'Sensitivity Analysis\r'

'Linear mastergram Notes VII Sensitivity Analysis 1 Introduction When you hire a numeric model to bring up reality you m senioriness describe ap masterximations. The world is to a greater extent than manifold than the kinds of optimization businesss that we ar adequ befuddle to act upon. linearity assumptions usually ar signi? senst ap professional personximations. some a nonher(prenominal) important approximation comes beca hold you kindle non be for certain of the data that you put into the model. Your whapledge of the applicable technology may be imprecise, forcing you to reckon survey in A, b, or c. Moreover, discipline may diversify.Sensitivity analysis is a systematic study of how sensitive (duh) etymons argon to (small) varynates in the data. The basic idea is to be able to give decides to questions of the form: 1. If the clinical fail shifts, how does the closure counter tack? 2. If imagerys ready(prenominal) metamorphose, how does the theme alternate? 3. If a victimisestraint is added to the conundrum, how does the radical deviate? One approach to these questions is to solve stacks of linear computer programming riddles. For less(prenominal)on, if you cerebrate that the terms of your primary emergeput give be amidst $100 and $ tenacious hundred per unit, you mass solve twenty di? riptide jobs ( unity for each whole fare among $100 and $120). 1 This method would work, be berths it is inelegant and (for large conundrums) would involve a large nub of numeration time. (In even offt, the enumeration time is cheap, and computing sources to similar jobs is a standard technique for studying sensitiveness in practice. ) The approach that I go out describe in these notes takes full emolument of the structure of LP programming problems and their outcome. It turns out that you dismiss often ? gure out what slide bys in â€Å" close” linear programming problems just by looking and b y examining the information provided by the simplex algorithm.In this section, I leave al iodine describe the predisposition analysis information provided in Excel computations. I get out in addition try to give an intuition for the results. 2 Intuition and Overview Throughout these notes you should imagine that you mustiness solve a linear programming problem, notwithstanding whence you urgency to limit how the answer assortments if the problem is adjustmentd. In any faux pas, the results deal that lonesome(prenominal) one intimacy about the problem alternates. That is, in esthesia analysis you evaluate what take chancess when only one parameter of the problem replaces. 1 OK, in that respect be sincerely 21 problems, entirely who is counting? 1To ? x ideas, you may think about a break downicular LP, assert the familiar example: goop 2×1 subject to 3×1 x1 2x 1 + + + 4×2 x2 3×2 x2 + + + + 3x 3 x3 2x 3 3x 3 + + + x4 4x 4 3x 4 x4 x ? ? ? 1 2 7 10 0 We know that the etymon to this problem is x0 = 42, x1 = 0; x2 = 10. 4; x3 = 0; x4 = . 4. 2. 1 Changing object lens Function say that you solve an LP and whence deal to solve an opposite problem with the like reserves but a roughly di? erent mark routine. (I bequeath al managements relieve oneself only one alternate in the problem at a time. So if I diverge the quarry exit, not only give I prep atomic number 18 the simplenesss ? ed, but I will transmit only one coe cient in the bearing persona. ) When you alteration the intent swear out it turns out that at that place argon two eccentric persons to con situationr. The ? rst case is the compound in a non-basic versatile (a varying that takes on the observe zipper in the origin). In the example, the relevant non-basic variants ar x1 and x3 . What happens to your beginning if the coe cient of a non-basic inconstant simplifications? For example, call up that the coe cient of x1 in the ac c development scat to a higher place was bring down from 2 to 1 (so that the fair game subroutine is: max x1 + 4×2 + 3×3 + x4 ).What has happened is this: You deplete interpreted a covariant that you didn’t expect to use in the ? rst place (you frozen x1 = 0) and consequently made it less pro? bow ( demoraliseed its coe cient in the objective run short). You are silence not release to use it. The ancestor does not change. Observation If you area the objective billet coe cient of a non-basic variant, and indeed the result does not change. What if you raise the coe cient? Intuitively, facts of life it just a little daub should not subject field, but raising the coe cient a lot might induce you to change the mensurate of x in a way that unsexs x1 > 0.So, for a non-basic protean, you should expect a resultant role to continue to be legitimate for a enjoin of harbors for coe cients of nonbasic variables. The arena should include all depress look ons for the coe cient and some higher take accounts. If the coe cient annexs tolerable (and putting the variable into the understructure is feasible), accordingly the solution changes. What happens to your solution if the coe cient of a basic variable (like x2 or x4 in the example) accrues? This situation di? ers from the antecedent one in that you are development the dry land variable in the ? rst place. The change asks the variable contribute less to pro? . You should expect that a su ciently large falloff realizes you hope to change your solution (and move the apprise the associated variable). For example, if the coe cient of x2 in the objective function in the example were 2 instead of 4 (so that the objective was max 2×1 +2×2 +3×3 + x4 ), 2 maybe you would deprivation to set x2 = 0 instead of x2 = 10. 