Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
IE 416: Operations Research I
Fall 2010
Source of the problem statement: W. L. Winston, " Operations Research, Application and Algorithms"
Solution by student groups (formulation was given)
Dr. Parisay’s comments are in red.
Post Office Problem – Problem 1, Pg. 75
Problem Statement
In the post office example, suppose that each full-time employee works 8 hours per day. Thus, Monday’s
requirement of 17 workers may be viewed as a requirement of 8(17) = 136 hours. The post office may meet its
daily labor requirements by using both full-time and part-time employees. During each week, a full-time
employee works 8 hours a day for five consecutive days, and part-time employee works 4 hours a day for five
consecutive days. A full-time employee costs the post office $15 per hour, whereas a part-time employee (with
reduced fringe benefits) costs the post office only $10 per hour. Union requirements limit part-time labor to
25% of weekly labor requirements. Formulate an LP to minimize the post office’s weekly labor costs.
Table 4 – Requirements for Post Office
Number of Full-time
Day
Employees Required
1 = Monday
17
2 = Tuesday
13
3 = Wednesday
15
4 = Thursday
19
5 = Friday
14
6 = Saturday
16
7 = Sunday
11
Summary Table (These tables are good ones for summary of the problem.)
The following information in the tables below was derived from the problem statement:
Summary Table 1 – Hours of Required Labor
Work Day
Required Labor Hours*
1 = Monday
136
2 = Tuesday
104
3 = Wednesday
120
4 = Thursday
152
5 = Friday
112
6 = Saturday
128
7 = Sunday
88
* To find the required labor hours, multiple the number of
full-time workers required for each day by the amount of
hours in a work day (8).
1
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
Type of
Employee
Table 2 –Employees Details
Hours per Day
Cost/Hour
Limitations
-
Full-Time
8
$15
-
Part-Time
4
$10
-
Must work 5
consecutive days
Must have 2
consecutive days off
Must work 5
consecutive days
Must have 2
consecutive days off
Only 25% of Total
Require Labor Hours can
be supplied by PartTime Employees
Decision Variables
These are the decision variables for this problem:
Decision Variable
XF1
XF2
XF3
XF4
XF5
XF6
XF7
XP1
XP2
XP3
XP4
XP5
XP6
XP7
Table 3 – Decision Variables
Description
Number of full-time employees starting on Monday
Number of full-time employees starting on Tuesday
Number of full-time employees starting on Wednesday
Number of full-time employees starting on Thursday
Number of full-time employees starting on Friday
Number of full-time employees starting on Saturday
Number of full-time employees starting on Sunday
Number of part-time employees starting on Monday
Number of part-time employees starting on Tuesday
Number of part-time employees starting on Wednesday
Number of part-time employees starting on Thursday
Number of part-time employees starting on Friday
Number of part-time employees starting on Saturday
Number of part-time employees starting on Sunday
The results from each of these decision variables specifically show the number of workers beginning their 5-day
shift on that particular day. For instance, the result for XF1 represents the number of full-time employees starting
2
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
their 5-day long shift on Monday. Furthermore, the result for XP1 represents the number of part-time employees
starting their 5-day long shift on Monday.
Objective Function (This is a good explanation of OF. Notice how objective function has changed from the
original example and included cost.)
The goal of this problem is to minimize the labor cost per week from hiring both full-time and part-time
employees to fulfill the Post Office’s daily workforce demand. Cost, in this problem, has been equated as:
Eqn. 1
Therefore, the following objective function has been made:
Eqn. 2
Eqn. 3
The result of this objective function will be the minimum total cost to the Post office needed to fulfill the daily
workforce demand.
Constraints (This is a good explanation of constraints and good summary table. Notice how the formulation of
constraints has changed from the original example.)
