HW Cht 13 Part 2 LP Formulation and Solution - Answer
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Answer to Homework Cht. 13, Part 2, LP Formulation and Solution
77 points, BSNS2120, J. Wang
Name _____________________
Page 511 – 518, Problems #9, #26, #32, #40
.
For each problem, define the variables, formulate a linear program, and solve it by using Excel Solver.
Problem #9 (16 pts)
Define the meanings of variables:
X
1
= number of Model 1 shoes to produce
X
2
= number of Model 2 shoes to produce
X
3
= number of Model 3 shoes to produce
Objective (in terms of X’s):
Maximize (total profit) = 50X
1
+44X
2
+40X
3
Constraints (in terms of X’s):
Constraint 1. Cardstock to be used <= cardstock available:
12X
1
+10X
2
+14X3 <= 1,200
Constraint 2. Satin to be used <= satin available:
24X
1
+20X
2
+15X
3
<= 2,000
Constraint 3. Plain fabric to be used <= plain fabric available:
40X
1
+40X
2
+30X
3
<= 7,500
Constraint 4. Leather to be used <= leather available:
11X1+11X
2
+10X
3
<= 1,000
Non-negative constraints:
X
1
, X
2
, X
3
>= 0
Enter the linear program in an Excel sheet, and solve it by Solver:
The optimal solution is:
X
1
= 66.67, X
2
= 0, X
3
= 26.67.
The optimal objective function value is: $4,400
With the optimal solution, which materials are fully used (i.e. LHS=RHS)?
Satin (as in constraint 2) and leather (as in constraint 4).
1
Problem #26 (16 pts)
Define the meanings of variables:
X
1
= number of Ferrari model cars to produce
X
2
= number of BMW model cars to produce
X
3
= number of Lotus model cars to produce X
4
= number of tesla model cares to produce
Objective: Maximize Total Profit = Max _350X
1
+330X
2
+270X
3
+255X
4
_________ (in terms of X’s):
Constraints. (For each of the five departments, [production time (in min) to be used] <= [time available].)
Write out the constraints in terms of X’s:
Molding dept.:
5X
1
+3.5X
2
+1X
3
+3X
4
<= 600,
Sanding dept.:
4X
1
+3.2X2+2X
3
+3.65X
4
<= 600,
Polishing dept.:
3.5X
1
+2X
2
+3X
3
+1X
4
<= 480,
Painting dept.:
3.75X
1
+3.25X
2
+1.75X
3
+2X
4
<= 480,
Finishing dept.:
4X
1
+1X
2
+2X
3
+3X
4
<= 480.
Non-negative constraints:
X
1
, X
2
, X
3
, X
4
>= 0.
Enter the linear program in an Excel sheet, and solve it by Solver:
The optimal solution is:
X
1
=0, X
2
=56.00933, X
3
=103.05718, X
4
=58.8098.
The optimal objective function value is: 61,305.018
.
If producing according to the optimal solution, how many production minutes are required in Molding Department?
Plug the optimal solution in the LHS of the first constraint (Modeling dept.):
5(0)+3.5(56/00933)+1(103.05718)+3(58.8098) = 475.5193
. Or,
as given in the results of Excel Solver
, the value of LHS of the first constraint: 475.5193
2
Problem #32 (25 pts)
Define the meanings of 11 variables:
X
1
= $ to invest in Large cap blend
X
2
= $ to invest in Small cap growth
X
3
= $ to invest in Green fund
X
4
= $ to invest in Growth and income
X
5
= $ to invest in Multicap growth,
X
6
= $ to invest in Midcap index,
X
7
= $ to invest in Multicap core,
X
8
= $ to invest in Small cap international,
X
9
= $ to invest in Emerging international,
X
10
= $ to invest in Money market fund,
X
11
= $ to invest in Savings account.
