How to Read Sensitivity Report Linear Programming

OR-Notes

J E Beasley

OR-Notes are a series of introductory notes on topics that autumn under the broad heading of the field of operations enquiry (OR). They were originally used by me in an introductory OR course I give at Royal College. They are now bachelor for apply by any students and teachers interested in OR bailiwick to the post-obit conditions.

A full list of the topics available in OR-Notes can be found hither.

Linear programming - sensitivity analysis - using Solver

Retrieve the product planning problem concerned with four variants of the same product which we formulated earlier every bit an LP. To remind you of information technology nosotros repeat beneath the problem and our formulation of information technology.

Product planning problem

A company manufactures four variants of the same product and in the final function of the manufacturing process there are assembly, polishing and packing operations. For each variant the fourth dimension required for these operations is shown beneath (in minutes) as is the profit per unit of measurement sold.

        Associates      Shine    Pack        Profit (£) Variant 1       2             3         2           1.50         two       4             2         three           2.50         3       three             iii         ii           iii.00         4       7             iv         5           4.50

Given the electric current state of the labour force the company estimate that, each yr, they have 100000 minutes of assembly time, 50000 minutes of polishing time and 60000 minutes of packing time bachelor. How many of each variant should the company make per year and what is the associated profit?
Suppose at present that the company is free to decide how much time to devote to each of the iii operations (assembly, polishing and packing) inside the full allowable time of 210000 (= 100000 + 50000 + 60000) minutes. How many of each variant should the company make per yr and what is the associated turn a profit?

Production planning solution

Variables

Permit:

x_i exist the number of units of variant i (i=1,ii,three,4) made per year

T_ass exist the number of minutes used in assembly per twelvemonth
T_{political leader} exist the number of minutes used in polishing per year
T_pac be the number of minutes used in packing per yr

where ten_i >= 0 i=i,two,3,4 and T_ass, T_politician, T_pac >= 0

Constraints

(a) functioning fourth dimension definition

T_ass = 2x₁ + 4x_ii + 3x₃ + 7x₄ (assembly)
T_pol = 3x₁ + 2x₂ + 3x₃ + 4x₄ (polish)
T_pac = 2x₁ + 3x₂ + 2x₃ + 5x₄ (pack)

(b) performance time limits

The operation fourth dimension limits depend upon the situation being considered. In the commencement situation, where the maximum time that can be spent on each operation is specified, we merely have:

T_donkey <= 100000 (assembly)
T_{political leader} <= 50000 (polish)
T_pac <= 60000 (pack)

In the second situation, where the only limitation is on the total time spent on all operations, we just accept:

T_ass + T_politico + T_pac <= 210000 (total time)

Objective

Presumably to maximise profit - hence we have

maximise ane.5x₁ + 2.5x₂ + 3.0x₃ + iv.5x₄

which gives united states the complete conception of the problem.

Solution - using Solver

Below we solve this LP with the Solver add-in that comes with Microsoft Excel.

If you click here you will be able to download an Excel spreadsheet called lp.xls that already has the LP we are considering set up.

Look at Sheet A in lp.xls and to use Solver do Tools and then Solver. In the version of Excel I am using (different versions of Excel have slightly different Solver formats) you volition get the Solver model as beneath:

but where at present we have highlighted (clicked on) two of the Reports available - Reply and Sensitivity. Click OK and you will observe that two new sheets have been added to the spreadsheet - an Answer Study and a Sensitivity Report.

As these reports are indicative of the information that is usually available when we solve a LP via a computer we shall deal with each of them in turn.

Answer Report

The answer report tin be seen below:

This is the most self-explanatory report.

We tin can see that the optimal solution to the LP has value 58000 (£) and that T_donkey=82000, T_pol=50000, T_pac=60000, X₁=0, X₂=16000, X_three=6000 and X₄=0.

Annotation that we had 3 constraints for total assembly, total polishing and total packing time in our LP. The associates fourth dimension constraint is declared to be 'Not Binding' whilst the other 2 constraints are declared to be 'Bounden'. Constraints with a 'Slack' value of zero are said to be tight or bounden in that they are satisfied with equality at the LP optimal. Constraints which are non tight are called loose or not bounden.

Sensitivity Report

The sensitivity report can be seen below:

This sensitivity report provides u.s. with information relating to:

irresolute the objective office coefficient for a variable
forcing a variable which is currently zippo to be not-zero
changing the right-mitt side of a constraint.

We deal with each of these in plow, and note here that the analysis presented beneath ONLY applies for a unmarried change, if two or more things change then nosotros effectively demand to resolve the LP.

Changing the objective function coefficient for a variable

To illustrate this suppose we vary the coefficient of X₂ in the objective part. How will the LP optimal solution change?

