#### Process Optimization

Total budget is \$4000 to spend. What is the best way to maximize profits out of the following items which need to fit on a 60" long by 30" wide table. SOLVE USING SIMPLEX METHOD Candles Cost: \$4, sell \$12 6" square space/unit Wigs Cost: \$10, Sell for 30 10" square space/unit Hat Clips Cost: \$10, Sell for \$24 2" square space/unit I am pretty sure the answer comes out to 125 wigs, 275 golf clips and zero candles for standard profit maximization

1: Consider the following problems from the first homework. For each, derive the dual problem and illustrate that strong duality holds for these problems by finding the optimal solution to the dual (You need to find the values of the slack variables aswell!). (Hint: You do not need to use the simplex method for this problem)

2: Solve the following linear program:

3: Use the dual-primal two-phase algorithm to solve :

4: Imagine you are a college student trying to live on a small budget. You want to minimize your food costs, but you know a diet consisting solely of instant ramen and beer may be harmful to your health. You look up recommended dietary allowance of several different nutrients to determine what you should eat to be healthy. Let i = 1...m denote the nutrients and let bi, i = 1...m denote the minimum daily requirement for each of the nutrients. Suppose make a list of your n favorite foods. Let j denote their index on your list and c; be the cost of the jh item. Let a denote the amount of the įth nutrient contained in the jth item on your food list. a: Formulate a linear problem to minimize your costs while still getting all of your daily required nutrients b: Formulate the dual to this problem c: Introduce another person to this story who is naturally interested in solving the dual problem. (Hint: Give semantic meaning to the dual problem) d: Name some real-world considerations that your model leaves out

Green Grass, Inc. just ran out of stock and suddenly has two emergency orders for grass seed blends: one is for 1500 pounds of normal, the other for 2300 pounds of special. At least each pound of normal should contain 60 percent annual seed, while each pound of special should contain at least 70 percent perennial seed. Green Grass has two input mixtures, A and B. Mixture A contains 80 percent perennial and 15 percent annual seed. Mixture B contains 70 percent annual and 25 percent perennial seed. Mixture A costs 90 cents per pound and mixture B costs 50 cents per pound. Provide a complete LP formulation for this problem in the standard format. DO NOT SOLVE.