One constraint in this problem is the specific number of employees needed per day, as shown:
3
ay
rs
da
y
ed
ne
sd
a
XF1
XP1
XF1
XP1
XP2
XF2
XP2
XF2
XP2
XF2
XP2
XF3
XP3
XF3
XP3
XF3
XP3
XF3
XP3
XF3
XP3
XF4
XP4
XF4
XP4
XF4
XP4
XF4
XP4
XF5
XP5
XF5
XP5
XF5
XP5
XF6
XP6
XF6
XP6
Start Wed
Start Thu
XF4
XP4
Start Fri
XF5
XP5
XF5
XP5
Start Sat
XF6
XP6
XF6
XP6
XF6
Part-Time
Full-Time S
un
da
y
Part-Time
Full-Time F
rid
a
XP6
y
XP1
XF2
Part-Time
XF1
XP2
Full-Time S
at
ur
d
XP1
XF2
Part-Time
XF1
Full-Time T
hu
Part-Time
XP1
Full-Time W
XF1
Start Tue
Full-Time T
ue
sd
ay
Part-Time
General Employee Schedule
Start Mon
Part-Time
Type of
Employee
Full-Time M
on
da
y
y
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
Start Sun
XF7
XP7
XF7
XP7
XF7
XP7
XF7
XP7
XF7
XP7
8(XF1 + XF4 + 8(XF1 + XF2 + 8(XF1 + XF2 + 8(XF1 + XF2 + 8(XF1 + XF2 + 8(XF2 + XF3 + 8(XF3 + XF4 +
XF5 + XF6 +
XF5 + XF6 +
XF3 + XF6 +
XF3 + XF4 +
XF3 + XF4 +
XF4 + XF5 +
XF5 + XF6 +
Sum of
XF7) + 4(XP1 + XF7) + 4(XP1 + XF7) + 4(XP1 + XF7) + 4(XP1 + XF5) + 4(XP1 + XF6) +4(XP2 + XF7) + 4(XP3 +
Employees XP4 + XP5 +
XP2 + XP5 +
XP2 + XP3 +
XP2 + XP3 +
XP2 + XP3 +
XP3 +XP4 +
XP4 + XP5 +
XP6 + XP7)
XP6 + XP7)
XP6 + XP7)
XP4 + XP7)
XP4 + XP5)
XP5 + XP6)
XP6 + XP7)
Min Labor
Hours Per
Day
136
104
120
152
112
128
88
Figure 1 - General Employee Schedule
One can interpret the chart as so: “For Monday, XF1, XF4, XF5, XF6, XF7, XP1, XP4, XP5, XP6, and XP7 employees
will be working. And, the required number of labor hours for Monday is 136 people. Therefore, the sum of XF1,
XF4, XF5, XF6, XF7, XP1, XP4, XP5, XP6, and XP7 times their respective daily working length (either 8 hours or 4
hours) must be at least 136 hours.” It is necessary to multiply the decision variables by their working length,
because the constraint is measured in hours per day. It should be noted that XF1 ≠ XF1, because XF1 means the
number of full-time employees beginning their 5-day long shift on Monday, and XF1 refers to the group of fulltime employees beginning their 5-day long shift on Monday. In other words, XF1 is the number of people in XF1.
By using the information in Figure 1, the S.T. Equations have been created, as shown below:
Day
Table 5 – S.T. Equations
Demand of Labor Hours
Monday
136
Tuesday
104
Wednesday
120
Thursday
152
Friday
112
Saturday
128
Sunday
88
S.T. Equations
4
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
Another constraint that should be taken into consideration is sign restriction, as shown below:
Eqn. 4
This constraint is necessary, because the essence of the problem can never allow a negative number of people.
With this restriction, it prohibits the optimum solution from having any negative values for the decision
variables.
One last constraint that should be taken into consideration is the part-time labor restriction, as shown below:
This is a tricky detail that should be paid attention to. It happens in many problems. It is true that the required
total hours are 840 hours, but considering the direction of inequalities, it is the minimum total value. That is in
the final solution you may have more than 840 hours. Then, 25% of it will be more than 210 hours. That is why
rather than formulating this constraint as below we should formulate it in a general form. You had similar case
in one of the examples in your book (page 89) and I highlighted it in class.
(Total hours during one week by part time labors)/(total hours during one week by ALL labors) <= 0.25
Eqn. 5
Eqn. 6
Eqn. 7
This constraint is necessary, because Union restrictions state that no more than 25% of weekly labor can be
supplied by part-time employees.
