Problem #4 on Page 97
Problem Statement
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Dr. Parisay’s comments are in red.
Variables:
X1 = Number of ounces of Product 1 sold
X2 = Number of ounces of Product 2 sold
X21 = Number of ounces of Product 2 sold after producing Product 1
X31 = Number of ounces of Product 3 sold after producing Product 1
X32 = Number of ounces of Product 3 sold after producing Product 2
X4 = Number of pounds of raw materials
As mentioned in class:
a) It is better to select variable names that assist in later analysis. For example, instead of X4 you can
have RAW.
b) Variable X21 is defined as product 2 sold after processing product 1. It is better to define variable as
X12 as it is more common. The first index usually refers to start point or origin and the second index
usually refers to the end point or final product.
c) This graphical presentation from problem statement is a very good idea. However needs some
modifications. First of all you do not want to use figures/nodes and presentation that implies
“Decision Tree”. This problem is quite different from decision tree. For example, at node “B” it is
not two paths that are “either/or”. The two paths will be taken “both” (unlike decision tree). Next,
please eliminate the path on top. It is not related to the problem. Last, as this graph is a summary of
the problem, you can add information on hours needed (or any other) on each path. Then the figure
will be a complete summary problem and will assist more in problem formulation.
d) You need to provide some explanation on graph. What are the end values? I guess you have
mistake there.
e) You can change the graph to include the demand of each product too.
Visual Aid:
The table below is a visual aid in an effort to facilitate clear communication
Note: Although this visual aid may appear to be a decision tree, it is not. It is simply a diagram that shows the
available network paths.
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Summary of the Problem
X1
X21
X4
X31
X2
X32
Proposed Solution
Objective Function:
OF: Z= Profit =Total Revenue – processing costs – purchase cost
Maximize Z = 10X1 + 20X2 + 20X21 + 30X31 + 30X32 – 26X4 –1X21 – 2X31 – 6X32
Constraints: This is a good practice to mention the units used in each equation. It prevents possible
mistakes.
Maximum amount of Product 1 that can be sold (ounces): X1 ≤ 5000
Maximum amount of Product 2 that can be sold (ounces): X2 + X21 ≤ 5000
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Maximum amount of Product 3 that can be sold (ounces): X31 + X32 ≤ 3000
Amt. of Product 1 available after raw material processing (ounces):3X4 = X1 + X21 + X31
Amount of Product 2 available after raw material processing (ounces): X4 = X2 + X32
Maximum labor hours available (hours): 2X4 + 2X21 + 3X31 + 1X32 ≤ 25000
Sign Constraint:
X1 ≥ 0, X2 ≥ 0, X21 ≥ 0, X31 ≥ 0, X32 ≥ 0, X4 ≥ 0
Practical Conversion:
1lb = 16oz.
Unit = 1 oz. of product (product # specified where applicable)
WinQSB Input: not needed at this level of class
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WinQSB Output:
Sensitivity Analysis Motivation & Justification (Basic Variable):
From the WinQSB output above (Prev. Page), one will notice that the variable “x21 (#oz prod 2 sold
after producing 1)” has a real allowable min and a real allowable max. As a result, sensitivity analysis will be
performed on this left hand side entity.
Parisay: This is not a good reason to select a BV for SA. You usually select BV based on which one has the
highest value in the solution, which means a bit of change in the unit profit will result in a major change in Z
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value. You may also select a BV that you think there is more probability for its unit profit to change or its
current unit profit is close to one of the limits (min or max) and makes it critical for analysis. Therefore, I
would prefer unit profit of X1.
Sensitivity Analysis Input: not needed at this level of class
Sensitivity Analysis Output:
The table above indicates that if the “#oz prod 2 sold after producing 1” increases from 19 to 24, the
maximum profit will increase to $156,500.00, a favorable difference of $8,750. The table above also indicates
that if the “#oz prod 2 sold after producing 1” decreases from 19 to 2 (Parisay: not a practical amount, the
difference is 17 out of 19!!), the maximum profit will decrease to $118,000.00, an unfavorable difference of
$29,750. The other values can be neglected in an effort to see what would happen “#oz prod 2 sold after
producing 1 goes to +/- infinity.
Sensitivity Analysis Motivation & Justification (Non-Basic Variable):
Varying x32 (#oz. of product 3 sold after producing product 2)
(Based on same WinQSB Inputs as above)
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x32 (#oz. of product 3 sold after producing product 2) is a non-basic variable given the existing
conditions because we are currently not producing any product 3 after producing product 2. The reason this is
true is because it is more cost effective to produce product 3 after producing product 1. After producing
product 1, product 3 can be produced for only $2/oz. On the other hand if product 3 is produced after
producing product 2, the cost will be $6/oz.
Parisay: This is not a strong reason, if we could sell more of product 3 then we would have had some value
for X32. The best reason to select this is that this is the only NBV! Increasing the profit from 24 (current
coefficient) to 29 can be possible.
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Sensitivity Analysis Motivation & Justification (RHS Binding Constraint):
Varying Constraint 3 (3 (Max # oz. of product 3 that can be sold). The reason sensitivity analysis will be
performed on this value is because it has the highest shadow price. Shadow price in this case is defined to be
the increase in total revenue by producing 1 extra ounce of product.
Based on the above WinQSB table, the value of Constraint 3 (Max # oz. of product 3 that can be sold) can vary
between 0 and 5,000 oz.). Now, sensitivity analysis will be performed on this value, and will be taken to the
maximum allowable value.
