Question

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 10% higher than the market price the distributor pays for the beans. The current market price is $0.47 per pound for Brazilian Natural and 062 per ound for Colombian Mild. The compositions of each coffee blend are as follows: RegularDeCaf Brazilian Natu 25% 60% Romans sells the Regular blend for $3.60 per pound and the DeCaf blend for $4.40 per pound. Romans would like to place an order for the Brazilian and Colombian coffee beans that will enable the production of 1000 pounds of Romans Regular coffee and 500 pounds of Romans DeCaf coffee. The production cost is so.80 per pound for the Regular blend. Because of the extra steps required to produce DeCaf, the production cost for the DeCaf blend is $1.05 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 af If required, round your answers to three decimal places. For subtractive or negative numbers use a minus sign even if there is a plus sign before the blank. (Example: -300) CD CR + BD+ BR + Max X CR X BR X CD- X BD 1 VBR BR, BD, CR, CD 2 0 What is the optimal solution and what is the contribution to profit? If required, round your answer to the nearest whole number Optimal solution: BR = CR CD = If required, round your answer to the nearest cent. value of the optimal solution = $

Answer 1

Solution:

Algebraic formulation Decision Variables: 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 Objective function: Max Z 2.033BR 2.583BD +1.868CR +2.418CD BR CR 1000 BD CD 500 0.25BR 0.75CR 0 Subject to constraints: BR, BD, CR, CD0

Q fx-SUMPRODUCT(D26:G26,SD$28:$G$28) H26 A B Solution using Excel Solver: 15 16 17 18 19 20 S Solver Parameters Market Disti Reqular DeCaf Price Marku Bean Brazilian Natural Colombian Mild Cost 0.47 | 10% | $0.517 6096 | 0.62 | 10% | $0.682 SHS26 Set Objective: To: By Changing Variable Cells: O Max O Min Value Of: O 25% SDS28 SG528 Subject to the Constraints: SHS31:SHS34- SIS31:SJS34 CR CD 3.600 4.40 3.600 0.800 1.050 0.8001.050 Selling Price Production Cost Packaging Cost Beans Cost Net Profit/pound 0.2500.250 0.250 0.5170.682 0.682 2.0330 2.5830 1.8680 2.4180 3,233.75 Change 0.517 26 Delete Decision Variable Xi -> 28 750 200 250 Reset Al 29 Constraints Production of Regular CR LHS 10001000 500 Demand CD Make Unconstrained Variables Non-Negative Select a Solving Method: Simplex LP Options 500 32 Production of Dec 0.25 -0.75 Solving Method 0.60 -0.40 Select the IPOPT Nonlinear engine for Solver Problems that are smooth nonlinear.Select the LP Simplex engine for linear Solver Problems. 35 Formulas H18-F18 (1+G18) D26 SUM D22:D25) H26-SUMPRODUCT D26:626SD$28:$6$28)H26, H31:H34 Copy to SokveClose

Optimal solution:

BR = 750

BD = 200

CR = 250

CD = 300

Total profit = $ 3233.75