WinQSB Formulation
To ease the WinQSB Formulation, the following Summary of the problem has been made:
Table 6 – Summary of Post Office Problem for WinQSB
Objective
Function
Subject-To
Equations
Labor Hours Per Day
Sign Restriction
Part-Time Labor
Restriction
5
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
There are a total of 14 decision variables and 22 constraints. Of the 22 constraints, only 8 constraints will be
used, because WinQSB has a function in which solutions will be positive. As a result, we will not need to input
the sign restriction constraints. With all the information in Table 6, it is now possible to apply WinQSB. Upon
loading a new problem, the following information should be entered into the Problem Specification window, as
shown below:
Figure 2 - WinQSB LP Problem Specification
Once the information has been entered and the “OK” button has been pressed, a new screen will load. Enter the
information in Table 6 into WinQSB. It should look similar to this: (Input and output will change if change the
last equation.)
6
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
Figure 3 – WinQSB LP Problem Input
Here is an explanation of the terms found in Figure 3:
Table 7 – Notation Explanation (Refer to Table 3, 4 and Equation 7)
St-FWrk-Mon Number of Full-Time Employees Beginning Their 5-day Shift on Monday = XF1
St-FWrk-Tue Number of Full-Time Employees Beginning Their 5-day Shift on Tuesday = XF2
St-FWrk-Wed Number of Full-Time Employees Beginning Their 5-day Shift on Wednesday = XF3
St-FWrk-Thu Number of Full-Time Employees Beginning Their 5-day Shift on Thursday = XF4
St-FWrk-Fri
Number of Full-Time Employees Beginning Their 5-day Shift on Friday = XF5
St-FWrk-Sat
Number of Full-Time Employees Beginning Their 5-day Shift on Saturday = XF6
St-FWrk-Sun Number of Full-Time Employees Beginning Their 5-day Shift on Sunday = XF7
St-PWrk-Mon
St-PWrk-Tue
St-PWrk-Wed
St-PWrk-Thu
St-PWrk-Fri
St-PWrk-Sat
St-PWrk-Sun
Number of Part-Time Employees Beginning Their 5-day Shift on Monday = XP1
Number of Part-Time Employees Beginning Their 5-day Shift on Tuesday = XP2
Number of Part-Time Employees Beginning Their 5-day Shift on Wednesday = XP3
Number of Part-Time Employees Beginning Their 5-day Shift on Thursday = XP4
Number of Part-Time Employees Beginning Their 5-day Shift on Friday = XP5
Number of Part-Time Employees Beginning Their 5-day Shift on Saturday = XP6
Number of Part-Time Employees Beginning Their 5-day Shift on Sunday = XP7
Dmd-Mon
Dmd-Tue
Dmd-Wed
Dmd-Thu
Dmd-Fri
Dmd-Sat
Dmd-Sun
Demand of Labor Hours for Monday = Monday Constraint
Demand of Labor Hours for Tuesday = Tuesday Constraint
Demand of Labor Hours for Wednesday = Wednesday Constraint
Demand of Labor Hours for Thursday = Thursday Constraint
Demand of Labor Hours for Friday = Friday Constraint
Demand of Labor Hours for Saturday = Saturday Constraint
Demand of Labor Hours for Sunday = Sunday Constraint
Lbr Rst
Labor Restriction for Part-Time Employees = Labor Restriction
After entering the information into the WinQSB, one must select “Solve the Problem,” which is under the “Solve
and Analyze” tab. It should result with the following outputs:
7
Solution 1
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
Solution 2
Figure 4 – WinQSB LP Problem Output 1
Solution 3
Figure 6 – WinQSB LP Problem Output 3
Figure 5 – WinQSB LP Problem Output 2
Solution 4
Figure 7 – WinQSB LP Problem Output 4
8
Solution 5
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
Solution 6
Figure 8 – WinQSB LP Problem Output 5
Solution 7
Figure 10 – WinQSB LP Problem Output 7
Figure 9 – WinQSB LP Problem Output 6
Solution 8
Figure 11 – WinQSB LP Problem Output 8
9
Solution 9
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
Solution 10
Figure 12 – WinQSB LP Problem Output 9
Solution 11
Figure 14 – WinQSB LP Problem Output 11
Figure 13 – WinQSB LP Problem Output 10
Solution 12
Figure 15 – WinQSB LP Problem Output 12
10
Solution 13
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
Solution 14
Figure 16 – WinQSB LP Problem Output 13
Solution 15
Figure 18 – WinQSB LP Problem Output 15
Figure 17 – WinQSB LP Problem Output 14
Solution 16
Figure 19 – WinQSB LP Problem Output 16
11
Solution 17
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
Solution 18
Figure 20 – WinQSB LP Problem Output 17
Solution 19
Figure 22 – WinQSB LP Problem Output 19
Figure 21 – WinQSB LP Problem Output 18
Solution 20
Figure 23 – WinQSB LP Problem Output 20
12
Solution 21
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
Solution 22
Figure 24 – WinQSB LP Problem Output 21
Figure 25 – WinQSB LP Problem Output 22
The optimum value for this problem, as seen from all the outputs, states that the minimum cost that the Post
Office will incur is $12,350 in order to fulfill the daily workforce requirement. However, there are many different
outputs because there are many different solutions that will yield the minimal cost of $12,350. The main
difference between each of the outputs is the scheduling of the workers, as shown below: (Good job in your
summary result of multi optimal solutions. You can use it in report to the manager. ALTERNATE SOLUTIONS
will provide FLEXIBILITY FOR DECISION MAKING.)