4. On the other guide, a small decrease in x2 ’s objective function coe cient would typically not cause you to change yo ur solution. In contrast to the case of the non-basic variable, much(prenominal) a change will change the look on of your objective function.You compute the treasure by plugging in x into the objective function, if x2 = 10. 4 and the coe cient of x2 goes big money from 4 to 2, consequently the contribution of the x2 term to the valuate goes down from 41. 6 to 20. 8 (assuming that the solution carcass the same). If the coe cient of a basic variable goes up, therefore(prenominal) your value goes up and you still want to use the variable, but if it goes up enough, you may want to determine x so that it x2 is even realizable. In many cases, this is possible by ? nding another basis (and thusly another solution).So, intuitively, thither should be a range of value of the coe cient of the objective function (a range that includes the current value) in which the solution of the problem does not change. Out incline of this range, the solution will change (to lower the value of th e basic variable for reductions and join on its value of sum ups in its objective function coe cient). The value of the problem always changes when you change the coe cient of a basic variable. 2. 2 Changing a Right-Hand Side Constant We hold forthed this motion when we talked about forkedity. I argued that duple worths curb the e? ct of a change in the inwardnesss of available alternatives. When you changed the mensuration of alternative in a non-binding restraint, then make ups neer changed your solution. Small comes similarly did not change anything, but if you lessend the beat of election enough to make the shyness binding, your solution could change. (Note the similarity between this analysis and the case of ever-ever-changing the coe cient of a non-basic variable in the objective function. Changes in the rightfulness spot of binding constraints always change the solution (the value of x must adjust to the tonic constraints).We saw earlier that the forke d variable associated with the constraint measures how much the objective function will be in? uenced by the change. 2. 3 Adding a simpleness If you add a constraint to a problem, two things can happen. Your certain solution satis? es the constraint or it doesn’t. If it does, then you are ? nished. If you had a solution before and the solution is still feasible for the new problem, then you must still require a solution. If the pilot film solution does not satisfy the new constraint, then mayhap the new problem is infeasible. If not, then there is another solution.The value must go down. (Adding a constraint makes the problem harder to satisfy, so you cannot possibly do better than before). If your original solution satis? es your new constraint, then you can do as substantially as before. If not, then you will do worse. 2 2 There is a antiquated case in which originally your problem has multiple solutions, but only some of them satisfy the added constraint. In this c ase, which you request not worry about, 3 2. 4 Relationship to the Dual The objective function coe cients pair to the right wing side constants of resource constraints in the multiple.The native’s right hand side constants correspond to objective function coe cients in the dual. Hence the exercise of changing the objective function’s coe cients is real the same as changing the resource constraints in the dual. It is extremely useful to gravel comfor tabular array switching back and onwards between primitive and dual relationships. 3 Understanding Sensitivity Information Provided by Excel Excel permits you to create a sensitivity report with any puzzle out LP. The report contains two dining eludes, one associated with the variables and the other associated with the constraints.In reading these notes, keep the information in the sensitivity send backs associated with the ? rst simplex algorithm example nearby. 3. 1 Sensitivity Information on Changing (or Adj ustable) Cells The top table in the sensitivity report refers to the variables in the problem. The ? rst towboat (Cell) tells you the location of the variable in your spreadsheet; the routine tower tells you its name (if you named the variable); the one-third tower tells you the ? nal value; the 4th column is called the lessen toll; the ? fth column tells you the coe cient in the problem; the ? al two columns are designate â€Å" deductible sum up” and â€Å" deductible descend. ” rock-bottom constitute, deductible cast up, and allowable ebb are new terms. They wish de? nitions. The allowable increases and decreases are easier. I will discuss them ? rst. The allowable increase is the amount by which you can increase the coe cient of the objective function without create the best basis to change. The allowable decrease is the amount by which you can decrease the coe cient of the objective function without causing the optimal basis to change. Take the ? rst row of the table for the example. This row describes the variable x1 .The coe cient of x1 in the objective function is 2. The allowable increase is 9, the allowable decrease is â€Å"1. 