Question

Q2. A commercial orchard grows, picks, and packs apples and pears. A peck (quarter bushel) of apples takes four minutes to pick and five minutes to pack; a peck of pears takes five minutes to pick and four minutes to pack. Only one picker and one packer are available. How many pecks each of apples and pears should be picked and packed every hour (60 minutes) if the profit is $4/peck for apples and $3/peck for pears? Solve graphically as a linear program and confirm analytically. Submission Date: Wednesday, 15/04/2019

Answer 1

Solution using graphical method is following:

Decision variables:Xpeck of apples to pick and pack every hour Y peck of pears to pick and pack every hour Objective function: Max Z 4X 3Y 4X5Y 60 5X4Y 60 X, Y20 C1 C2 Constraints: X-axis and Y-axis Tabulate values of X & Y for above constraints and plot them on the graph 16 14 12 10 12 15 15 6 Feasible 12 4 Region obj (Z) 16 0 0 2 46 8 10 12 14 16 12 The yellow shaded region in the graph is the feasible region Obj. (Z) Corner Points (C2, X-axis) C1, C2 C1, Y-axis 48Optimal Solution 12 6.67 6.67 12 46.67 36

Optimal solution:

Number of pecks of apples to pick and pack (X) every hour = 12

Number of pecks of pears to pick and pack (Y) every hour = 0

Total profit per hour = $ 48

Solution confirmation by analytical solver method is following:

Q fx-SUMPRODUCT(B3:C3,$B$9:$C$9) D E F D3 SSolver Parameters 3 Profit 4 48 Set Objective SDS3 ⓔMaxO Min O Value Of: 0 LHS 48 60 RHS By Changing aiable Cells: 4 5 60 SBS9:SCS9 5 4 60 Subject to the Constraints: SDS6:SDS7SFS6:SFS7 Add Result: 12 Change Delete 12 13 Reset All Make Unconstrained Variables Non-Negative 15 16 17 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. 19 20 21 Solve Close

EXCEL FORMULAS:

D3 =SUMPRODUCT(B3:C3,$B$9:$C$9) copy to D6:D7

The solution is same using analytical method