or F (False)
2. A prerequisite for applying linear programming is that there should be an objective which is clearly identifiable and which maY, or may not, be measurable in quantitative terms. Mark the statement as T (True) or F (False)
3. Proportionality property in the linear programming context implies that per unit contribution of a variable in the objective function is independent of the size of the variable. Mark the statement as T (True) or F (False)
4. The solution to an LPP implicitly assumes that the variables are continuous, which may take fractional as well as integer values in the solution. Mark the statement as T (True) or F (False)
5. All constraints in an LPP as well as its objective function must be linear in nature. Mark the statement as T (True) or F (False)
6. A typical linear programming problem is characterised by an objective function to be maximised or minimised, and a set of constraints, and non-negativity condition. Mark the statement as T (True) or F (False)
7. For an n variables linear programming problem, there must be an equal number of constraints. Mark the statement as T (True) or F (False)
8. An LPP can have only two decision variables. Mark the statement as T (True) or F (False)
9. Linear programming is probabilistic in nature. Mark the statement as T (True) or F (False)
10. In real life, no variable can be unrestricted in sign. Mark the statement as T (True) or F (False)
11. An LPP must have all constraints of the "~" or ";;:;::" type. Mark the statement as T (True) or F (False)
12. A feasible solution is one which meets at least one of the constraints of the problem. Mark the statement as T (True) or F (False)
13. An optimal solution to a linear programming problem is a feasible solution which optimises. Mark the statement as T (True) or F (False)
14. The feasible region of a linear programming problem must be a convex set. Mark the statement as T (True) or F (False)
15. The graphic approach to the solution to LPPs cannot handle problems with more than three variables. Mark the statement as T (True) or F (False)
16. An iso-cost line cannot be parallel to the line of any constraint. Mark the statement as T (True) or F (False)
17. !so-profit lines on a graph of an LPP would always be parallel to each other. Mark the statement as T (True) or F (False)
18. Replacing the ·~, sign of constraint by a '=' sign would improve the value of the objective function. Mark the statement as T (True) or F (False)
19. The feasible region for a constraint is restricted if its•;;:;::• or·~· sign is replaced by a'=' sign. Mark the statement as T (True) or F (False)
20. A constraint 4x1 + 7x2 ;;:;:: 57 of an LPP is replaced by the constraint 4x1 + 7x2 ;;:;:: 40. This would make the LPP more restrictive in nature. Mark the statement as T (True) or F (False)
21. Exclusion of a redundant constraint does not affect the optimal solution to an LPP. Thus, a redundant constraint represents an abundant resource. Mark the statement as T (True) or F (False)
22. A linear programming problem cannot have more than one redundant constraint. Mark the statement as T (True) or F (False)
23. For a linear programming model, the feasible region may change if non-binding constraints are deleted. Mark the statement as T (True) or F (False)
24. Every linear programming problem has a unique optimal solution. Mark the statement as T (True) or F (False)
25. It is possible for the objective function value of an LPP to be the same at two distinct extreme points. Mark the statement as T (True) or F (False)
26. When infeasibility does not exist, it is always possible to determine the optimal solution from a knowledge of all the extreme points of the polygon of the feasible region. Mark the statement as T (True) or F (False)
27. Changes in the objective function coefficients shall always result in changing the optimal values of the decision variables. Mark the statement as T (True) or F (False)
28. Infeasibility indicates that there are very few feasible solutions to an LPP and it is, therefore, difficult to say which of these is optimal. Mark the statement as T (True) or F (False)
29. An LPP with an unbounded feasible region would obviously have unbounded solution. Mark the statement as T (True) or F (False)
30. For a linear programming problem to be unbounded, its feasible region must be unbounded. Mark the statement as T (True) or F (False)
1. What is a linear programming problem? Discuss the scope and role of linear programming in solving management problems.
2. Discuss and describe the role of linear programming in managerial decision-making bringing out limitations, ifanY. (MBA, Delhi, 1999)
3. "Linear Programming is one of the most frequently and successfully used Operations Research technique to managerial and business decisions." Elucidate this statement with some examples.
4. Give the mathematical and economic structure of the linear programming problem. What requirements should be met in order that the linear programming may be applied?
5. Briefly explain the major applications oflinear programming in business.
6. What are the components of an LPP? What does the non-negativity restriction mean?
7. Give a general statement of a linear programming problem. Is it correct to say that the constraints should be of 'less than or equal to' form for the maximisation problems and of 'more than or equal to'form for the minimisation problems?
8. Discuss the assumptions of proportionality, additivity, continuity, certainty and finite choices in the context ofLPPs.
9. In relation to linear programming, explain the implications of the following assumptions of the model:(a) linearity ofobjective function and constraints,(b) continuous variable,(c) certainty.
10. What steps are required in solving LPPs by graphic method? Discuss in brief.
11. What is feasibility region? Is it necessary that it should always be a convex set?
12. Define iso-profit line. How does it help to obtain solution to the linear programming problems?
13. What is a redundant constraint? What does it imply? Does it affect the optimal solution to an LPP?
14. How would you know whether the solution to a linear programming problem is unique or not? In this connection, state the conditions that should be satisfied for more than one optimal solution to a problem to exist.
15. With the help of suitable sketches, define convex, non-convex and infeasible regions in relation to the graphical solution of a linear programming problem.
2. A company manufactures 3 types of parts which use precious metals platinum and gold. Due to shortage of these precious metals, the government regulates the amount that may be used per day. The relevant data with respect to supply, requirements, and profits are summarised in the table as
3. A manufacturer of purses makes four styles of purses: a three-compartment bag which takes 45 minutes to assemble; a shoulder-strap bag, taking one hour to assemble; a tote bag, needing 45 minutes for assembly, and pocket purse requiring 30 minutes to assemble. There are 32 hours of assembly time
4. An electronics company is engaged in the production of two components C1 and C2, used in radio sets.Each unit of C1 costs the company Rs 5 in wages and Rs 5 in materials, while each unit of C2 costs the company Rs 25 in wages and Rs 15 in materials. The company sells both products on one-period
5. At the beginning of a month, a lady has Rs 30,000 available in cash. She expects to receive certain revenues at the beginning of the months 1, 2, 3 and 4 and pay the bills after that, as detailed here:It is given that any money left over may be invested for one month at the interest rate of0.5%;
6. Formulate the following as a linear programming problem:A publishing house publishes three weekly magazines-Daily life, Agriculture Today, and Surf's Up.Publication of one issue of each of these magazines requires the following amounts of production time and paper.Each week the publisher has
7. A Mutual Fund company has Rs 20 lakhs available for investment in Government bonds, blue chip stocks, speculative stocks and short-term bank deposits. The annual expected return and risk factors are given as follows:Mutual Fund is required to keep at least Rs 2 lakhs in short-term deposits and
8. A company produces three types of parts for automatic washing machines. It purchases castings of the parts from a local foundry and then finishes the parts on drilling, shaping, and polishing machines. The selling prices of parts A, Band C respectively, are Rs 8, Rs 10 and Rs 14. All parts made
9. Dr Shilpa Soni, the head administrator at XYZ hospital must determine a schedule for nurses to make sure there are enough nurses on duty throughout the day. During the day, the demand for nurses varies.Dr. Shilpa has broken the day in to 12 two-hour periods. The slowest time of the day
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