Question

7.1) Billy Idell is trying to determine how many units of two types of lawn mowers to produce each day. One of these is the Standard model, while the other is the Deluxe model. The profit per unit on the Standard model is $60, while the profit per unit on the Deluxe model is $40. The Standard model requires 20 minutes of assembly time, while the Deluxe model requires 35 minutes of assembly time. The Standard model requires 10 minutes of inspection time, while the Deluxe model requires 15 minutes of inspection time. The company must fill an order for 6 Deluxe models. There are 450 minutes of assembly time and 180 minutes of inspection time available each day. How many units of each product should be manufactured to maximize profits? 7.2) Chewy's produces a low-calorie dog food for overweight dogs. This product is made from beef products and grain. Each pound of beef costs $0.90, and each pound of grain costs 0.60. A pound of the dog food must contain at least 9 units of Vitamin 1 and 10 units of Vitamin 2. A pound of beef contains 10 units of Vitamin 1 and 12 units of Vitamin 2. A pound of grain contains 6 units of Vitamin 1 and 9 units of Vitamin 2. Formulate this as an LP problem to minimize the cost of the dog food. How many pounds of beef and grain should be included in each pound of dog food? What is the cost and vitamin content of the final product? 7.2) Chewy's produces a low-calorie dog food for overweight dogs. This product is made from beef products and grain. Each pound of beef costs $0.90, and each pound of grain costs 0.60. A pound of the dog food must contain at least 9 units of Vitamin 1 and 10 units of Vitamin 2. A pound of beef contains 10 units of Vitamin 1 and 12 units of Vitamin 2. A pound of grain contains 6 units of Vitamin 1 and 9 units of Vitamin 2 Formulate this as an LP problem to minimize the cost of the dog food. How many pounds of beef and grain should be included in each pound of dog food? What is the cost and vitamin content of the final product?

Answer 1

7.1)

Fill all the detail in the question to excel as:

Now open the SOLVER from DATA in excel and enter the following constraints:

A D E G 1 Formula Used Allocated Maximize 0 D6*H4+E6*14 Standard Deluxe 4 Standard Deluxe S 60 $ Profit 40 20 35 7 Assembly Time Inspection Time 10 15 Constraints Deluxe 10 Standard Min Deluxe Order 11 6 12 13 Total Assembly Time Total Inspection Time 14 Available 450 180 Allocated 15 0 D9*H4+E9*14 Formula Used =H4*D8+I4*E8 16 17

Solver Parameters $C$3 Set Objective: To: 0 Min Value Of Max By Changing Variable Cells: $H$4:$I$4 Subject to the Constraints: $C$14 $D$15 < $D$14 $I$4 >=$E$11 $C$15 Add Change Delete Reset All Load/Save Make Unconstrained Variables Non-Negative Select a Solving GRG Nonlinear Options Method: 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. Help Solve Close

Enter solve button,we get the following results :

E F G H A Formula Used 1 Maximize Allocated $ 780D6*H4+E6*|4 Standard Deluxe 6 Standard Deluxe 60 $ Profit 40 Assembly Time 20 35 Inspection Time 10 15 Deluxe 10 Constraints Standard Min Deluxe Order 6 11 12 13 Total Assembly Time Total Inspection Time Available 14 450 180 Allocated 15 390 180 Formula Used '=H4*D8+14*E8 D9*H4+E9* 14 16 17

The total units for standard = 9 units

The total units for Deluxe = 6 units

Therefore, the total profit after maximization is $780.

7.2)

Fill all the detail in the question to excel as:

Let the quantity of beef be X1 and Grain be X2 Objective function is Zmin= 0.9 Xi+0.6 X2 Subject to constraints Vitamin A-10 X1+6 X2 Vitamin D-12 X1+9 X2 Where X1, X220 Now solve the uation using excel frSUMPRODUCT(C4:D4, C5: D5) E5 B A C F G H 1 2 3 X1 X2 4 Cost 0.9 0.6 C 6 7 8 Vitamin A 10 6 0 9 > 9 Vitamin D 12 C 10 > 10 Nm n N00 n

Formula for cell F5 = SUMPRODUCT(C8:D8,C4:D4)

Now open the SOLVER from DATA in excel and enter the following constraints:

Now use solver to calculate optimum value Solver Parameters Set Objective: $F$5 0 To: Value Of: O Min Max By Changing Variable Cells: $C$4:$D$4 Subject to the Constraints: SE$8: $E$9> $G$8:$G$9 Add Change Delete Beset All Load/Save Make Unconstrained Variables Non-Negative Simplex LP Select a Solving Method: 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. Help Solve Close

Enter solve button,we get the following results :

A. C D E G H 1 2 3 X1 X2 0.9 5 Cost 0.9 0.6 0.81 6 7 8 Vitamin A 10 6 9 > Vitamin D 12 9 10.8 10 > 10 11 stLn LO N00

SO the company should use 0.9 units of beef in its dog food to reduce the cost