1.(a)Beng Yew Wine Company produces two kinds of wine - Valley Nectar and Valley Red.The wines are produced from 64 tons of grapes the company has acquired this season.A 1,000-litre batch of Nectar requires 4 tons of grapes; and a 1,000-litre batch of Red requires 8 tons.

However, production is limited by the availability of 55 cubic metres of storage space for aging and 128 hours of processing time.A batch of each type of wine requires 3 cubic metres of storage space.The processing time for batch of Nectar is 14 hours, and the processing time for a batch of Red is 9 hours.

Demand for each type of wine is limited to seven batches.

The profit for a batch of Nectar is $9,000, and a batch of Red is $12,000.The company wants to determine the number of 1,000-litre batches of Nectar (X1) and Red (X2) to produce in order to maximize profit.

(i) Write down an expression for the objective function.

(ii)Formulate the constraints.

(b)Consider the following linear programming problem:

Minimize cost = 110X1 +75X2

Subject to:

2X1 +X2 40

-6X1+ 8X2120

70X1 + 105X2 2100

X1 5

X2 40

(i)Determine the feasible region graphically.

(ii)Find all corner points.

(iii)What is the minimum cost and optimal solution?

(iv)Which constraint is redundant?