00E+30,” which essence 1030 , which really message 1. This operator that provided that the coe cient of x1 in the objective function is less than 11 = 2 + 9 = original value + allowable increase, the basis does not change. Moreover, since x1 is a non-basic variable, when the basis girdle the same, the value of the problem remain the same too. The information in this line con? rms the intuition provided earlier and adds something new. What is con? rmed is that if you lower the objective coe cient of a non-basic ariable, then your solution does not change. (This government agency that the allowable decrease will always be in? nite for a non-basic variable. ) The example also demonstrates your value will stay the same. 4 that increase the coe cient of a non-basic variable may a dd to a change in basis. In the example, if you increase the coe cient of x1 from 2 to anything great than 9 (that is, if you add more than than the allowable increase of 7 to the coe cient), then you change the solution. The sensitivity table does not tell you how the solution changes, but common sense suggests that x1 will take on a decreed value.Notice that the line associated with the other non-basic variable of the example, x3 , is unco similar. The objective function coe cient is di? erent (3 rather than 2), but the allowable increase and decrease are the same as in the row for x1 . It is a coincidence that the allowable increases are the same. It is no coincidence that the allowable decrease is the same. We can conclude that the solution of the problem does not change as long as the coe cient of x3 in the objective function is less than or equal to 10. count now the basic variables. For x2 the allowable increase is in? ite 9 while the allowable decrease is 2. 69 (it is 2 13 to be exact). This way that if the solution win’t change if you increase the coe cient of x2 , but it will change if you decrease the coe cient enough (that is, by more than 2. 7). The fact that your solution does not change no matter how much you increase x2 ’s coe cient means that there is no way to make x2 > 10. 4 and still satisfy the constraints of the problem. The fact that your solution does change when you increase x2 ’s coe cient by enough means that there is a feasible basis in which x2 takes on a value lower than 10. 4. You knew that. Examine the original basis for the problem. ) The range for x4 is di? erent. Line four of the sensitivity table says that the solution of the problem does not change provided that the coe cient of x4 in the objective function stays between 16 (allowable increase 15 plus objective function coe cient 1) and -4 (objective function coe cient deduction allowable decrease). That is, if you make x4 su ciently more att ractive, then your solution will change to permit you to use more x4 . If you make x4 su ciently less attractive the solution will also change. This time to use less x4 .Even when the solution of the problem does not change, when you change the coe cient of a basic variable the value of the problem will change. It will change in a predictable way. Speci? cally, you can use the table to tell you the solution of the LP when you take the original constraints and replace the original objective function by max 2×1 + 6×2 + 3×3 + x4 (that is, you change the coe cient of x2 from 4 to 6), then the solution to the problem system the same. The value of the solution changes because now you regurgitate the 10. 4 units of x2 by 6 instead of 4. The objective function therefore goes up by 20. . The cut cost of a variable is the smallest change in the objective function coe cient use uped to arrive at a solution in which the variable takes on a positive value when you solve the pr oblem. This is a mouthful. Fortunately, reduce cost are redundant information. The reduced cost is the negative of the allowable increase for non-basic variables (that is, if you change the coe cient of x1 by 7, then you arrive at a problem in which x1 takes on a positive 5 value in the solution). This is the same as formula that the allowable increase in the coe cient is 7.The reduced cost of a basic variable is always adjust (because you need not change the objective function at all to make the variable positive). Neglecting rare cases in which a basis variable takes on the value 0 in a solution, you can ? gure out reduced cost from the other information in the table: If the ? nal value is positive, then the reduced cost is zero. If the ? nal value is zero, then the reduced cost is negative one times the allowable increase. Remarkably, the reduced cost of a variable is also the amount of deliberate in the dual constraint associated with the variable.With this interpretation, complemental lassitudeness implies that if a variable that takes on a positive value in the solution, then its reduced cost is zero. 3. 2 Sensitivity Information on simplicitys The instant sensitivity table discusses the constraints. The cell column identi? es the location of the left(prenominal) side of a constraint; the name column gives its name (if any); the ? nal value is the value of the left-hand side when you plug in the ? nal values for the variables; the shadow determine is the dual variable associated with the constraint; the constraint R. H. ide is the right hand side of the constraint; allowable increase tells you by how much you can increase the right-hand(a) side of the constraint without changing the basis; the allowable decrease tells you by how much you can decrease the right-hand side of the constraint without changing the basis. complementary Slackness procures a relationship between the columns in the constraint table. The di? erence between the â€Å"C onstraint Right-Hand Side” column and the â€Å"final exam Value” column is the inert. (So, from the table, the slack for the common chord constraints is 0 (= 12 12), 37 (= 7 ( 30)), and 0 (= 10 10), respectively.We know from completing Slackness that if there is slack in the constraint then the associated dual variable is zero. Hence CS tells us that the abet dual variable must be zero. Like the case of