3.3Linear Programming:
Objective function z = ax + by (this is what you are trying to ______or ______)
Linear Constraints
Optimal Solution/Value:
Objective Function:
Optimal values will occur at ______ of the feasible region.
Ex:Minimize:
Subject to:
Ex:Maximize:z = 12x + 9y
Subject to: 4x + 3y 36
8x + 3y 48
Linear Programming Steps:
1.5.
2.6.
3.7.
4.8.
Ex 1. An appliance store manager plans to offer a special on washers and dryers. The storeroom capacity is limited to 50 items. Each washer requires 2 hours to unpack and set up, and each dryer requires 1 hour. The manager has 80 hours of employee time available for unpacking and setup. Washers sell for $300 each, and dryers sell for $200 each. How many of each should the manager order to obtain the maximum revenue?
Ex 2:Lamps Inc. makes desk lamps and floor lamps. The company has 1200 hours of labor and $4200 to purchase materials each week. It takes 0.8 hour of labor to make a desk lamp and 1.0 hour to make a floor lamp. The materials cost $4 for each desk lamp and $3 for each floor lamp. The company makes a profit of $2.65 on each desk lamp and $3.15on each floor lamp. How many of each should be made each week to maximize profit?
3.3 Practice Sheet (Homework)
1. Maximize z = 2x + 3y, subject to
2. Maximize z = 10x + 15y, subject to
3. Find the Maximum and Minimum values of
z = 5x + 12y, subject to
4. The maximum production of a soft-drink bottling company is 5000 cartons per day. The company produces two kinds of soft drinks, regular and diet. It costs $1.00 to produce each carton of regular and $1.20 to produce each carton of diet. The daily operating budget is $5400. The profit is $0.15 per carton on regular and $0.17 per carton on diet drinks. How much of each type of drink is produced to obtain the maximum profit? (Do Not Solve, answer the following:))
X = Y =
What is the problem asking you to find? Which numbers have to do with this?
Write the Objective Function:Write your Constraints:
#5-6 Solve the following by showing the 8 steps of Linear Programming.
5. A T-shirt company has 3 machines, I, II, and III, which can be used to produce two types of T-shirts, standard design and custom design. The following table shows the number of minutes required on each machine to product the designs
Machine
I II III
Standard 1 1 1
Custom 1 4 5
For efficient use of equipment, the company uses machine I at least 240 min. per day, machine II at least 660 min per day, and machine III at least 1000 min per day. Each standard T-shirt costs $3, and each custom T-shirt costs $4. Find the number of each type of T-shirt that should be produced to minimize costs.
6. A sewing machine operator may sew coats or trousers. The trousers require 3 minutes of sewing time, and operator receives $0.50 each. A coat requires 8 minutes of sewing time, and the operator receives $1.00 each. The operator must sew at least three coats per hour. How many coats and how many trousers should the operator sew each hour to maximize hourly income?