Leeds School of Business | University of Colorado Boulder
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.
Table 3
|Supplier |Cost Per Valve ($) |Percent Large |Percent Medium |Percent Small |
|1 |5 |40 |40 |20 |
|2 |4 |30 |35 |35 |
|3 |3 |20 |20 |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 z = 5x1 + 4x2 + 3x3
s.t. .4x1 + .3x2 + .2x3 ( 500 (Receive enough large valves)
.4x1 + .35x2 + .2x3 ( 300 (Receive enough medium valves)
.2x1 + .35x2 + .60x3 ( 300 (Receive enough small valves)
x1 ( 700, x2 ( 700, x3 ( 700
x1 ( 0, x2 ( 0 x3 ( 0
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)
s.t. 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 ................
................
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