In a market research survey on use of bathing soaps, the findings were as follows; Out of those who used lux at last purchase, 10% switched to Jai, 10% switched to Liril. and 10% switched to Hamam. Out of those who used Jai, 20% switched to Liril, and 10% to Hamam. Out of those who were using Liril, only 10% switched to Hamam. Out of those who used Hamam, 20% switched to Lux, 10% to Jai, and 10% to Liril. Assuming that the bathing soaps' switching patterns continue, determine; a) A transition probability Matrix. b) The long-run Market shares. A and B play a game as follows: They simultaneously and independently write one of the three numbers 1, 2, 3, If the sum of the numbers written is even, Player B pays player A this sum in pounds. If it is odd, player A pays the sum to B in pounds. Develop a pay off matrix with respect to player A and find the optimal strategies for the players. - Consider the following activities, associated normal time and cost, together with the extra cost of saving a day on selected activities. Activity Normal time in days A to B A to C A to D B to C B to E B to F C to D C to F D to F E to F Normal cost of activity () 10,000 12,000 5.000 6,000 9,000 5,000 4,000 3,000 5,000 6,000 5 7 9 10 8 6 4 3 7 12 Extra cost of reducing normal time by one day () 3,000 2,000 800 600 2,000 900 700 200 2,000 500 No single activity can be reduced by more than one day. In addition, site costs will be incurred at the rate of 3,500 for every day that work is in progress on the site irrespective of how many activities are in progress. Determine and state. (10marks) a) The network diagram, critical path, using normal time, the time taken and the total cost of completing the project. b) The shortest time in which the project can be completed, the associated cost and the critical path. c) The lowest cost for which the project can be completed and the associated cost and time A farmer is raising pigs for market and wishes to determine the quantities of the available types of feed that should be given to each pig to meet certain nutritional requirements at a minimum cost. The numbers of each type of basic nutritional ingredient contained within a kilogram(kg) of each feed type is given in the following table along with the daily nutritional requirements and the feed costs: __(15marks) Nutrients Carbohydrates Protein Vitamins Costs (in kshs) Corn 9 3 1 7 Tankage 1 8 2 6 What is the best solution to the farmer? Alfalfa 4 6 6 5 Minimum requirement. 20 18 15