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Simplex Method and Sensitivity Analysis-Operation Research-Lecture Handout, Exercises of Operational Research

Chhattisgarh Swami Vivekanand Technical UniversityOperational Research

Gagan Rudrani provided this handout for Operation Research subject at Chhattisgarh Swami Vivekanand Technical University. It includes: Simplex, Method, Sensitivity, Optimality, Range, Analysis, LP, , Model, Parameters, TOYCO

Typology: Exercises

2011/2012

Uploaded on 07/13/2012

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Course: Operations Research
Spring
2012
Topic 3: Simplex Method and Sensitivity Analysis
Sensitivity and post-optimality analysis
In LP, the parameters of the model can change within certain limits without causing the
Optimal solution to change. This is referred to as Sensitivity analysis.
In contrast, Post-optimality analysis deals with determining the new Optimal solution
resulting from making targeting changes in the input data.
Sensitivity analysis
In LP model, parameters are usually not exact. With sensitivity analysis, one can
ascertain the impact of this uncertainty on the quality of the optimum solution. For example, if
sensitivity analysis reveals that the optimum remains the same for a ±10% change in unit ‘Ci’, on
cane conclude that the solution is more robust than in case where indifference range is only ±10
percent.
There are two major types of sensitivity analysis:
(i) Optimality Range Analysis or Objective-Function Range Analysis
(ii) Feasibility analysis or Right Hand Side (RHS) Range Analysis
Taking an example:
TOYCO assembles three types of toys - trains, trucks and cars - using three operations.
The limits on the available times for the three operations are 430, 460 and 420 minutes,
respectively, and the revenue per unit of train, truck and car are $3, $2 and $5, respectively. The
assembly times for train per three operations are 1, 3 and 1 minutes, respectively. The
corresponding times per truck and car are (2, 0, 4) and (1, 2, 0) minutes (where a zero time
indicates that the operation is not used).
Letting X1, X2 and X3 represent the daily number of units assembled of trains, trucks and
cars, respectively, the associated LP model is given as:
Maximize Z = 3 X1 + 2 X2 + 5 X3
Subject to: X1 + 2 X2+ X3 < 430 (operation 1)
3 X1 + 2 X3 < 460 (operation 2)
X1 + 4 X2 < 420 (operation 3)
X1, X2, X3 > 0
Using X4, X5 and X6 as the slack variables for the constraints of operations 1, 2 and 3,
respectively, the optimum tableau is:
Basic
X1
X2
X3
X4
X5
X6
Z
4
0
0
1
2
0
X2
-1/4
1
1
½
-1/4
0
X3
3/2
0
0
0
½
0
X6
2
0
0
-2
1
1
The result from above tableau can be summarized, as follows.
X* = (X1, X2, X3) = (0, 100, 230)
Z* = $1350
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2012

Topic 3: Simplex Method and Sensitivity Analysis

Sensitivity and post-optimality analysis

In LP, the parameters of the model can change within certain limits without causing the Optimal solution to change. This is referred to as Sensitivity analysis. In contrast, Post-optimality analysis deals with determining the new Optimal solution resulting from making targeting changes in the input data.

Sensitivity analysis

In LP model, parameters are usually not exact. With sensitivity analysis, one can ascertain the impact of this uncertainty on the quality of the optimum solution. For example, if sensitivity analysis reveals that the optimum remains the same for a ±10% change in unit ‘Ci’, on cane conclude that the solution is more robust than in case where indifference range is only ± percent. There are two major types of sensitivity analysis: (i) Optimality Range Analysis or Objective-Function Range Analysis (ii) Feasibility analysis or Right Hand Side (RHS) Range Analysis

Taking an example: TOYCO assembles three types of toys - trains, trucks and cars - using three operations. The limits on the available times for the three operations are 430, 460 and 420 minutes, respectively, and the revenue per unit of train, truck and car are $3, $2 and $5, respectively. The assembly times for train per three operations are 1, 3 and 1 minutes, respectively. The corresponding times per truck and car are (2, 0, 4) and (1, 2, 0) minutes (where a zero time indicates that the operation is not used). Letting X 1 , X 2 and X 3 represent the daily number of units assembled of trains, trucks and cars, respectively, the associated LP model is given as:

Maximize Z = 3 X 1 + 2 X 2 + 5 X 3 Subject to: X 1 + 2 X 2 + X 3 < 430 (operation 1) 3 X 1 + 2 X 3 < 460 (operation 2) X 1 + 4 X 2 < 420 (operation 3) X1, X 2 , X 3 > 0

Using X 4 , X 5 and X 6 as the slack variables for the constraints of operations 1, 2 and 3, respectively, the optimum tableau is:

Basic X 1 X 2 X 3 X 4 X 5 X 6 Solution Z 4 0 0 1 2 0 1350 X 2 - 1/4 1 1 ½ - 1/4 0 100 X 3 3/2 0 0 0 ½ 0 230 X 6 2 0 0 - 2 1 1 20

The result from above tableau can be summarized, as follows. X* = (X 1 , X 2 , X 3 ) = (0, 100, 230) Z* = $

2012

S = (S1, S2, S 3 ) = (0, 0, 20)  Slacks Y = (Y1, Y2, Y 3 ) = (1, 2, 0)  Shadow prices

If we further investigate into the output of this problem, we get data on optimality and feasibility ranges. Optimality Ranges (Ci) Reduced Cost For X 1 (Trains): - ∞ < 3 < 7 4. X 2 (Trucks): 0 < 2 < 10 0. X 3 (Cars): 2.33 < 5 < ∞ 0.

Feasibility Ranges (bi) Dual Price Operation 1: 230 < 430 < 440 1. Operation 2: 440 < 460 < 860 2. Operation 3: 400 < 420 < ∞ 0.

Optimality Ranges The above reported data on ‘ Optimality Ranges ’ suggest:

  1. Any one unit addition of X 1 will reduce value of Z by $ 4.00 (Reduced cost); note X 1 is already zero, due to the same reason.
  2. Range for C 1 (revenue from X 1 ) is (-∞ < C 1 < 7), which indicates that any value of C 1 , less than 7, will not help X 1 to enter into ‘optimum’ solution.
  3. The definition of ‘Reduced Cost’ is: Reduced cost = (cost of consumed resources per unit output) – (revenue per unit output)
  4. Sub paragraphs (2) and (3) suggest that X 1 (presently unprofitable at C 1 = $3.00) can be made profitable in one of the two ways; either increase its revenue (above 7) or reduce its cost. In real life, the former option is not valid as unit price of output is determined in the market, the later option may be implemented through adopting efficient way of production.
  5. In summary, the ‘optimality ranges’ show that the current solution remain optimal (does not change) until one keeps changes in Ci within indicated ranges.

Feasibility Ranges In the same token, one can interpret the ‘Feasibility Ranges’ , namely:

  1. The given ‘ Dual Prices ’ suggest that one minute increase in Operation 1 will increase value of Z by $1.00; similarly, one minute increase in Operations 2 and 3 will increase value of Z by $2.00 and $0.00, respectively.
  2. On the basis of values of dual prices, it seems better to give priority to increase time on Operation 2, relative to other two operations.
  3. ‘Feasibility Ranges’ suggest that one should remain within specified ranges to have ‘optimum’ solution unchanged; to remain within ranges should mean to keep dual prices and optimal solution valid.