Question

CNY Coffee Inc. uses three types of coffee beans (Columbia, Java, and Brazil) as mixtures to manufacture its two coffee products (Energy, and Everyday). The cost per pound, the availability, and rating information of the coffee beans are as follows.

Coffe Bean	Cost Per Pound	Caffeine Rating	Taste Ratting	Aroma Rating	Available Pounds
Columbia	$2.25	68	90	88	695
Java	$1.70	78	63	77	880
Brazil	$1.95	85	80	95	955

Coffee product tests are used to provide ratings on a scale of 0-100, with higher ratings indicating higher product performance. Here are the two coffee products and their product information:

Coffee Product	Selling Price per Pound	Product Segment	Product Information
Energy	$4.95	High Caffeine	Caffeine level rates at least 75 or higher; Taste level rates at least 77 or higher; No Aroma level requirement
Everyday	$3.15	Low Price	No requirement on caffeine level; Tastes level rates between 70 and 75; Aroma level rates no more than 85.

Due to the popular demand of the low-price segment, at least 920 pounds of Everyday must be produced.

Set up and list the linear programming model for the above problem. (You do not need to solve the listed model).

Answer 1

The objective should be to minimize the cost price of "Everyday" cofee product :

Let the number of pounds of Columbia, Java & Brazil Coffee Bean used be X₁ , X₂ & X₃ respectively to make "Everyday" cofee product.

Then we have to minimize the following function in order to keep the cost price lowest:

z = 2.25X₁ + 1.7X₂ + 1.95X₃

The main constraints are as follows :

1) 0 X₁ 695 (Maximum available pounds for Columbia)

2) 0 X2 880 (Maximum available pounds for Java)

3) 0 X3 955 (Maximum available pounds for Brazil)

4) X₁ + X₂ + X₃ 920 (Atleast 920 pounds of "Everyday" cofee must be produced)

5) 88X₁ + 77X₂ + 95X₃ (85*920 = 78200) (Aroma level rate should not be more than 85)

6) (70*920 = 64400) 90X₁ + 63X₂ + 80X₃ (75*920 = 69000) (Tastes level rate between 70 & 75)