AGEC_622_LP_Homework 3.docx

AGEC 622 LP Homework 3 ■ Total Points Possible : 150 points 1. (65 pts) A dairy processes milk using capacity, labor, and refrigerator space and sells low fat milk, high fat milk, and butterfat. In addition, the dairy has the option of using butterfat to make butter, which is then sold. The process of converting butterfat to butter is constrained by churn time. Essentially, the dairy must be processed using one of the 3 processes. The first process uses raw milk to produce low fat milk, the second process uses raw milk to produce high fat milk, and the third process uses butterfat from the first two processes to produce butter. Process 1 Process 2 Process 3 Use raw milk to make low fat milk Use raw milk to make high fat milk Use butterfat to make butter Raw Milk (gal) 500 500 Low fat milk yield (gal) 450 High fat milk yield (gal) 491 Butterfat yield (gal) 50 9 Butterfat used (gal) 3000 Butter yield (pounds) 8000 Labor used (hours) 3.2 2.2 2.4 Refrigerator space (cu ft) 70 70 1000 Churn use (hours) 1.5 Processing cost ($) .15 per gal raw milk .16 per gal raw milk .45 per gal butterfat Packaging cost ($) .15 per gal raw milk .15 per gal raw milk .30 per gal butterfat The raw milk costs $0.50 per gallon. The company has a contract to sell 200,000 gallons of high-fat milk to one of its customers and it can sell the dairy products for the following prices: Sales price Low fat milk ($/gal) 2.00 High fat milk ($/gal) 1.20 Butterfat ($/gal) 2.20 Butter ($/pound) 2.50 The resources used in production are constrained to the following: Availability Labor available (hours) 2,300 Raw milk inventory (gal) 1,500,000 Refrigerator space (cu ft) 200,000 Churn time (hours) 120 1) (30 pts) Set up a linear programming model to determine the optimal production pattern for this dairy company. 2) (10 pts) Provide a discussion verifying that the model you have set up satisfies the homogeneity of unit criteria (just two examples, one for column, one for row).

3) (15 pts) Provide the solution of the LP model and interpret the results (objective function value, 2 decision variables, 2 shadow prices, 2 reduced costs). 4) (10 pts) You should have used at least 3 different types of constraints in your model. Describe the different types of constraints and their purpose in the model. 2. (35 points) Skylar has a beverage packing company packing canned soda into varying 36-can packages. The packing process consumes labor, and it is limited by the machine’s capacity. Assume the capacity of the packing machine is 6000 packages. The machine can switch to any package style without extra cost. The packing company has already agreed with a wholesale company to sell a certain number of each package. The rest will be sold on the market at a fixed price. The company has 90 labor hours. The cost of purchasing a can of beverage is $0.3 (same for Pepsi, Diet Pepsi, and 7up). There are four packing options. Pepsi package Diet Pepsi package 7up package Mix package Packing Cost 1 1.1 1 1.2 Pepsi 36 12 Diet Pepsi 36 12 7up 36 12 Labor 0.01 0.01 0.01 0.02 Yield (36 cans beverage package) 1 1 1 1 The prices of beverage packages are Price ($/package) Sales agreement (package) 36-can Pepsi 12 1000 36-can Diet Pepsi 12.5 1000 36-can 7up 11.5 1000 Mix package 13 2500 1) (20 pts) Set up a linear programming model to determine the optimal production pattern for this packing company. 2) (10 pts) Provide the LP model solution and interpret the results (objective function value, 2 decision variables, 2 shadow prices, 2 reduced costs). 3) (5 pts) If the wholesale company cancels the sales agreement of the 7up packages, how will the net profit change?

Only 900 chickens can be cut up per day. There are 2,750 units of labor available to the firm per day. 1) (35 pts) Set up the tableau 2) (15 pts) Solve the model and interpret objective function, decision variables (2), shadow prices (2) and reduced costs (2) Back (piece) $0.2 0 Breasts (piece) $1.6 0 Meat (lb) $1.0 0 Breast Quarters (piece) $1.7 5 Leg Quarter (piece) $1.4 0