Table 8 – Optimal Solution Summary 1
# of Full-Time Workers Starting Monday
# of Full-Time Workers Starting Tuesday
# of Full-Time Workers Starting Wednesday
# of Full-Time Workers Starting Thursday
# of Full-Time Workers Starting Friday
# of Full-Time Workers Starting Saturday
# of Full-Time Workers Starting Sunday
# of Part-Time Workers Starting Monday
# of Part-Time Workers Starting Tuesday
# of Part-Time Workers Starting Wednesday
# of Part-Time Workers Starting Thursday
# of Part-Time Workers Starting Friday
# of Part-Time Workers Starting Saturday
# of Part-Time Workers Starting Sunday
Solution 1
6.33
5
0.33
2.0833
0
3.33
0
0
0
0
10.5
0
0
0
Solution 2
6
5.33
0
2.0833
0
3.33
0.33
0
0
0
10.5
0
0
0
Solution 3
0.75
5.33
0
7.333
0
3.33
0.33
10.5
0
0
0
0
0
0
Solution 4
6
0.83
0
7.33
0
3.33
0.33
0
10.5
0
0
0
0
0
Solution 5
6.33
0.083
0
7.33
0
3.33
0
0
9.83
0.67
0
0
0
0
Solution 6
6.33
5
0
2.4167
0
3.33
0
0
0
0.67
9.83
0
0
0
Solution 7
6.33
5
0
5.75
0
0
0
0
0
0.67
3.167
0
6.67
0
13
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
Table 9 – Optimal Solution Summary 2
# of Full-Time Workers Starting Monday
# of Full-Time Workers Starting Tuesday
# of Full-Time Workers Starting Wednesday
# of Full-Time Workers Starting Thursday
# of Full-Time Workers Starting Friday
# of Full-Time Workers Starting Saturday
# of Full-Time Workers Starting Sunday
# of Part-Time Workers Starting Monday
# of Part-Time Workers Starting Tuesday
# of Part-Time Workers Starting Wednesday
# of Part-Time Workers Starting Thursday
# of Part-Time Workers Starting Friday
# of Part-Time Workers Starting Saturday
# of Part-Time Workers Starting Sunday
Solution 8
6
5.33
0
5.75
0
0
0
0
0
0
3.167
0
6.67
0.67
Solution 9
4.4167
5.33
0
7.33
0
0
0
0
0
0
0
0
6.67
3.83
Solution 10
4.4167
3.33
2
7.33
0
0
0
0
0
0
0
0
6.67
3.83
Solution 11
1.33
3.33
2
7.33
0
3.0833
0
0
0
0
0
0
0.5
10
Solution 12
1.33
3.33
2
7.33
0
0
3.0833
0
0
0
0
0
6.67
3.83
Solution 13
0
3.33
2
7.33
0
0
4.4167
2.67
0
0
0
0
6.67
1.167
Solution 14
0
2.75
2
7.33
0
0
5
2.67
1.167
0
0
0
6.67
0
Table 10 – Optimal Solution Summary 3
# of Full-Time Workers Starting Monday
# of Full-Time Workers Starting Tuesday
# of Full-Time Workers Starting Wednesday
# of Full-Time Workers Starting Thursday
# of Full-Time Workers Starting Friday
# of Full-Time Workers Starting Saturday
# of Full-Time Workers Starting Sunday
# of Part-Time Workers Starting Monday
# of Part-Time Workers Starting Tuesday
# of Part-Time Workers Starting Wednesday
# of Part-Time Workers Starting Thursday
# of Part-Time Workers Starting Friday
# of Part-Time Workers Starting Saturday
# of Part-Time Workers Starting Sunday
Solution 15
0
3.33
1.4167
7.33
0
0
5
2.67
0
1.167
0
0
6.67
0
Solution 16
0
3.33
2
6.75
0
0
5
2.67
0
0
1.167
0
6.67
0
Solution 17
0
5.33
0
7.33
0
0
4.4167
2.67
0
0
0
0
6.67
1.167
Solution 18
0
5.33
0
7.33
0
0
4.4167
3.83
0
0
0
0
6.67
0
Solution 19
4.0833
5.33
0
7.33
0
0
0.33
3.833
0
0
0
0
6.67
0
Solution 20
4.167
5
0.33
7.33
0
0
0
3.83
0
0
0
0
6.67
0
Solution 21
1.083
5
0.33
7.33
0
3.33
0
10.5
0
0
0
0
0
0
Solution 22
0.75
5.33
0
7.33
3.33
0.33
10.5
0
0
0
0
0
0
0
Since there are differences with scheduling, the reduced cost and shadow prices will be different for each.
Sensitivity Analysis – Change Objective Function Coefficient for Monday
You should specify which one of optimal solutions have been used for the following.
The objective function coefficient that was chosen to be analyzed is the one that affects the number of full-time
employees starting on Monday, XF1. Currently, the coefficient is 600. The reason for choosing this objective
