Toy Problem, Version 2, Sensitivity Analysis
Source of the problem: page 336, Jotin Khisty and Jamshid Mohammadi, "Fundamentals
of Systems Engineering with Economics, Probability, and Statistics", 1/e, Prentice Hall,
2001, ISBN: 0-13-010649-6
By: Parisay
Problem statement changed from the original:
Consider a small toy company making two kinds of toys X1 and X2 on an hourly basis.
Toy X1 produces a profit of $5 while X2 produces $3 in profit. The total production
should be more than 9 units (toys) per hour. There are 36 hours of labor available that
should be used completely. At most 18 parts can be used. The resources needed to
produce each toy are as below. The company wants to know the number of toys of type
X1 and X2 that should be produced per hour to maximize profit.
Table 1: Resources per toy
Resources
Labor (hour)
Parts
Table 2: Summary of Input Data
Toy 1 Toy 2
Profit $/toy
5
3
Labor hour/toy 6
2
Parts/toy
2
1
Toy 1
6
2
Toy 2
2
1
Use all available 36 hours
At most 18 parts available
Total production more than 9
Xi = number of Toy i per hour
Objective Function: max Z = 5 X1 + 3 X2
Subject To:
Labor: 6 X1 + 2 X2 = 36
Part: 2 X1 + 1 X2 =< 18
Total: X1 + X2 => 9
X1 and X2 => 0
Table 3: Input Data (WinQSB) Original Problem:
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Table 4: Solution (WinQSB):
This solution shows that to maximize the profit, considering our constraints, we should
produce no Toy 1 (X1 is NBV) and 18 of Toy 2 (X2 is BV). Then the maximum profit
will be $54 (OF). Table 4 also shows that we have used all of our labor and parts
resources (slack = 0, binding constraint). We will produce a total of 18 toys that is more
than the required minimum total number (is more by 9 units, surplus or excess variable is
9, non-binding constraint).
If we would like to be able to produce Toy 1 (now X1 is a NBV and we want it to
become BV) we need to use Reduced Cost information (RC = -4) from Table 4. RC, in
this case, means to increase Toy 1’s unit profit by at least $4 per toy. The Table 5 uses a
profit of $9.5 per unit for X1 and the solution shows that we can produce 4.5 units of Toy
1.
We can obtain more useful information from “Allowable min” and “Allowable max”
columns of Table 4. For example the unit profit for each Toy 2 (BV) is currently $3. If
this profit changes anywhere from $1.667 to infinity (M in the table) we still will be
producing 18 units of Toy 2 and no Toy 1. Of course the total profit will change
correspondingly. We can draw a sensitivity line segment for this range of unit profit
value. To draw this line we can use one point of ($3 profit per unit on X axis, $54 total
OF on Y axis) and the slope of X2=18.
Shadow Price (SP) for labor constraint is 1.5. Currently a total of 36 hours is available.
If this total hour changes in a range of 18 to 36 hours, still Toy 2 will be produced (the
number may change) and no Toy 1 will be produced. The total profit will change
correspondingly. We can draw a sensitivity line segment for this range of RHS. To draw
this line we can use one point of (36 available hours on X axis, $54 total OF on Y axis)
and the slope of SP=1.5. Table 6 provides an example with total hours being 20. Notice
that 10 Toy 2 will be produced.
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Moreover, SP of 1.5 indicates that if we want to add to the total hours by using some
other sources such as overtime, for each extra hour used the total profit will increase by
$1.5. However, usually we need to pay extra for overtime. So SP indicates how much
extra we are willing to pay for the extra time used. That is if we need to pay extra $1,
over the regular rate, for each overtime hour, it is fine as there will remain 1.5-1 = 0.5
$/hour extra profit for us. Of course, in the case of this problem as we have used the
maximum allowable RHS (36) we cannot add to it anymore.
Table 5: Solution (WinQSB) for changed profit of NBV:
Table 6: Solution (WinQSB) for changed RHS:
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Sensitivity Analysis for Toy version 2 Problem
Change of optimal OF value for changes in profit of Toy 2.
Change of optimal OF value for changes in available hours of labor.
Prepared: 4-12-05
Updated: 1-1-07
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