Question

1. A small computer manufacturing company pro. duces both laptop and desktop computers. The production of a laptop computer requires 4 hours of assembly and 1 hour of testing time. The pro- duction of a desktop computer requires 4 hours of assembly and 1.5 hours of testing time. The profit margin for a laptop computer is $400, and the profit margin for a desktop computer is $500. There are a total of 5,000 hours of assembly time and 1,000 hours of testing time available every week. The company has decided to produce at least twice as many laptop computers as desktop computers. a. Assuming unlimited demand for both laptop and desktop computers, formulate a linear programming problem. b. Solve the problem using the graphical method. c. Solve the problem using Microsoft Excel Solver.

Answer 1

(a)

Decision Variables Let number of laptop computers produced be Let number of desktop computers produced be Constraints Assembly time constraint 4*X 4-Y VI 5000 ...1 Testing Time Constraint 1.0*X 1.5*Y s 1000 ....2 Ratio of Laptop to Desktop X 2-Y Objective Function Maximize Profit 400*X+ 500*Y

(b)

Graphical solution 1400 1200 plotting the constraints assuming equality of LHS and RHS X Y Assembly constraint 0 1250 1250 0 750 500 1000 -Assembly constraint 800 Testing Time Constraint -Testing Time Constraint 600 0 1000 250 666.67 0 500 400 Ratio of laptop to Desktop Ratio of Lap top to Desktop 200 0 1000 571.42 0 500 285.71 0 0 500 1000 1500 The fesible solution has three points : Y Point1 Point2 Point3 х 0 571.4 1000 0 285.71 0 Profit $ $ 3,71,423.00 $ 4,00,000.00 So the point X= 1000 and Y=0 i.e. Laptop = 1000 and Desktop = 0 gives maximum profi. Maximum Profit = $ 4,00,000.00

(c)

Initial formulation:

B D E F G H х Y 0 0 A 28 29 Objective Function 30 31 Maximize Profit 400*X+ 500*Y 32 Profit = $ 33 34 Excel Model 35 Solver Model 36 Assembly constraint 37 Testing Time Constraint 38 Ratio of Lap top to Desktop Constraint Total 0 LHS 5000 1000 0 0 29

D E F G H Y X 0 0 А. B с 28 29 Objective Function 30 31 Maximize Profit : 400*X+ 500"Y 32 Profit = =400*G31+H31*500 33 34 Excel Model 35 Solver Model 36 Assembly constrain 37 Testing Time Const 38 Ratio of Lap top to 39 Constraint LHS s Total =G31*4+H31-4 =G31.1+H31.1.5 =G31-2-H31 5000 1000

Solver constraints:

Solver Parameters х Set Objective: SBS32 To: Max Min O Value of: 0 By Changing Variable Cells: SG$31:5H531 Subject to the Constraints: SE$36:$E$37 <= $G$36:$G$37 SES38 >= SG$38 Add Change Delete Reset All Load/Save 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. Help Solve Close

Clicking solve, we get

X Y 1000 0 29 Objective Function 30 31 Maximize Profit 400*X+ 500*Y 32 Profit = $ 4,00,000.00 33 34 Excel Model 35 Solver Model 36 Assembly constraint 37 Testing Time Constraint 38 Ratio of Lap top to Desktop Constraint Total 4000 1000 1000 LHS 5000 1000 0 s

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