function coefficient is because Monday is the start of the week, and there are many instances in which people
request Monday off due to prolonging their weekends. So it would be of interest to see how the optimum value
changes as the objective function coefficient changes. This is a wrong motivation. Notice that this problem has
different concept for its OF coefficients. Here 600 includes: cost per hour, number of hours per day, and 5 days
14
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
per week. A good motivation would be cost per hour. You can argue that cost per house can change and so
the value of 600. However, be careful to assume that the cost per hour changes only for Monday.
The problem specifications will remain the same in WinQSB, as will the input. However, it is necessary to
perform a Parametric Analysis, as shown below:
Figure 21 - WinQSB LP Problem Parametric Analysis 1
The following screen will appear, after performing the Parametric Analysis:
Figure 27 - WinQSB LP Problem Parametric Analysis Output 1
15
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
The following is the result of selecting Graphic Parametric Analysis, as shown below:
Figure 28 - WinQSB LP Problem Graphical Parametric Analysis Output 1
This is wrong. >> This analysis shows how a change in the minimum number of required hours will affect the
solution to the objective function. Monday was again selected and shown in the above calculations and
graphical representation.
Sensitivity Analysis – Change Objective Function Coefficient for Thursday
You should specify which one of optimal solutions have been used for the following.
This should completely changed based on the comments on previous sensitivity analysis.
The objective function coefficient that I’ve chosen to analyze is the one that affects the number of employees
starting on Thursday, X4. Currently, the coefficient is 600. My reason for choosing this objective function
coefficient is because Thursday has the largest demand in terms of workforce size, so it would be of interest to
see how the optimum value changes as the objective function changes.
16
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
The problem specifications will remain the same in WinQSB, as will the input. However, it is necessary to
perform a Parametric Analysis, as shown below:
Figure 29 - WinQSB LP Problem Parametric Analysis 2
The following screen will appear, after performing the Parametric Analysis:
Figure 30 - WinQSB LP Problem Parametric Analysis Output 2
This shows us that the optimal solution (z) increases as the objective function coefficient for Thursday increases.
In fact when the coefficient changes from 600 to 900 or greater, then the optimal solution changes from
$12,350 to $12,975. Similarly, if the coefficient changes from 600 to 0, then the new optimal solution is $7,950.
17
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
The following is the result of selecting Graphic Parametric Analysis, as shown on the next page:
Figure 31 - WinQSB LP Problem Graphic Parametric Analysis 2
This demonstrates that the objective function coefficient for Thursday increases, which will cause the optimal
solution (z) to increase as well. It should be noted that the coefficient for this objective function is comprised of
3 things: labor rate, hours per day, and days per week. If the coefficient increases above 600, then the cost to
the company will change less dramatically than if the coefficient decreases below 600.