Currently Ten_i=0, X₂=16000, 10_iii=6000 and Ten_four=0. The current solution value for 10_ii of 16000 is in cell B3 and the electric current objective office coefficient for X_two is 2.v. The Allowable Increase/Decrease columns tell u.s. that, provided the coefficient of X_two in the objective role lies between 2.5+ii = 4.v and 2.5 - 0.142857143 = 2.3571 (to four decimal places), the values of the variables in the optimal LP solution will remain unchanged. Note though that the bodily optimal solution value will alter as the objective function coefficient of X_two is irresolute.

In terms of the original problem we are effectively maxim that the decision to produce 16000 of variant 2 and 6000 of variant three remains optimal even if the turn a profit per unit on variant 2 is not really two.5 (but lies in the range 2.3571 to 4.50). Similar conclusions can be drawn near X₁, X₃ and X₄.

Forcing a variable which is currently zilch to be non-naught

For the variables, the Reduced Cost column gives us, for each variable which is currently nada (X₁ and X_iv), an judge of how much the objective role will change if we brand (force) that variable to be non-nix. Notation here that the value in the Reduced Cost column for a variable is frequently chosen the 'opportunity cost' for the variable.

Hence we have the table

Variable                             X_i        X_ivReduced Cost (opportunity toll)      1.five     0.2 New value (= or >=)                  X_one=A    10₄=B                                   or X_i>=A   10_iv>=B Estimated objective function alter  1.5A    0.2B

where we ignore the sign of the reduced cost when constructing the higher up table. The objective role will e'er become worse (become downward if we have a maximisation problem, become upward if we accept a minimisation trouble) past at least this guess. The larger A or B are the more inaccurate this estimate is of the exact change that would occur if we were to resolve the LP with the corresponding constraint for the new value of X₁ or X₄ added.

Notation hither than an alternative (and every bit valid) interpretation of the reduced cost is the amount by which the objective office coefficient for a variable needs to change earlier that variable volition become not-zero.

Hence for variable Ten₁ the objective role needs to change by 1.five (increment since we are maximising) earlier that variable becomes non-zero. In other words, referring dorsum to our original situation, the profit per unit on variant 1 would need to increase by i.5 earlier information technology would exist profitable to produce any of variant one. Similarly the profit per unit on variant iv would need to increase by 0.ii before it would exist assisting to produce any of variant four.

Changing the right-mitt side of a constraint

For each constraint the cavalcade headed Shadow Price tells the states exactly how much the objective function will modify if we change the right-mitt side of the corresponding constraint inside the limits given in the Allowable Increase/Subtract columns

Hence we tin form the table

Constraint                                  Associates    Polish    Pack Opportunity (Reduced) Cost (ignore sign)    0           0.80      0.xxx Change in correct-hand side                   a           b         c Objective role change                   0           0.80b     0.30c Lower limit for right-hand side             82000       40000     33333.34 Current value for right-hand side           100000      50000     60000 Upper limit for right-mitt side             -           90000     75000

For case for the polish constraint, provided the right-hand side of that constraint remains between 50000 + 40000 =90000 and 50000 - 10000 = 40000 the objective office change volition be exactly 0.80[change in correct-hand side from 50000].

The direction of the alter in the objective role (up or down) depends upon the management of the modify in the right-manus side of the constraint and the nature of the objective (maximise or minimise).

To decide whether the objective function will go upward or down use:

constraint more than (less) restrictive subsequently modify in right-hand side implies objective part worse (amend)
if objective is maximise (minimise) then worse means down (upwards), ameliorate means upwards (downwards)

Hence

if yous had an actress 100 hours to which functioning would y'all assign it?
if you lot had to take fifty hours away from polishing or packing which one would you choose?
what would the new objective office value be in these two cases?

The value in the column headed Shadow Price for a constraint is often chosen the 'marginal value' or 'dual value' for that constraint.

Note that, as would seem logical, if the constraint is loose the shadow cost is nix (as if the constraint is loose a small change in the right-manus side cannot alter the optimal solution).

Comments

Different LP packages have different formats for input/output but the same information as discussed in a higher place is withal obtained.
You may have plant the above confusing. Substantially the interpretation of LP output is something that comes with exercise.
Much of the information obtainable (as discussed above) as a by-production of the solution of the LP problem can be useful to management in estimating the outcome of changes (e.grand. changes in costs, production capacities, etc) without going to the hassle/expense of resolving the LP.
This sensitivity information gives us a measure of how robust the solution is i.due east. how sensitive it is to changes in input data.

Note hither that, as mentioned above, the assay given above relating to:

irresolute the objective function coefficient for a variable; and
forcing a variable which is currently zero to exist non-aught; and
changing the right-manus side of a constraint

is merely valid for a single modify. If two (or more) changes are made the state of affairs becomes more circuitous and information technology becomes appropriate to resolve the LP.

How to Read Sensitivity Report Linear Programming

Source: http://people.brunel.ac.uk/~mastjjb/jeb/or/lpsens_solver.html

Wren Whounner