Question

Problem 4-13 (Algorithmic) 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 Coffee and Romans Decaf Coffee. These coffees are blends of Brazilian Natural and Colombian Mild coffee beans, which are purchased from a distributor located in New York City. Because Romans purchases large quantities, the coffee beans may be purchased on an as-needed basis for a price 11% higher than the market price the distributor pays for the beans. The current market price is $0.48 per pound for Brazilian Natural and $0.69 per pound for Colombian Mild. The compositions of each coffee blend are as follows: Bean Blend Regular DeCaf 70% 40% 30% 60% Brazilian Natural Colombian Mild Romans sells the Regular blend for $3.5 per pound and the DeCaf blend for $4.4 per pound. Romans would like to place an order for the Brazilian and Colombian coffee beans that will enable the production of 900 pounds of Romans Regular coffee and 450 pounds of Romans DeCaf coffee. The production cost is $0.77 per pound for the Regular blend. Because of the extra steps required to produce DeCaf, the production cost for the DeCaf blend is $1.09 per pound. Packaging costs for both products are $0.25 per pound. Formulate a linear programming model that can be used to determine the pounds of Brazilian Natural and Colombian Mild that will maximize the total contribution to profit. Let BR = pounds of Brazilian beans purchased to produce Regular BD = pounds of Brazilian beans purchased to produce DeCaf CR = pounds of Colombian beans purchased to produce Regular CD = pounds of Colombian beans purchased to produce DeCaf If required, round your answers to four decimal places. For subtractive or negative numbers use a minus sign even if there is a plus sign before the blank. (Example: -300)

Answer 1

& frame the By using Constrains the following information as follows: | Blend Bean Regular Delaf Brazilian Natusel 704 40:1. Colambian Mird/ 301.601. The ratio of BR and CR is 70:30 The ratio of BD and CD is 40:60 The following constraints can be formed 0.70 0.30 BR -0.70CR = 0 , 60 = 40 0.6BD -0.460-0 other constraints BD+CD450 BR+CR € 900 BR, BD, CR, CD 20 Formulate an Ercel sheet as follows:

F Decision variable BR CR BD CD Composition constraints =(0.3*B2)-(0.7*B3) =(0.6*B4)-(0.4*B5) = 0 = 0 OVU AWN- DeCaf Production constraints =B4+B5 =B2+B3 <= 450 <= 900 Regular Revenue 3.5 9 Production cost 0.77 10 Packaging cost 0.25 4.4 1.09 0.25 Total production cost =B9*(B2+B3)+C9*(B4+B5) Brazalian natural Colombian mild 13 Bean cost in $) =(1+0.11)*0.48 =(1+0.11)*0.69 Total packging cost =B10*(B2+B3)+C10*(B4+B5) Total bean cost =B13*(B2+B4)+C13*(B3+B5) Total cost =D11+D14+D17 Max revenue =B8*(B2+B3)+C8*(B4+B5)-F17

the optimal solution Use Excel "solver" to get as follows:

Solver Parameters Set Objective: $D$20 To: O Max O Min value of: By Changing Variable Cells: $B$2:$B$5 Subject to the Constraints: $B$2:$B$5 >= 0 $D$3:$D$4 = 0 $D$7:$D$8 <= $F$7:$F$8 Add Change Delete Reset All Load/Save V Make Unconstrained variables Non-Negative Select a Solving Method: Simplex LP Options Solving Method Select the GRG Nonlinear engine for Solver Problems that are smooth nonlinear. Select the LP Simplex engine for linear Solver Problems, and select the Evolutionary engine for Solver problems that are non-smooth.

Thus, the optimal solution (units) is

Thus, the optimal solution (units) is E F Composition constraints 2 AB Decision variable 2 BR 630 3 CR 270 4BD 180 5 CD 270 = = 0 0 | Production constraints 450 900 450 900 Revenue 9 Production cost 10 Packaging cost Regular $3.50 $0.77 $0.25 DeCaf $4.40 $1.09 $0.25 Total production cost $1,183.50 11 12 13 Bean cost (in %) Brazalian natural Colombian mild 0.5328 Total packging cost $337.50 14 Total bean cost 845.154 Total cost $2,366.15 Max revenue $2,763.85

Optimal solution: BR = 630 CR=270 BD = 180 CD=270 Fi contribution to profit = $2,763.85]