Sensitivity Analysis – Change Right Hand Side for Monday
You should specify which one of optimal solutions have been used for the following.
The right hand side, also known as R.H.S., that was chosen to be analyzed is the one that affects the number of
employees needed for Monday, the Monday Constraint. Currently, the R.H.S. value is 17 people. The reason for
choosing this objective function coefficient is because Monday is the start of the week, and there are many
instances in which people request Monday off due to prolonging their weekends. (This is not a good motivation.
The total number required on any day should not depend on how employees want to work, but on the workload
and job requirements for that day.) Therefore, the demand for Monday may change. So it would be of interest
to see how the optimum value changes as the objective function coefficient changes.
18
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
The problem specifications will remain the same in WinQSB, as will the input. However, it is necessary to
perform a Parametric Analysis, as shown on the next page:
Figure 32 - WinQSB LP Problem Parametric Analysis 3
The following screen will appear, after performing the Parametric Analysis, as shown below:
Figure 33 - WinQSB LP Problem Parametric Analysis Output 3
19
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
The following is the result of selecting Graphic Parametric Analysis, as shown below:
Figure 34 - WinQSB LP Problem Parametric Analysis Output 3
This graph only represents a bug in the WinQSB software! You needed to repeat the steps and get the correct
graph. I mentioned it in one of my handouts!
This analysis shows how a change in the minimum number of required hours will affect the solution to the
objective function. Monday was again selected and shown in the above calculations and graphical
representation.
Sensitivity Analysis – Change Right Hand Side on Thursday
You should specify which one of optimal solutions have been used for the following.
(Please refer to my comments for previous SA. This is not right.)
The right hand side, also known as R.H.S. that I’ve chosen to analyze is the one that affects the number of labor
hours needed for Thursday, the Thursday Constraint. Currently, the value is 152 hours. My reason for choosing
this objective function coefficient is because Thursday has the largest demand in terms of workforce size, and it
would be interesting to see how the optimum solution changes as the R.H.S. changes.
20
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
The problem specifications will remain the same in WinQSB, as will the input. However, it is necessary to
perform a Parametric Analysis, as shown below:
Figure 35 - WinQSB Parametric Analysis Input 4
The following screen will appear, after performing the Parametric Analysis:
Figure 36 - WinQSB LP Problem Parametric Analysis Output 4
The following is the result of selecting Graphic Parametric Analysis, as shown on the next page:
21
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
Figure 37 - WinQSB LP Problem Parametric Analysis Output 4
This graph only represents a bug in the WinQSB software! You needed to repeat the steps and get the correct
graph. I mentioned it in one of my handouts!
This graph shows that if the demand for labor hour decreases from 152 hours, then the optimum value
decreases substantially, which is a positive thing. However, if the labor hour requirement increases past 152
hours, then the optimum value trend increases.
Report to Manager
You should specify which one of optimal solutions have been used for the following.
With the current demands of the Post Office and the union restrictions, the Post Office will have
to spend $12,350 to fully optimize the demand for one week. And the number of worker required is
listed In the table below.
Shift-Start
Day
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
Sunday
Table - Optimum Schedule
Number of FullNumber of PartTime Employees
Time Employees
0
6.333
0
5
0
.3333
10.5
7
0
0
0
2.083
0
0
The problem with the data above is that there is no way this company says “we need six and a
third of a person to work on Monday.” In order to find the best solution that makes sense: we have to
eliminate fraction in our calculation. When we eliminate fraction by using integer values; the Post Office
will incur a cost of $12,600 (Where is the source of information? You need to show the details of
22
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
calculation or any other source before the report.) in order to satisfy the daily requirements in terms of
daily labor hours with the number of worker required is listed in the table below (Where is the source
of information?)
Table - Optimum Schedule
Shift-Start
Number of FullNumber of PartDay
Time Employees
Time Employees
1
Monday
5
0
Tuesday
6
1
Wednesday
0
0
Thursday
7
1
Friday
0
6
Saturday
0
0
Sunday
0
This is a good discussion for now, though we will learn about ILP later on. You need to talk
about alternate solutions and how it provides flexibility in decision making.