WinQSB Input
WinQSB Output
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Based on the new WinQSB outputs above, one will notice that two output items change:
1. Total profit changes from $147,750 to $191,250 which is a $43,500 gain.
2. Max labor hours available does not change, but the number of labor hours used changes from
19,000 to 25,000 (which is the maximum value allowed based on constraints defined in the
problem statement)
When the noted change is made in the WinQSB input, total profit increases, and worker
“utilization” increases, and the maximum time available will be utilized. This is a key indicator that if
the constraint of the maximum # of oz. of product 3 that can be sold increases to 5,000 from 3,000, the
company will benefit. The team realizes that this constraint may be out of the company’s control, but is
worth mentioning for the sake of communication and what-if scenarios.
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Sensitivity Analysis Motivation & Justification (Basic Variable): (Parisay: not needed, especially after future
discussions on SA)
Increase selling price of X31 from $28/oz. to $30. The motivation for performing this analysis on this
value is because product 3 (produced after product 1) has (Originally) the highest profit per unit. With this in
mind, the team was interested in seeing what would happen if the company decided to raise its selling price
by $2. The findings are that Total profit will increase by $6000.
WinQSB Input:
WinQSB Output:
Here, the team would like to indicate the how changing the value for a basic variable towards positive
infinity will affect the overall solution. Based on the original WinQSB output, one will notice that the allowable
range of values for the selling price for product 3 after producing product 1 is from $23/oz. to $infinity.
The justification for performing this sensitivity analysis is to demonstrate how changing this value
towards $infinity will result in no change of the final solution (with the exception of total profit).
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Report to Manager (I expected better report!!)
Dear Chemco Management,
The Industrial Engineering team has concluded their study involving the production of the three
manufactured perfumes produced at this facility. It is important to note that the analysis performed is to not
impact the form, fit, function or safety of the finished good. Based on the information provided; the team
performed various forms of analysis, and the findings are presented in this report.
Parisay: First mention Z and the original solution!!
Solution Summary
# lb of Raw Materials
3,250 lb
# ounces product 1 sold
5,000 oz.
# ounces product 2 sold
3,250 oz.
# ounces product 1
1,750 oz.
# ounces product 1
# ounces product 2
product 2
product 3
product 3
3,000 oz.
None
Maximum profit : $147,750.00
The first analysis that was performed was that of adjusting the selling price of x21 (Don’t use
variable in report!!) (# of oz. of product 2 sold after producing product 1) this adjustment in selling price is to
lead to an increase of profit to $19 to $24 (rewrite! Explain why 24, that is 26% increase in unit profit. Is that
practical?). The $24 value for profit/unit sold is justified by the WinQSB software used by the team to perform
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their analysis. Based on calculations performed by this software, this is the maximum allowable (Does not
make sense for a manager. Why not more? Explain that the solution will change completely.) profit per unit
sold for product 2 after the production of product 1. After analyzing the data for this variable, the team
presents the following:
(This is not a clear sentence) “#oz prod 2 sold after producing 1” increases from 19 to 24, the
maximum profit will increase to $156,500.00, a favorable difference of $8,750. The table above (where is the
table? Use better reference.) also indicates that if the “#oz prod 2 sold after producing 1” decreases from 19
to 2 (justify such a major decrease from practical point of view), the maximum profit will decrease to
$118,000.00, an unfavorable difference of $29,750. The other values can be neglected in an effort to see what
would happen as the “#oz prod 2 sold after producing 1” goes to +/- infinity.
The rest need modification as well.
A second analysis was performed on varying x32 (#oz. of product 3 sold after producing product 2). It is
important to note that x32 (#oz. of product 3 sold after producing product 2) is a non-basic variable given the
existing conditions because we are currently not producing any product 3 after producing product 2. The
reason this is true is because it is more cost effective to produce product 3 after producing product 1. After
producing product 1, product 3 can be produced for only $2/oz. On the other hand if product 3 is produced
after producing product 2, the cost will be $6/oz which is less favorable.
A third analysis was performed on varying constraint 3 (3 (Max # oz. of product 3 that can be sold). The
value of Constraint 3 (Max # oz. of product 3 that can be sold) can vary between 0 and 5,000 oz.). Now,
sensitivity analysis will be performed on this value, and will be taken to the maximum allowable value. The
reason sensitivity analysis will be performed on this value is because it has the highest shadow price. Shadow
price in this case is defined to be the increase in total revenue by producing 1 extra ounce of product. When
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the noted change is made in the WinQSB input, total profit increases, and worker “utilization” increases, and
the maximum time available will be utilized. This is a key indicator that if the constraint of the maximum # of
oz. of product 3 that can be sold increases to 5,000 from 3,000, the company will benefit. The team realizes
that this constraint may be out of the company’s control, but is worth mentioning for the sake of
communication and what-if scenarios.
Furthermore, a fourth analysis was performed for the sake of thoroughness. Here, the team would like
to indicate the how changing the value for a basic variable towards positive infinity will affect the overall
solution. Based on the original WinQSB output, one will notice that the allowable range of values for the
selling price for product 3 after producing product 1 is from $23/oz. to $infinity.
The justification for performing this sensitivity analysis is to demonstrate how changing this value
towards $infinity will result in no change of the final solution (with the exception of total profit).
The motivation for performing this analysis on this value is because product 3 (produced after product
1) has (Originally) the highest profit per unit. With this in mind, the team was interested in seeing what would
happen if the company decided to raise its selling price by $2. The findings are that Total profit will increase by
$6000.
In closing, the team would like to emphasize their goal of performing analysis without changing the
original path network (Your study is not about changing any path of the process, it is given in the problem
statement based on production conditions.). Additionally, the team would like to communicate to
management that the initial conditions result in a total profit of $147,750.Based on analysis performed using
WinQSB software, the team has determined that if management is willing to consider a different production
plan and/or pricing scheme, the outcome of our analysis would vary.
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Please inform of any further questions or feedback.
Thanks,
The Industrial Engineering Team
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