Question

i just want the answer to include decision variable , objective function and the constrains

"6 Chapter 2 Modeling with Linear Programming 4-48. Four products are processed sequentially on three machines. The following table gives The pertinent data of the problem. Manufacturing time (ht) per unit ya Product Product 2 Product 3 Product 4 - A Machine Cost per hr ($) Capacity (hr) 500 380 450 10 Unit selling price ($) 75 70 com 55 45 Formulate the problem as an LP model and find the optimum solution using AMPL, Solver,

Answer 1

Profit per unit of each product is determined by the following formula

Profit = Unit selling price - SUMPRODUCT of Manufacturing time on each machine and Cost per hr

Profit of product 1 = 75 - (2*10+3*5+7*4) = $ 12

Profit of product 2 = 70 - (3*10+2*5+3*4) = $ 18

Profit of product 3 = 55 - (4*10+1*5+2*4) = $ 2

Profit of product 4 = 45 - (2*10+2*5+1*4) = $ 11

LP model is formulated as under:

Decision variables:

Let X1, X2, X3, X4 be the number of product 1, 2, 3, 4 to be produced respectively

Objective function:

Max 12X1+18X2+2X3+11X4 (total profit)

Constraints:

2X1+3X2+4X3+2X4 <= 500   (total time on machine 1)

3X1+2X2+1X3+2X4 <= 380   (total time on machine 2)

7X1+3X2+2X3+1X4 <= 450   (total time on machine 3)

X1, X2, X3, X4 >= 0

Create the spreadsheet model, enter Solver Parameters as shown below. Then click Solve button on the Solver parameters window. Result will appear in cells B10:E10

@ fx C A =SUMPRODUCT(B3:E3,$B$10:$E$10) D E F G x3 x4 H I S Solver Parameters B X1 XP X2 Maximize: 12 18 2 11 2950 Set Objective: SF53] To: Max Min value of: O By Changing Variable Cells: $B$10:SE$10 Subject to the Constraints: SF$5:SF$7 <= SH$5:SH$7 2 5 6 7 8 Machine 1 Machine 2 Machine 3 L 3 4 3 2 1 7T3T 2 1 2 1 500 366.6667 | <= 450 <= 500 380 450 Add X3 X4 Change x1 0 X2 133.33333 Result: Delete Reset All Make Unconstrained Variables Non-Negative Select a Solving Method: Simplex LP - Options Solving Method Select the IPOPT Nonlinear engine for Solver Problems that are smooth nonlinear. Select the LP Simplex engine for linear Solver Problems Solve Close

EXCEL FORMULAS:

F3 =SUMPRODUCT(B3:E3,$B$10:$E$10) copy to F5:F7

Solution:

X1 = 0

X2 = 133.33

X3 = 0

X4 = 50

Total profit = $ 2,950