Pet Food Company A pet food company wants to find the optimal mix of ingredients, which will minimize the cost of a batch of food, subject to constraints on nutritional content. There are two ingredients, P1 and P2. P1 costs $5/lb. and P2 costs $8/lb. A batch of food must contain no more than 400 lbs. of P1 and must contain at least 200 lbs. of P2. A batch must contain a total of at least 500 lbs. What is the optimal (minimal cost) mix for a single batch of food? Linear Programming

Pet Food Company – Linear Equations Linear Programming

Pet Food Company – Graph Solution Linear Programming

Pet Food Company – Graph Solution Constraint 1 Linear Programming

Pet Food Company – Graph Solution Constraint 1 Linear Programming

Pet Food Company – Graph Solution Constraint 2 Linear Programming

Pet Food Company – Graph Solution Constraint 1 & 2 Linear Programming

Pet Food Company – Graph Solution Constraint 3 Linear Programming

Pet Food Company – Graph Solution Constraint 1, 2 & 3 Linear Programming

Pet Food Company – Solve Linear Equations Linear Programming

Pet Food Company – Graph Solution Linear Programming

Pet Food Company – Solve Linear Equations Linear Programming

Pet Food Company – Solve Linear Equations Linear Programming

Pet Food Company – Graph Solution Optimal Point (300, 200) Linear Programming

Pet Food Company – Slack/ Surplus Calculation Linear Programming

Pet Food Co. – Linear Equations Slack/ Surplus Variables Min 5P1 + 8P2 + 0S1 + 0S2 + 0S3 s.t. 1P1 + 1S1 = 400 1P2 - 1S2 = 200 1P1 + 1P2 - 1S3 = 500 P1, P2, S1 ,S2 ,S3 >= 0 Linear Programming

Pet Food Co. – Slack Variables For each ≤ constraint the difference between the RHS and LHS (RHS-LHS). It is the amount of resource left over. Constraint 1; S1 = 100 lbs. Linear Programming

Pet Food Co. – Surplus Variables For each ≥ constraint the difference between the LHS and RHS (LHS-RHS). It is the amount bt which a minimum requirement is exceeded. Constraint 2; S2 = 0 lbs. Constraint 3; S3 = 0 lbs. Linear Programming