A product mix example Quick-Screen is a clothing manufacturing company that spec
ID: 366451 • Letter: A
Question
A product mix example
Quick-Screen is a clothing manufacturing company that specializes in producing commemorative shirts immediately following major sporting events such as the World Series, Super Bowl, and Final Four. The company has been contracted to produce a standard set of shirts for the winning team, either State University or Tech, following a college football bowl game on New Year's Day. The items produced include two sweatshirts, one with silk-screen printing on the front and one with print on both sides, and two T-shirts of the same configuration. The company has to complete all production within 72 hours after the game, at which time a trailer truck will pick up the shirts. The company will work around the clock. The truck has enough capacity to accommodate 1,200 standard-size boxes. A standard-size box holds 12 T-shirts, and a box of 12 sweatshirts is three times the size of a standard box. The company has budgeted $25,000 for the production run. It has 500 dozen blank sweatshirts and T-shirts each in stock, ready for production. The company wants to know how many dozen (boxes) of each type of shirt to produce in order to maximize profit
Explanation / Answer
Decision Variables
This problems contains four decision variables, representing the number of dozens (boxes) of each type of shirts to produce.
X1 = sweatshirts, front printing
X2 = sweatshirts, front and back printing
X3 = T-shirts, front printing
X4 = T-shirts, front and back printing
The objective function
The company’s objective is to maximize profit. The total profit is the sum of the individual profits gained from each type of shirt. The objective is expressed as
Maximize Z = $90x1 + 125x2 + 45x1 + 65x4
Model Constraints
The first constraint is for processing time. The total available processing time is the 72-hour period between the end of the game and the truck pickup:
0.10x1 + 0.25x2 + 0.08x3 + 0.21x4 72 hr
The second constraint is for the available shipping capacity, which is 1,200 standard-size boxes. A box of sweatshirts is three times the size of a standard-size box. Thus, each box of sweatshirts is equivalent in size to three boxes of T-shirts. This relative size differential is expressed in the following constraint:
3x1 + 3x2 + x3 + x4 1,200 boxes
The third constraint is for the cost budget. The total budget available for production is $25,000:
$36x1 + 48x2 + 25x3 + 35x4 25,000
The last two constraints reflect the available blank sweatshirts and T-shirts the company has in storage:
x1 + x2 500
x3 + x4 500
Model Summary
Maximize Z = $90x1 + 125x2 + 45x1 + 65x4
Subject to
0.10x1 + 0.25x2 + 0.08x3 + 0.21x4 72 hr
3x1 + 3x2 + x3 + x4 1,200 boxes
x3 + x4 500
x1 + x2 + x3 + x4 0
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