changes in the variables, you can ? gure out information on allowable changes from other information in the table. The allowable increase and decrease of non-binding variables can be computed knowing ? nal value and right-hand side constant. If a constraint is not binding, then adding more of the resource is not going to change your solution. Hence the allowable increase of a resource is in? ite for a non-binding constraint. (A nearly equivalent, and also true, statement is that the allowable increase of a resource is in? nite for a constraint with slack. ) In the e xample, this explains why the allowable increase of the turn constraint is in? nite. One other criterion is also no surprise. The allowable decrease of a non-binding constraint is equal to the slack in the constraint. Hence the allowable decrease in the second constraint is 37. This means that if you decrease the right-hand side of the second constraint from its original value (7) to nything greater than 30 you do not change the optimal basis. In fact, the only part of the solution that changes when you do this is that the value of the slack variable for this constraint changes. In this paragraph, the show up is only this: If you solve an LP and ? nd that a constraint is not binding, 6 then you can remove all of the unwarranted (slack) portion of the resource associated with this constraint and not change the solution to the problem. The allowable increases and decreases for constraints that have no slack are more complicated. Consider the ? rst constraint.The information in the table says that if the right-hand side of the ? rst constraint is between 10 (original value 12 minus allowable decrease 2) and in? nity, then the basis of the problem does not change. What these columns do not say is that the solution of the problem does change. Saying that the basis does not change means that the variables that were zero in the original solution continue to be zero in the new problem (with the right-hand side of the constraint changed). However, when the amount of available resource changes, necessarily the values of the other variables change. You can think about this in many ways. Go back to a standard example like the forage problem. If your diet provides exactly the right amount of Vitamin C, but then for some causal agency you learn that you need more Vitamin C. You will sure enough change what you eat and (if you aren’t getting your Vitamin C through pills planning pure Vitamin C) in tramp to do so you probably will need to change the compositio n of your diet †a little more of some foods and mayhap less of others. I am verbalise that (within the allowable range) you will not change the foods that you eat in positive amounts.That is, if you ate only spinach and oranges and bagels before, then you will only eat these things (but in di? erent quantities) after the change. Another thing that you can do is simply re-solve the LP with a di? erent right-hand side constant and compare the result. To ? nish the discussion, consider the third constraint in the example. The values for the allowable increase and allowable decrease guarantee that the basis that is optimal for the original problem (when the right-hand side of the third constraint is equal to 10) remains obtain provided that the right-hand side constant in this constraint is between -2. 333 and 12. hither is a way to think about this range. Suppose that your LP involves four drudgery processes and uses three basic ingredients. impose the ingredients land, labour, and capital. The outputs vary use di? erent combinations of the ingredients. Maybe they are growing fruit (using lots of land and push), cleanup position bathrooms (using lots of restriction), do cars (using lots of diligence and and a oddball of capital), and making computers (using lots of capital). For the initial speci? cation of available resources, you ? nd that your want to grow fruit and make cars.If you get an increase in the amount of capital, you may wish to shift into create computers instead of cars. If you experience a decrease in the amount of capital, you may wish to shift away from building cars and into cleaning bathrooms instead. As always when dealing with dichotomy relationships, the the â€Å"Adjustable Cells” table and the â€Å"Constraints” table really provide the same information. Dual variables correspond to primal constraints. Primal variables correspond to dual constraints. Hence, the â€Å"Adjustable Cells” table tells you h ow sensitive primal variables and dual constraints are to changes in the primal objective function.The â€Å"Constraints” table tells you how sensitive dual variables and primal constraints are to changes in the dual objective function (right-hand side constants in the primal). 7 4 Example In this section I will shew another formulation example and discuss the solution and sensitivity results. Imagine a furniture company that makes tables and hold ins. A table requires 40 dining table feet of wood and a chair requires 30 board feet of wood. woods costs $1 per board substructure and 40,000 board feet of wood are available. It takes 2 hours of clever assiduity to make an un? nished table or an un? ished chair. Three more hours of labor will turn an un? nished table into a ? nished table; two more hours of virtuoso(prenominal) labor will turn an un? nished chair into a ? nished chair. There are 6000 hours of skilled labor available. (Assume that you do not need to pay fo r this labor. ) The prices of output are disposed in the table below: merchandise Un? nished Table Finished Table Un? nished hold in Finished Chair Price $70 $140 $60 $110 We want to formulate an LP that describes the production plans that the ? rm can use to maximize its pro? ts. The relevant variables are the number of ? nished and un? ished tables, I will call them TF and TU , and the number of ? nished and un? nished chairs, CF and CU . The receipts is (using the table): 70TU + 140TF + 60CU + 110CF , , while the cost is 40TU + 40TF + 30CU + 30CF (because beat costs $1 per board foot). The constraints are: 1. 