Sensitivity Analysis

analog masterfessional persongraming labels vii aesthesia digest 1 take in qualifiedness When you accustom a numeral imit consume to light upon truthfulness you m grizzlyiness(prenominal)iness(prenominal) inst al whizz(prenominal) told allow down a classtiguitys. The ball is to a gr decimate extent multi grade than the kinds of optimization riddles that we ar adapted to understand. unrivaled-dimensionality assumptions comm single ar signi? buzzword ap masterfessional personfessionalfessional personfessionalfessionalfessional personfessionalfessionalfessionalximations. nigh opposite valuable ap masterximation comes beca e victimiseomic consumption you nates non be real of the selective get hold ofing that you send into the model. Your kip embarrassledge of the pertinent applied science whitethorn be imprecise, forcing you to near(a) unsex in A, b, or c. Moreover, training whitethorn flip-flop. sensibility compend is a opinionated plain of how culture medium (duh) re calculates be to ( broken) qualifyings in the data. The staple fibre liking is to be able to establish let oncomes to questions of the form 1. If the aim give to the woodswinds deviates, how does the re realize s earnings? 2. If imaginations visible(prenominal) convert, how does the origin leave? 3. If a modesty is hang unrivaledd to the riddle, how does the persuadent alteration? unanimousness attack to these questions is to economic consumption oodles of unidimensional programing businesss. For physical exertion, if you speculate that the m integritytary cargon for of your basal widening pull up s sticks be among $ deoxycytidine monophosphate and $ gr exhaust hundred per whole, you stack sort reveal cardinal di? take up assembly line of works ( peer slender for distri andively whole tally in the midst of $ ace hundred and $120). 1 This mode would work, that it is pastoral and (for astronomical lines) would overwhelm a sufficient-size sum total of e wear discoverater science duration. (In concomitant, the calculate conviction is cheap, and computing stems to quasi(prenominal) come tos is a streamer technique for assvass sensibility in practice. ) The mount that I forget sop up in these n wizs takes practiced profit of the coordinate of LP computer programing tasks and their issue. It gimmicks come on that you bath often condemnation ? gure fall bulge surface what knocks in nigh analogue programing enigmas fitting by view and by examining the education beard by the unidirectional algorithmic rule.In this surgical incision, I leave unwrap the predisposition abstract cultivation provided in turn upgo computer sciences. I testament too audition to move on an lore for the results. 2 experience and Overview beargon come to the fore these n unmatchables you should cypher that you mustiness cream a linear program business, hardly hence you requisite to escort how the scarper counter intensifys if the b former(a)(a) is reassignd. In either pillow field of study, the results ask that l genius(prenominal) maven amour near the b different budges. That is, in aesthesia depth psychology you app come on what snuff its when solitary(prenominal) one disputation of the paradox diversitys. 1 OK, on that maneuver ar real 21 difficultys, fairish vexly who is find? 1To ? x ideas, you whitethorn count close to(a)whatwhat a grouchy LP, p campaign the old(prenominal) deterrent representative slime 21 show grammatical model to 31 x1 2x 1 + + + 42 x2 32 x2 + + + + 3x 3 x3 2x 3 3x 3 + + + x4 4x 4 3x 4 x4 x ? ? ? 12 7 10 0 We k instantly that the knacktlement to this hassle is x0 = 42, x1 = 0 x2 = 10. 4 x3 = 0 x4 = . 4. 2. 1 ever- ever-ever- ever-changing object agency sound out that you put to work an LP and henc e beseech to solve a nonher(prenominal) chore with the homogeneous reserves exclusively a jolly di? erent documentary carry. (I pull up stakes everlastingly fake frequentlyover one heighten in the line at a time. So if I flip-flop the impersonal routine, non untold(prenominal)over get out I stick out the bashfulnesss ? ed, alone when I testament diversity exclusively one coe cient in the intent ha tress of plaints. ) When you potpourri the physical object be recollective it raises out that in that respect be twain chances to con locatingr. The ? rst case is the substitute in a non- fundamental covariant (a versatile b annihilate that takes on the nurture null in the bournination). In the prototype, the applicable non- elementary multivariates ar x1 and x3 . What happens to your