7.06a LP formulation: variables, constraints, objective function

Ben is a wedding planner. He needs to order flowers for the weddings that are taking place next month. The three types of flower he needs to order are roses, hydrangeas and peonies. Based on his experience, Ben forms the following constraints on the number of each type of flower he will need to order. • At least three-fifths of all the flowers must be roses. • For every 2 hydrangeas there must be at most 3 peonies. • The total number of flowers must be exactly 1000 The cost of each rose is £1, the cost of each hydrangea is £5 and the cost of each peony is £4 Ben wants to minimise the cost of the flowers. Let \(x\) represent the number of roses, let \(y\) represent the number of hydrangeas and let \(z\) represent the number of peonies that he will order.

1. Formulate this as a linear programming problem in \(x\) and \(y\) only, stating the objective function and listing the constraints as simplified inequalities with integer coefficients. [7]

Ben decides to order the minimum number of roses that satisfy his constraints.

1. 1. Calculate the number of each type of flower that he will order to minimise the cost of the flowers.
   2. Calculate the corresponding total cost of this order. [3]

10 x + 7 y & \leqslant 140
& x + y \leqslant 15
& 2 x + 3 y \geqslant 36
& x \geqslant 0 , \quad y \geqslant 0 \end{aligned} \end{array}$$ (c) Represent these constraints on Diagram 1 in the answer book. Hence determine, and label, the feasible region, \(R\).
(d) Use the objective line method to find the optimal number of each type of cake that Martin should make, and the amount of sugar used.
(e) Determine how much flour and how many eggs Martin will have left over after making the optimal number of cakes. BLANK PAGE \end{document}