8. Susie is preparing for a triathlon event that is taking place next month. A triathlon involves three activities: swimming, cycling and running. Susie decides that in her training next week she should

• maximise the total time spent cycling and running
• train for at most 39 hours
• spend at least \(40 \%\) of her time swimming
• spend a total of at least 28 hours of her time swimming and running

Susie needs to determine how long she should spend next week training for each activity. Let

• \(x\) represent the number of hours swimming
• \(y\) represent the number of hours cycling
• \(z\) represent the number of hours running
  1. Formulate the information above as a linear programming problem. State the objective and list the constraints as simplified inequalities with integer coefficients.
     (5)

Susie decides to solve this linear programming problem by using the two-stage Simplex method.

Set up an initial tableau for solving this problem using the two-stage Simplex method. As part of your solution you must show how

• the constraints have been made into equations using slack variables, exactly one surplus variable and exactly one artificial variable
• the rows for the two objective functions are formed
  (6)

The following tableau \(T\) is obtained after one iteration of the second stage of the two-stage Simplex method.

b.v.	\(x\)	\(y\)	\(z\)	\(s _ { 1 }\)	\(\mathrm { S } _ { 2 }\)	\(S _ { 3 }\)	Value
\(y\)	0	1	0	1	0	1	11
\(s _ { 2 }\)	0	0	5	-2	1	-5	62
\(x\)	1	0	1	0	0	-1	28
\(P\)	0	0	-1	1	0	1	11

Obtain a suitable pivot for a second iteration. You must give reasons for your answer.

Starting from tableau \(T\), solve the linear programming problem by performing one further iteration of the second stage of the two-stage Simplex method. You should make your method clear by stating the row operations you use.