# Numericals on Formation and Graphical Method of Linear Programming

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(Formation and Graphical Method)

1.         A firm buys castings of P and Q type of parts and sells them as finished products after machining, boring and polishing. The purchasing cost for castings are ` 3 and ` 4 each for parts P and Q and selling costs are ` 8 and ` 10 respectively. Per hour capacity of machines used for machining, boring and polishing for 2 products is given below.

 Capacity Parts (per hour) P Q Machining 30 50 Boring 30 45 Polishing 45 30

The running costs for machining, boring and polishing are ` 30, ` 22.5 and ` 22.5 per hour respectively. Formulate the LPP to find out the product mix to maximize the profit.

2.       Mr. A. P. Ravi wants to invest ` 1,00,000 in two companies A and B so as not to exceed ` 75,000 in either of the company. The company A assures average return of 10% whereas average return for company B is 20%. The risk factor of company A is 4 on 0 to 10 scale whereas the risk factor rating for B is 9 on similar scale. As Mr. Ravi wants to maximize his returns, he will not accept an average rate of return below 12% or a risk factor above 6. Formulate this as a LPP and solve it graphically.

3.       XYZ factory manufacture two articles A & B. To manufacture article A, a certain machine has to work for 1.5 hours and in addition a craftsman has to work for 2 hours.  To manufacture article B, the machine has to work for 2.5 hours and in addition a craftsman has to work for 1.5 hours. In a week factory can avail of 80 hours of machine time and 70 hours of craftsman’s time. The profit on each article A is ` 50 and that on each article B is ` 40. If the entire article produced can be sold away, find how many of each kind should be produced to earn the maximum profit each week. Formulate the problem as L.P. model using graphical method or simplex method of linear programming.

4.       Agashe & Co. plans to reach target audiences belonging to two different monthly income groups, the first with incomes greater than ` 15,000 and the second with income of less than ` 15,000. The total advertising budget is ` 2,00,000. Advertising on TV costs ` 50,000 for one program, whereas advertising on Radio costs             ` 20,000 for one program. For contract reasons atleast 3 programmes must be given on TV and the No. of Radio programmes are limited to 5 only. One TV programme covers 4,50,000 audiences belonging to income group having more than ` 15,000 monthly income where as it reaches to 50,000 audiences belonging to below ` 15,000 income group. Similarly one Radio program reaches to 20,000 and 80,000 audiences belonging to above ` 15,000 and below ` 15,000 monthly income groups respectively. Formulate the LPP and using graphical method, determine the media mix so as to maximize the total number of target audience. Comment on the solution.

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