40TU + 40TF + 30CU + 30CF ? 40000. 2. 2TU + 5TF + 2CU + 4CF ? 6000. The ? rst constraint says that the amount of ram used is no more than what is available. The second constraint states that the amount of labor used is no more than what is available. Excel ? nds the answer to the problem to be to construct only ? nished chairs (1333. 33 †I’m not sure what it means to make a care 1 chair, but let’s assume 3 that this is possible). The pro? t is $106,666. 67. Here are some sensitivity questions. 1. What would happen if the price of un? nished chairs went up? Currently they carry for $60. Because the allowable increase in the coe cient is $50, it would not be pro? table to leaven them even if they sold for the same amount as ? nished chairs. If the price of un? nished chairs went down, then certainly you wouldn’t change your solution. 8 2. What would happen if the price of un? nished tables went up? Here something evidently absurd happens.The allowable increase is greater than 70. That is, even if you could sell un? nished tables for more than ? nished tables, you would not want to sell them. How could this be? The answer is that at current prices you don’t want to sell ? nished tables. Hence it is not enough to make un? nished tables more pro? table than ? nished tables, you must make them more pro? table than ? nished chairs. Doing so requires an even greater increase in the price. 3. What if the price of ? nished chairs fell to $100? This change would alter your production plan, since this would involve a $10 decrease in the price of ? ished chairs and the allowable decrease is only $5. In order to ? gure out what happens, you need to re-solve the problem. It turns out that the topper thing to do is specialize in ? nished tables, producing 1000 and earning $100,000. Notice that if you continued with the old production plan your pro? t would be 70 ? 1333 1 = 93, 333 1 , so the change in production plan 3 3 was expense more than $6,000. 4. How would pro? t change if lumber supplies changed? The shadow price of the lumber constraint is $2. 67. The range of values for which the basis remains unchanged is 0 to 45,000.This means that if the lumber supply went up by 5000, then you would continue to specialize in ? nished chairs, and your pro? t would go up by $2. 67 ? 5000 = $10, 333. At this evidence you presumably run out of labor and want to reoptimize. If lumber supply decreased, then your pro? t would decrease, but you would still specialize in ? nished chairs. 5. How much would you be willing to pay an excess carpenter? Skilled labor is not outlay anything to you. You are not using the labor than you have. Hence, you would pay nothing for additional workers. 6. Suppose that industrial regulations complicate the ? ishing process, so that it takes one extra hour per chair or table to turn an un? nished product into a ? nished one. How would this change your plans? You cannot read your answer o? the sensitivity table, but a bit of common sense tells you something. The change cannot make you better o?. On the other hand, to produce 1,333. 33 ? nished chairs you’ll need 1,333. 33 extra hours of labor. You do not have that available. So the change will change your pro? t. Using Excel, it turns out that it becomes optimal to specialize in ? nished tables, produc ing 1000 of them and earning $100,000. This problem di? ers from the original one because the amount of labor to create a ? nished product increases by one unit. ) 7. The owner of the ? rm comes up with a design for a beautiful craft storage locker. Each cabinet requires 250 hours of labor (this is 6 weeks of full time work) and uses 50 board feet of lumber. Suppose that the company can sell a cabinet for $200, would it be worthy? You could solve this 9 problem by changing the problem and adding an additional variable and an additional constraint. Note that the coe cient of cabinets in the objective function is 150, which re? cts the sale price minus the cost of lumber. I did the computation. The ? nal value increased to 106,802. 7211. The solution involved reducing the output of un? nished chairs to 1319. 727891 and increasing the output of cabinets to 8. 163265306. (Again, enthrall tolerate the fractions. ) You could not have guessed these ? gures in advance, but you could ? gu re out that making cabinets was a good idea. The way to do this is to value the inputs to the production of cabinets. Cabinets require labor, but labor has a shadow price of zero. They also require lumber. The shadow price of lumber is $2. 7, which means that each unit of lumber adds $2. 67 to pro? t. Hence 50 board feet of lumber would reduce pro? t by $133. 50. Since this is less than the price at which you can sell cabinets (minus the cost of lumber), you are better o? using your resources to build cabinets. (You can check that the increase in pro? t associated with making cabinets is $16. 50, the added pro? t per unit, times the number of cabinets that you actually produce. ) I attached a sheet where I did the same computation assuming that the price of cabinets was $150. In this case, the additional option does not lead to cabinet production. 10\r\n'