It should be noted that with a workforce size of 27 people, there will be some days out of the
week where the workforce size will exceed the daily demand for employees. In particular, there will be
16 additional labor hours for Tuesday (Where is the source of information?), and there will be 44
additional labor hours for Friday; the other days of the week will have exactly the necessary labor hours
per day. Also, because there are only 9 part time employees, the Post Office will only utilize 180 hours
out of the maximum 210 hours that can be supplied by part time employees. And lastly, if the labor hour
demands for Saturday increases from 128 hours to 129 hours; then the new cost to the Post Office will
be $12,650 (Where is the source of information?), which is an additional $50. However; we do not
know when and which day the additional worker needs to start working, and if they are full time or part
time.
In this report we also went over additional information for the company if they are not satisfied
with the data. In this letter we only provided data if the company wants to change the number of worker
needed or the number of worker who start working on Monday and Thursday. We focus our attention on
23
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
Monday since most full-time employees prefer to begin there week on Mondays to the fact that our
employees are more obligated to be involved in a religion activity or have the desire to spend time with
their family on the traditional weekend. That’s why we want to change either the amount of work
required on Monday or the amount or worker start working on Monday. We also focus our attention on
Thursday since Thursday has the largest workload, and we have some information if the company wants
to reduce the workload on Thursday.
For Monday:
1. We can’t change the number of full time and part time worker that start working on Monday
without using the Win QSB program. (Where is the source of information? It seems to be a
wrong statement.)
2. We also cannot change the number of hours required without using WinQSB program
We can’t change both (When performing LP Sensitivity Analysis, you consider only one change and its
effect on solution. What do you mean by “both”?) full time and part time worker that start on Monday
since they are complementary worker to fulfill the requirements (Where is the source of information?
Not clear or wrong.)
For Thursday: Please refer to the previous comments.
1. We can’t change the number of full time and part time worker that start working on Thursday
without using the Win QSB program.
2. We also cannot change the number of hours required without using WinQSB program
We can’t change both full time and part time workers that start on Thursday. Full time workers that start
on Thursday are complementary workers to fulfill the requirements; while the Part time workers are
critical to minimize the cost of labor, but there is no wiggle space to add or subtract worker.
24
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
Since we haven’t discuss integer values; we also round up the decimals numbers from solution 1
and we have this answers:
Shift-Start
Day
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
Sunday
Table - Optimum Schedule
Number of FullNumber of PartTime Employees
Time Employees
0
7
0
5
0
1
11
7 should be 3
0
0
0
3 should be 4
0
0
Based on this data; we have 20 fulltime workers and 11 part time workers. This is not the
maximum ?? cost is more than $? 12,600. With this combination; the company will have to spend
$?160,000 each week. On the other hand, this number still fulfills the union labor requirement.
Additional information to the manager for Monday:
1. We can’t change the cost of full time and part time worker that start working on Monday without
using the Win QSB program.
2. If we change the number of hours required between 116 to 140 hours; the amount of full time
and part time worker needed will change; but the days that part time and fulltime workers start
working will stay the same. Full time worker will start working on Monday, Tuesday,
Wednesday, Thursday, and Saturday. Part time worker will need to start working on Thursday.
The minimum cost will change between 159500 (160000 - 25* (136-116) and 160100 (160000 +
25* (140-136))
Additional information to the manager for Monday (you mean Thursday!):
25
Benjamin Tran, Caselyn Iglesias, Chad Creason, Kristianto Andresen, Tiran Alando
IE416
10/19/10
1. We can’t change the cost of part time worker for Thursday (why not?) but we can change cost of
Fulltime worker. If the cost of full time worker change between $15 (related to 600) and $22.5
(related to 900); the amount of full time and part time worker needed will NOT change; but the
days that part time and fulltime workers start working will stay the same. Full time worker will
start working o Monday, Tuesday, Wednesday, Thursday, and Saturday. Part time worker will
need to start working on Thursday. The minimum cost will change between 160000 and
162100(160000+ 7(900-600))
2. We can’t change the hours required for Thursday without using the WinQSB programs. You can
change hours and answer some questions (i.e. will have same starting days, but with different
values.) but for other changes of solution (i.e. specific numbers starting each day) need to use
WinQSB.
26