1. The table at the bottom gives a composition of various foods used in making cereals. The other material in each food is fiber, water, etc. The company blends these food materials and makes two kinds of cereals. In the process of blending, 3% of protein, 5% of starch, and 10% of minerals and vitamins are completely lost from the mix. For each 100 kg of foods added in the blend, the blending process adds 5 kg of other material (mainly water and fat). Cereal type 1 sells for $1.50/kg. It should contain at least 22% protein, 2% of minerals and vitamins, and at most 30% of starch by weight. Cereal type 2 sells for $1.00/kg. It should contain at least 30% starch by weight. Percentage Content (by Weight) Other Material Price ($/kg) Availability Food Protein Starch Minerals, Vitamins, etc. per Day (kg) 1 2 3 4 6722 45 12 4 39 0.68 1500 38 1 54 0.27 500 12 25 2 61 0.31 1000 40 3 30 0.45 2000 Write the LP and the optimal product mix for the company. (Use an LP solver. For example, I was able to use the solvers http://www.phpsimplex.com/simplex/simplex.htm?l=en http://comnuan.com/cmnn03/cmnn03004/ https://www.easycalculation.com/operations-research/simplex-method-calculator. php Note that in these solvers all variables are automatically constrained to be non-negative.) 2. Use the two-phase method to solve the following linear program. max 3x1 + x2 subject to x1-22-1 -21-22-3 2x+2≤2 21, 20 3. Use the two-phase method to solve the following linear program. max=3x1 + x2 subject to 1-2-1 -21-22-3 2x1-22 1,20 Note the difference in the third constraint from the problem above. 4. Use the two-phase method to solve the following linear program. max z= 3x2+x3 subject to 1+ 2x2 + 3 ≤2 2x12 3-1 - 3x+2x+33 21, 22, 23 20 5. The Transportation Problem (known also as the Hitchcock problem) is as follows. There are m sources of some commodity, each with a supply of a; units, i = 1,...,m and n terminals, each of which has a demand of b; units, j = 1,..., n. The cost of sending a unit from source i to terminal j is c, and a = b. We want to find a cheapest way to satisfy all demands. State this problem as an LP. (The general answer can be expressed in a compact form.)