PRACTICE PROBLEMS
25
Chapter 7 Optimization Practice Problems
Use Excel and VBA to solve the following problems. Document your solutions using the Expert Problem
Solving steps outlined in Table 1.2.
1. You plan to bake cakes and cookies. Each cake requires 5/2 cups of flour and 2 cups of sugar. A
batch of cookies uses 1 cup of flour and ½ cup of sugar. You only have 70.5 cups of flour and 40.5
cups of sugar. If you can sell each cake for $10 and each batch of cookies for $3, how many cakes
and how many batches of cookies should you make to maximize your income? Show your work.
2. Your company manufactures two products A and B. Three machines, M1, M2, M3 are available to
process either product. The processing times in hours per item A or B on each machine are given in
the table below.
M1 M2 M3
A 0.5 0.4 0.2
B 0.25 0.3 0.4
The available production time of the machines M1, M2, and M3 are 40 hours, 36 hours, and 36 hours,
respectively, each week. The profit per item of A and B is $5 and $3 respectively. Determine the
weekly production amounts of A and B, which will maximize profits. Show your work.
3. A company manufactures two products that use the same two raw materials A and B. Each unit of
product 1 requires 8 lb of A and 7 lb of B; each unit of product 2 requires 3 lb of A and 6 lb of B.
The maximum amount of A available per week is 1200 lb, and the maximum amount of B available
per week is 2100 lb. The company can sell as many units of product 1 and product 2 as they can
make, but the selling price of each product depends on how many units of both products are sold.
The profit for one unit of product 1 is 800 – x1 – x2 and the profit for one unit of product 2 is 2000 –
x1 – 3x2, where x1 and x2 are the weekly production totals for the products 1 and 2 respectively. Find
the optimal quantities of the two products to produce each week that maximizes the weekly profit.
Show your work.
4. A chemical manufacturer sells three products with monthly production rates, x, y, and z. The monthly profit function is f = 10x + 4.4y2 + 2z. It is necessary to impose the following limits on production
rates and raw materials:
x2
0.5z 2  y 2  3
x  4 y  5z  32
x  3 y  2 z  29
Determine the best combination of production rates for x, y and z. Use Excel or Mathcad.
5. A candy company makes two kinds of candy bars from caramel and chocolate. Bar A has 3 oz of
caramel and 1 oz of chocolate and sells for $0.30. Bar B has 2 oz of caramel and 2 oz of chocolate
and sells for $0.54. The company has stocked 90 lb of chocolate and 144 lb of caramel. How many
units of each type of candy bar should be made to maximize the income? (There are 16 oz in a lb.)
6. Find the location and value of the minimum of the following function with constraints.
2
2
2
f  100  x13 x2    200   x1 x2    300  x1 x23 


2
0  x1  5
0  x2  5
6. Use Excel, PNM2Suite to find the location and value of the global minimum of the following function
26
PRACTICAL NUMERICAL METHODS
2
2
f  x, y    cos  x  cos  y  exp    x  1   x  2  


7. You may use Excel to solve this problem. A three stage compressor is used to raise a feed pressure
from P0 = 1 bar to a third stage discharge pressure of P3 =78 bar. The total work need is described by
 P 2 7  P 2 7  P 2 7 
W  C  1    2    3  
 P0 
 P1 
 P2  
where P0, P1, P2, and P3 are the feed and discharge pressures from stages 1 through 3.
a. Plot the work function in surface and contour plots and identify the approximate location of
the optimum.
b. Determine the optimal discharge pressures P1 and P2 for the first and second stages to minimize the work.
8. Use surface and contour plots in Excel to analyze the following objective function. Find the values for
x1, x2, and f at the minimum.
f   x1  1   x2  1  x1 x2
9. Optimize f(x) for A = 7.5, B = 0.855, using the following methods. Use a VBA user-defined function
of the following equation:
2
2
x
f  x   x 2  A sin  
B
a. Use your plot from Problem 1 to estimate the locations and value for f(x) of the global minimum and maximum of f(x).
b. Find the location in the range 0 < x ≤ 5 and value of f(x) of the global minimum using the
Solver.
c. Find the location in the range 0 < x ≤ 5 and value of f(x) of the global maximum using the
Solver.
Answers: (b) x=1.12, f=5.99 (c) x=5, f = 28.16