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Inequalities and Linear Programming?

"LEEMING ENTERPRISES LTD" operate the bus services for the new town of Twittering on a charter basis. The cost of hiring a bus is £15 a journey and they operate fixed charges of 40p per journey for the first class passengers and 20p for second class passengers. The seating accomodation demands that: i) First class passengers must not exceed second class by more than 30. ii) The number of second class passengers must not exceed 50% nor fall below 25% of the total number of passengers. iii) The max seating capacity of the bus is 80. The firm do not start a journey unless there are at least 50 passengers in all but the local council guarantees at least 30 (first class) passengers for each journey. a) List all the constraints involved in terms of x, the number of second class passengers and y, the number of first class passengers. b) How do you draw a graph of the feasible region for the simultaneous constraints in (a) and label all the vertices?

Public Comments

  1. You might like to look at my answer to the question "Use graphical methods to solve . . . " from two weeks ago. This covered the same topic in some detail.
  2. The linear program I used requires a zero on the right. If you don’t need that then it’s probably easier to leave the numbers on the right. 1) First class passengers must not exceed second class by more than 30: Y – X – 30 less than or equal to zero 2) The number of second class passengers must not exceed 50%: X – Y less than or equal to zero 3) nor fall below 25%: 3X – Y greater than or equal to zero 4) The max seating capacity of the bus is 80: X+Y-80 less than or equal to zero 5) The firm do not start a journey unless there are at least 50 passengers in all: X + Y – 50 greater than or equal to zero 6) The local council guarantees at least 30 (first class) passengers for each journey: Y – 30 greater than or equal to zero This leaves you the equation for 6 lines. Plot them on the same graph. As you plot each line shade the area that does not satisfy that line. Once you have all 6 lines plotted & shaded the unshaded area is the feasible region.
  3. Yah, I too would recommend you the graphical method to solve all the problems related to linear inequalities
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