MKTG 7825, Spring 2010
Homework #1, Answer Key
1. Winston (1994), Section 3.4, Problem 2
U.S. Labs manufactures mechanical heart valves from the heart valves of pigs. Different
heart operations require valves of different sizes. U.S. Labs purchases pig valves from
three different suppliers. The cost and size mix of the valves purchased from each
supplier are given in Table 3. Each month, U.S. Labs places one order with each supplier.
At least 500 large, 300 medium, and 300 small valves must be purchased each month.
Because of limited availability of pig valves, at most 700 valves per month can be
purchased from each supplier. Formulate an LP that can be used to minimize the cost of
acquiring the needed valves; and use Excel Solver to solve this problem.
Supplier
1
2
3
Cost Per Valve
($)
5
4
3
Table 3
Percent Large
40
30
20
Percent
Medium
40
35
20
Percent Small
20
35
60
Solution
Let x1 = Number of valves ordered each month from supplier 1.
x2 = Number of valves ordered each month from supplier 2.
x3 = Number of valves ordered each month from supplier 3.
Then a correct formulation is
min
s.t.
z = 5x1 + 4x2 + 3x3
.4x1 + .3x2 + .2x3  500
.4x1 + .35x2 + .2x3  300
.2x1 + .35x2 + .60x3  300
x1  700, x2  700, x3  700
x1  0, x2  0 x3  0
(Receive enough large valves)
(Receive enough medium valves)
(Receive enough small valves)
The optimal orders are (x1, x2, x3) = (700,700,50), and the minimal cost is $6450.
2. Winston(1994), Section 3.5, Problem 3
Suppose the post office can force employees to work one day of overtime each week.
For example, an employee whose regular shift is Monday to Friday can also be required
to work on Saturday. Each employee is paid $50 a day for each of the first five days
worked during a week and $62 for the overtime day (if any). Formulate an LP whose
solution will enable the post office to minimize the cost of meeting its weekly work
requirements. (Note that this problem is asking you to change the linear program given in
Example 7 on pg 74.); and use Excel Solver to solve this problem.
Solution
Let x1 = Number of employees who start work on Sunday and work 5 days, x2 =
Number of employees who start work on Monday and work 5 days, ... x7 = Number
of employees who start work on Saturday and work 5 days.
Also let y1 = Number of employees who start work on Sunday and work 6 days, ... ,
y7 = Number of employees who start work on Saturday and work 6 days. Then the
appropriate LP is
min z=5*50*(x1+x2+...x7)+(5*50+62)*(y1+y2+...y7)=250(x1+x2+...x7)
312(y1+y2+...y7)
s.t.
+
x1+x4+x5+x6+x7+y1+y3+y4+y5+y6+y711 (Sunday)
x1+x2+x5+x6+x7+y1+y2+y4+y5+y6+y717 (Monday)
x1+x2+x3+x6+x7+y1+y2+y3+y5+y6+y713 (Tuesday)
x1+x2+x3+x4+x7+y1+y2+y3+y4+y6+y715 (Wednesday)
x1+x2+x3+x4+x5+y1+y2+y3+y4+y5+y719 (Thursday)
x2+x3+x4+x5+x6+y1+y2+y3+y4+y5+y614 (Friday)
x3+x4+x5+x6+x7+y2+y3+y4+y5+y6+y716 (Saturday)
All variables nonnegative
Alternate Solution:
Let x1 = Number of employees who start work on Sunday, x2 = Number of
employees who start work on Monday, ... x7 = Number of employees who start work
on Saturday.
Let y1 = # of employees who work overtime on Sunday, y2 = # of employees who
work overtime on Monday, …, y7 = # of employees who work overtime on Saturday
