Notes on Sensitivity Analysis

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Notes on Sensitivity Analysis
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  Notes on Sensitivity Analysis 1  INDRODUCTION The linear programming approach to optimization problems includes the assumption that input data is known and is not subject to changes. In real life this assumption may be found inaccurate. For example, cost estimates are sometimes subject to errors, and to changes oer time due to dynamic behaior of the enironment! emand reflects market  behaior, which in itself is unpredictable to some degree! #esource aailability may change when management changes its preferences. $o, a %uestion about the sensitiity of the optimal solution to changes in input parameters seems to be alid, and important for the sake of making informed decisions. This is the topic dealt with in these notes.$ensitiity analysis allows for only one parameter change at a time. $ince in reality seeral changes may occur simultaneously, we&ll extend the discussion to the multiple changes case later. For now, two types of changes are considered within the framework of a linear programming model. 'i()hanges in one objectie*function coefficient.'ii()hanges in one constraint right*hand*side.First let us present a decision problem to be soled using linear programming. This  problem will then sere as the ehicle with which we demonstrate the sensitiity analysis concepts. Example )+I manufactures a standard dining chair used in restaurants. The demand forecasts for chairs for %uarter 1 and %uarter  are -// and 0//, respectiely. The chair contains an upholstered seat that can be produced by )+I or purchased from +. + currently charges 21.3 per seat, but has announced a new prices of 21-.3 effectie the second %uarter. )+I can produce at most -4// seats per %uarter at a cost of 21/.3 per seat. $eats produced or purchased in %uarter 1 can be stored in order to satisfy demand in %uarter .  seat cost )+I 21.3/ each to hold in inentory, and maximum inentory cannot exceed -// seats. Find the optimal make*or*buy plan for )+I.The problem is formulated as follows56 1  7 8umber of seats produced by )+I in %uarter 1.6   7 8umber of seats purchased from + in %uarter 1.6 -  7 8umber of seats carried in inentory from %uarter 1 to .6 0  7 8umber of seats produced by )+I in %uarter .6 3  7 8umber of seats purchased from + in %uarter .The linear programming model is proided next59inimize 1/.36 1 :1.36  :1.36 - :1/.36 0 :1-.36 3 $ubject to5 6 1 : 6   7 -//: 6 - 6 - : 6 0 :6370//6 1 ≤ -4//6 0 ≤ -4//6 - ≤ -//6 1 , 6  , 6 - , 6 0 , and 6 3  are non*negatie  The linear programming model was run using $;<=># and the output results are gien in the attached printout5 a. ?hat is the optimal solution including the optimal alue of the objectie function@ X 1 X 2 X 3 X 4 X 5 -4///1//-4//-//The total cost 'objectie function( 7 24,13.9anagement is interested in the analysis of a few changes that might be needed for arious reasons. For example, the per*unit inentory cost may change from 21.3/ to 2.3/ due to an expected increase in the interest rate and the insurance costs. Aow will this change affect the optimal production plan@ In addition, if )+I is considering increasing storage space such that 1// more seats can be stored, what is the maximum it should be willing to pay for this additional space@ Buestions like these can be answered  by performing sensitiity analysis. <et us discuss the releant concepts and then return to this problem to answer a few interesting %uestions. Can!in! te val e o# one o$%e&tive ' # n&tion &oe##i&ient )hanging the alue of one coefficient in the objectie function, makes the ariable associated with this coefficient more attractie or less attractie for the optimization mechanism. For example, if we look for the solution that maximizes the objectie 36 1  : 06  , when the coefficient 0 becomes C 'max 36 1  : ( 6  (, the ariable 6   becomes more attractie. Therefore, one would expect the maximization mechanism to increase the alue of 6   in the optimal solution. It turns out that it is not necessarily so.  change might occur in the structure of the optimal solution, but this depends on the amount by which the coefficient is changing. This is discussed next. Statement 1) Te Ran!e o# Optimality The optimal solution of a linear programming model does not change if a single coefficient of some ariable in the objectie function changes within a certain range. Thisrange is called the range of optimality.  Note, that only one coefficient is allowed to change for the range of optimality to apply. ?e can find the range of optimality for each objectie coefficient in the $;<=># output.  Adjustable Cells  Final Reduced Objective    Allowable AllowableCell Name Value Cost Coefficient Increase Decrease $B$2 X1 3800 0 10.25 2 1E+30$C$2 X2 0 0.25 12.5 1E+30 0.25$D$2 X3 100 0 1.5 2 0.25$E$2 X4 3800 0 10.25 3.5 1E+30$F$2 X5 300 0 13.75 0.25 2 <et us return to our example and answer a few sensitiity %uestions related to the range of optimality.-   #ange of optimality5Dpper bound 7 1/.3: 7 1.3<ower bound 7 1/.3 E infinity 7 *infinity  * estion 1)  If the per*unit inentory cost increased from 21.3/ to 2.3/, would the optimal solution change@nswer5 First look for the changing parameter in the model. The coefficient 1.3/ of the ariable 6 -  in the objectie function is changing to .35 '1/.36 1 :1.36  : 1+5 6 - :1/.36 0 :1-.36 3 (. ?e need to look for the range of optimality of the coefficient 1.3. From the output 'see below( the range of optimality is5<ower bound 71.3 E /.3 7 1.3Dpper bound 7 1.3 :  7 -.3 Objective Allowable AllowableCoefficient Increase Decrease 1.5 2 0.25 Interpretation5 s long as the coefficient of 6 -  in the objectie function 'currently e%uals to 1.3( falls in the interal 1.3, -.3G the current optimal solution does not change. $ince the alue .3 does fall in this range, there will be no change in the optimal solution 'in terms of the ariable aluesH(. Aoweer, the objectie alue changesH  8ew objectie alue 7 )urrent objectie alue : ')hange in coefficient alue('the ariable 6 - ( 7 4,13 : '.3 E 1.3('1//(7 4,3. $o, in spite of the increasing cost of holding inentory, it remains optimal to store 1// chairs at the end of %uarter 1. * estion 2)  If + reduced the selling price per seat in %uarter 1 from 21.3/ to 21./, should )+I consider the purchase of seats in %uarter 1 'note that currently no seat is purchased in %uarter 1(@nswer5 The parameter changing is the coefficient of 6   '1.3( in the objectie function. It is changing to 1.3. The range of optimality is5<ower bound 7 1.3 E /.3 7 1.3Dpper bound 7 1.3 : infinity 7 Infinity Objective Allowable AllowableCoefficient Increase Decrease 12.5 1E+30 0.25 Interpretation5 $ince 21./ falls below the lower bound of the range of optimality, there will be a change in the optimal solution, and seats will be purchased at this price 'to rephrase, 6   becomes sufficiently attractie, so the minimization mechanism will make it a part of the optimal plan(. 8otice, that the objectie alue is likely to change, because the ariables are optimized at different alues. Aoweer, we cannot calculate the new objectie alue without re*running the model. )omment5 If the changing coefficient falls exactly on the boundary of the range of optimality, there will be more than one optimal solution with the same objectie function alue 'called the multiple optimal or the alternate optimal solution case(. For example, assume the coefficient 1.3/ just discussed becomes 1.3. The two optimal solutions are5 Sol tion 1)  the current solution! Sol tion 2)  a new solution, shown next50
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