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Sensitivity analysis
• It refers to the study of coefficient coefficient changeschanges of decision variables that would not affect to the optimal solution we obtained
• What does it mean?
• Tutorial
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• Consider the following LP problem:
Solution: z= 2,240, x1=4, x2=8, and s3=48
Question: what range of Cj values can be changed and yet the optimal solution remains unchanged?
Two groups of coefficients could be studied …….(to p3)
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Sensitivity Analysis
Two groups of coefficients could be changed:
1. The objective function cost cj
2. The constraint quantities bi
• Semantic view • Procedural steps for sensitivity analysis
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Semantic view
Cj
bi
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Procedural Steps
1. Change of objective function cost cj
2. Change of constraint quantity bi
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Change of objective function cost cj
Consider the following optimal tableau:
Here, we are to study what value can Cj be changed and yet the above optimal solution remains unchanged.
Let consider C1, and let ▲ be the change of C1, thenC1 = 160 + ▲, replacing this to the above tableau, we have …
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Cj analysis
Again, we follow the normal procedure and compute the solution until all values of cj-zj ≤ 0, then we have
Solve these two equations …. (to p8)
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c1
We can repeat these steps for c2 too (to p9)
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c2
Solve these two inequalities, we have …
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c2
Important note! (to p11)
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Important note! For c1 and c2
Thus, we can only consider them ONE at a time only!
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Change of constraint quantity bi
Consider the following:
Here, we can analyze b1, b2, and b3,and its corresponding slackvariables are s1, s2, and s2 (why?)
Then, we analyze the changes of b1, b2, and b3 independently since each has a different slack variablesHow? ….
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Analysis of bi
• Three bi can be analyzed:
– b1– b2– b3
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b1Here, we multiple s1 to the total quantity, and solve ▲
Now, all bi ≥ 0, then
Continue..
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b1
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b2
Continue …
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b3
In a similar vein, we get ..
Please try q3 at home!
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Tutorial
• See the following slides for these full questions:
– 48 b-f, 49a-e, 50 b-d– Also run and compare their computer outputs!
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48
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49
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50
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