Min z = 5*50*(x1+x2+...x7)+62(y1+y2+...y7) = 250(x1+x2+...x7)+62(y1+y2+...y7)
x1+x4+x5+x6+x7+y1 11 (Sunday)
x1+x2+x5+x6+x7+y2 17 (Monday)
x1+x2+x3+x6+x7+y3 13 (Tuesday)
x1+x2+x3+x4+x7+y4 15 (Wednesday)
x1+x2+x3+x4+x5+y5 19 (Thursday)
x2+x3+x4+x5+x6+y6 14 (Friday)
x3+x4+x5+x6+x7+y7 16 (Saturday)
y1 <= x3 (Only employees who start work on Tuesday are allowed to work overtime
on Sunday)
s.t.
y2 <= x4 (Only employees who start work on Wednesday are allowed to work
overtime on Monday)
y3 <= x5 (Only employees who start work on Thursday are allowed to work
overtime on Tuesday)
y4 <= x6 (Only employees who start work on Friday are allowed to work overtime
on Wednesday)
y5 <= x7 (Only employees who start work on Saturday are allowed to work
overtime on Thursday)
y6 <= x1 (Only employees who start work on Sunday are allowed to work overtime
on Friday)
y7 <= x2 (Only employees who start work on Monday are allowed to work overtime
on Saturday)
All variables nonnegative
The optimal number of regular employees are (x1, x2, x3, x4, x5, x6, x7) = (3,0,2,4,0,0,0);
The optimal number of overtime employees are (y1, y2, y3, y4, y5, y6, y7) = (5,0,0,0,0,5,0);
and the minimal salary is $5370.
3. On the given set S, determine whether each function is convex, concave, or neither.
3.1. f(x) = x3; S = [0, ∞ )
3. 2 f(x) = x3; S = R1
3.3 f(x) = (1/x); S = (0, ∞ )
3. 4 f(x) = xa (0 <= a <= 1); S = (0, ∞ )
3.5 f(x) = ln x; S = (0, ∞ )
3.1.
f''(x) = 6x0, (for x0) so f(x) is convex on S.
3.2.
f''(x)>0 for x>0 and f''(x)<0 for x<0, so f(x) is neither convex nor concave.
3. 3.
f''(x) = 2x-3>0 (for x>0). Thus f(x) is convex on S.
3.4.
f''(x) = a(a - 1)xa-20 so f(x) is concave on S.
3.5.
f''(x) = -x-2<0, so f(x) is a concave function on S.
4. On the given set S, determine whether each function is convex, concave, or neither.
4.1. f(x1, x2) = x13 + 3x1x2 + x22; S = R2
4.2. f(x1, x2) = x12 + x22; S = R2
4.3. f(x1, x2) = -x12 – x1x2 – 2x22; S = R2
4.4. f(x1, x2, x3) = -x12 – x22 – 2x32 + .5x1x2; S = R3
4.5 .For what values of a,b, and c will ax12 + bx1x2 + cx22 be a convex function R2?
4. 6. For what values of a,b, and c will ax12 + bx1x2 + cx22 be a concave function on R2?
Solution
4.1.
4.2.
4.3.
4.4.

3
H  6 x1 
 3 2
When x1>3/4, the first and second principal minors are all positive. Thus f(x1, x2) is
convex when x1>3/4.
f(x1, x2) is the sum of convex functions and is therefore a convex function.
 2  1 
H 

  1  4
First principal minors are -2 and -4(both <0) and second principal minor = 7>0. Thus
f(x1, x2) is a concave function.
0
 2 .5

H   .5  2 0 
 0
0  4
1st PM's are -2, -2, -4 which are all <0. 2nd PM's are all >0, as shown below.
 2 .5 
det 
  3.75
 .5  2
 2 0 
det 
 8
 0  4
 2 0 
det 
 8
 0  4
3rd order PM = -4((-2)(-2)-(.5)(.5)) = -15<0. Thus f(x1, x2, x3) is concave on R3
because all nonzero principal minors have the same sign as (-1)k.
4.5. and 4.6.
First, find the Hessian.
 2a b 
H 

 b 2c 
The function will be convex if 2c0, 2a0, and 4ac- b20. These conditions ensure
that all principal minors will have nonnegative determinants.
The function will be concave if 2a0, 2c0 and 4ac - b20. These conditions ensure
that both first principal minors are non-positive and the second principal minor is
nonnegative.