Solution

Romans Food Market, located in Saratoga, New York, carries a variety of specialty foods from around the world. Two of the stores leading products use the Romans Food Market name! Romans Regular "offee and Romans #e"af "offee. These coffees are $lends of  %ra&il %ra&ilian ian Natura Naturall and "olom$ "olom$ian ian Mild Mild coffee coffee $eans, $eans, which which are purcha purchased sed from from a distri$utor located in New York "ity. %ecause Romans purchases large 'uantities, the coffee $eans may $e purchased on an as(needed $asis for a price )*+ higher than the market price the distri$utor pays for the $eans. The current market price is *.- per   pound for %ra&ilian Natural and *./0 per pound for "olom$ian Mild. The compositions of each coffee $lend are as follows!

.!. Romans sells the Regular $lend for 1./* per pound and the #e"af $lend for -.-* per   pound. Romans would like to place an order for the %ra&ilian and "olom$ian coffee  $eans that will ena$le the production of )*** pounds of Romans Regular coffee and 2**  pounds of Romans #e"af coffee. The production cost is *.3* per pound for the Regular   $lend. %ecause of the e4tra steps re'uired to produce #e"af, the production cost for the #e"af $lend is ).*2 per pound. 5ackaging costs for $oth products are *.02 per pound. Formulate a linear programming model that can $e used to determine the pounds of  %ra&ilian Natural and "olom$ian Mild that will ma4imi&e the total contri$ution to profit. 6hat is the optimal solution and what is the contri$ution to profit7

SOLUTION:

8et

%R

9

pounds of %ra&ilian $eans purchased to produce Regular  

%#

9

pounds o off %r %ra&ilian $e $eans purchased to to pr produce #e #e"af 

"R

9

pounds of "olom$ian $eans purchased to produce Regular 

"#

9

pounds o off "o "olom$ian $e $eans pu purchased to to pr produce #e #e"af 

Type of Bean %ra&ilian "olom$ian

Cost per pound ($) ).)*:*.-; 9 *.2) ).)*:*./0; 9 *./30

Total revenue 9 1./*:%R < "R; < -.-*:%# < "#;

Total cost of $eans 9 *.2):%R < %#; < *./30:"R < "#;

Total production cost 9 *.3*:%R < "R; < ).*2:%# < "#;

Total packaging cost 9 *.02:%R < "R; < *.02:%# < "#;

Total contri$ution to profit 9 :total revenue; ( :total cost of $eans; ( :total production cost;

∴

Total contri$ution to profit 9 0.*11%R < 0.231%# < ).3/3"R < 0.-)3"#

Regular % constrant

%R 9 *.2:%R < "R; *.02%R ( *.2"R 9 *

!eCaf % constrant

%# 9 *.-*:%# < "#; *./*%# ( *.-*"# 9 *

5ounds of Regular! %R < "R 9 )***

5ounds of #e"af! %# < "# 9 2**

The complete linear program is

Ma4

0.*11%R < 0.231%#

< ).3/3"R

< 0.-)3"#

s.t. *.02%R

( *./*%#

 

%R

< %#

*.2"R

9 * ( *.-*"# 9 * "R 9 )*** < "# 9 2** %R, %#, "R, "# ≥ *

=sing The Management Scientist, the optimal solution is %R 9 2*, %# 9 0**, "R 9 02*, and

"# 9 1**.

The value of the optimal solution is 1011.2