A large textile firm has two manufacturing plants, two sources of raw material, and three market centers. The transportation costs between the sources and the plants and between the plants and the markets are as follows:

Plant A Plant B

Source 1 $1/ton $1.5/ton

Source 2 $2/ton $1.5/ton

Market 1 Market 2 Market 3

Plant A $4/ton $2/ton $1/ton

Plant B $3/ton $4/ton $2/ton

Ten tons are available from source 1 and 15 tons from source 2. The three market centers require 8 tons, 14 tons, and 3 tons. The plants have unlimited processing capacity

i) Formulate the problem of finding the shipping patterns from sources to plants to markets that minimizes the total transportation cost.

ii) Reduce the problem to a single standard transportation problem with two sources and three destinations. (Hint: Find minimum cost paths from sources to markets.)

iii) Suppose that plant A has a processing capacity ofo 8 tons, and plant B has a processing capacity of 7 tons. Show how to reduce the problem to two separate standard transportation problems.