Objective: Maximize total net $ return
= Max 0.1627X
1
+ 0.1984X
2
+ 0.2560X
3
+ 0.1468X
4
+ 0.1888X
5
+ 0.2188X
6
+ 0.2692X
7
+ 0.3446X
8
+ 0.3493X
9
+ 0.0475X
10
+ 0.01X
11
Hint: (Net return %) = (average return %) – (expense %)
Constraints (there should be totally 15 constraints):
(1) X
1
+ X
2
+ X
3
+ X
4
+ X
5
+X
6
+X
7
+X
8
+ X
9
+ X
10
+ X
11
<= 100,000 (total up to $100,000)
(2) X
11
>= 5000 (>=$5,000 in savings)
(3) X
10
>= 14,000 (>=14% of $100,000 in money market fund)
(4) X
8
+ X
9
>= 16,000 (>=16% of $100,000 in international funds)
(5) X
1
+ X
2
+ X
3
+ X
4
>=35,000 (Keep 35% of funds in current holdings)
(6) X
5
+ X
6
+ X
7
+ X
8
+ X
9
+ X
10
+ X
11
>= 30,000, (>=30% of $100,000 in new investment)
Next 9 constraints: (<=20% of $100,000 in each fund except for money market and savings)
(7) X
1
<= 20000, (8) X
2
<= 20000,
(9) X
3
<= 20000,
(10) X
4
<= 20000,
(11) X
5
<= 20000,
(12) X
6
<= 20000,
(13) X
7
<= 20000,
(14) X
8
<= 20000,
(15) X
9
<= 20000.
Enter the linear program in an Excel sheet, and solve it by Solver:
The optimal solution is:
X
2
=$15,000, X
3
=$20,000, X
7
=$6,000. X
8
=$20,000, X
9
=$20,000, X
10
=$14,000, X
11
=$5,000, other X’s = $0.
The optimal objective function value is: $24,304 (total net return).
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Problem #40 (20 pts)
Define the meanings of variables:
X
1
= number of pounds of Grade 1 plastic to make,
X
2
= number of pounds of Grade 2 plastic to make,
X
3
= number of pounds of Grade 3 plastic to make,
X
4
= number of pounds of Grade 4 plastic to make.
Objective (in terms of X’s):
Max
total profit = Max 2X
1
+1.7X
2
+1.5X
3
+2.8X
4
Constraints (in terms of X’s):
(1) 0.4X
1
+0.37X
2
+0.34X
3
+0.9X
4
<= 100,000
(Additive A, in pounds)
(2) 0.3X
1
+0.33X
2
+0.33X
3
<= 90,000
(Additive B, in pounds)
(3) 0.2X
1
+0.25X
2
+0.33X
3
<= 40,000
(Additive C, in pounds)
(4) 0.1X
1
+0.05X
2
+0.1X
4
<= 10,000
(Additive D, in pounds)
(5) X
1
+X
2
<= 75%(X
1
+X
2
+X
3
+X
4
)
(total of Grade 1 and 2 <= 75% of total)
that is: X
1
+X
2
−0.75X
1
−0.75X
2
−0.75X
3
−0.75X
4
<= 0,
that is: 0.25X
1
+0.25X
2
−0.75X
3
−0.75X
4
<= 0
.
(6) X
4
>= 40%(X
1
+X
2
+X
3
+X
4
)
(at least 40% of total must be Grade 4)
that is: X
4
− 0.4X
1
−0.4X
2
−0.4X
3
−0.4X
4
>= 0,
that is: −0.4X
1
−0.4X
2
−0.4X
3
+0.6X
4
>= 0
.
Non-negative constraints: X
1
, X
2
, X
3
, X
4
>= 0.
Enter the linear program in an Excel sheet, and solve it by Solver:
The optimal solution is:
X
1
=30,370.37, X
2
=0, X
3
=74,074.074, X
4
=69,629.6296
The optimal objective function value is: $366,814.81
.
In Excel Solver’s results, LHS of constraint 2 is _
33,555.555
_____, while RHS of constraint 2 is _
90,000
____________. What does the difference between RHS and LHS mean in constraint 2? (i.e., what does [RHS]−[LHS] mean in constraint 2?)
90,000-33555.555 = 56,444.445 is pounds of additive B that are not used.
4