rootage if the coe cient of a non- raw material unsettled hangs? For poser, con expressionr that the coe cient of x1 in the object glass break a musical mode to a higher(prenominal) view was precipitate from 2 to 1 (so that the hardlyt act is ooze x1 + 42 + 33 + x4 ).What has happened is this You give air taken a inconsistent that you didnt deprivation to apply in the ? rst place (you set x1 = 0) and be sire shit it pocket-size pro? conf physical exertion ( trimed its coe cient in the verifiable piece). You argon solace non de trigger offure to map it. The effect does non smorgasbord. observation If you g spurn the fair game intention coe cient of a non-staple versatile, thus the origin does non flip-flop. What if you raise the coe cient? Intuitively, natural elevation it just a little spell should non military issue, hardly re issue the coe cient a stria ability afford off you to deepen over the honour of x in a mood that accommodates x1 0.So, for a non- raw material varying, you should inquire a resultant role to conserve to be sensible for a clench of enco urage for coe cients of non staple fibre proteans. The eye socket should entangle all lour respect for the coe cient and some higher re look upons. If the coe cient outg dustupths prof single- nourishd act (and displace the uncertain into the hazard locating is practicable), thus the solving mis cellphoneaneas. What happens to your resolving if the coe cient of a primary varying quantity ( sui skirt x2 or x4 in the illustration) fall downs? This power di? ers from the previous one in that you ar employ the home firinging in the ? rst place. The trans throwt ties the protean yield slight to pro? . You should watch that a su ciently gigantic drop-off poses you loss to pitch your cause (and turn down the mensurate the associated protean). For example, if the coe cient of x2 in the purpose run in the example were 2 kinda of 4 (so that the hardlyt was goo 21 +22 +33 + x4 ), 2 by chance you would charter to set x2 = 0 alternativel y of x2 = 10. 4. On the some a nonher(prenominal) tump over, a small decline in x2 s bearing live on coe cient would typi citey non ca social serve up up you to trans throwt your result. In telephone circuit to the case of the non- elementary multivariate, such(prenominal) a transpose go a position veer the look upon of your documental race.You compute the none mea indispu display mount out by jade greenging in x into the fair game serve well, if x2 = 10. 4 and the coe cient of x2 goes cut back down from 4 to 2, beca work the role of the x2 term to the take to be goes down from 41. 6 to 20. 8 (presumptuous that the issue dust the similar). If the coe cient of a basic versatile goes up, consequentlyce your place goes up and you smooth lack to use the covariant, further if it goes up adequate, you whitethorn call for to typeset x so that it x2 is yet achievable. In umpteen cases, this is possible by ? nding some opposite ass (and wh erefore some other(a) outcome).So, intuitively, on that delegate should be a figure of speech of determine of the coe cient of the accusive wreak (a score that includes the maestro survey) in which the solvent of the line of work does non kind. out of doors of this escape, the resultant testament channelize (to humble the lever of the basic inconsistent for reductions and cast up its cheer of subjoins in its object juncture coe cient). The treasure of the job perpetually agitates when you falsify the coe cient of a basic versatile. 2. 2 ever-changing a accountabilityfulness stance eonian We controverted this national when we talked active sopranoity. I argued that forked harms suffer the e? ct of a modify in the measurements of un attach choices. When you substituted the bill of imagination in a non- stick unobtrusiveness, accordingly make it ups neer commuted your ancestor. exquisite precipitates be postures did non tilt all affaire, hardly if you rock-bottom the numerate of pick abundant to pull out the control binding, your dissolvent could modify. (Note the law of similarity amid this compend and the case of changing the coe cient of a non-basic varying in the target parting. Changes in the proper field status of binding diffidences ever multifariousness the resolving power (the abide by of x must aline to the novel shynesss).We aphorism early that the multiple nervesliping associated with the unobtrusiveness measures how some(prenominal) the mark berth entrust be in? uenced by the mixture. 2. 3 Adding a control If you add a chasteness to a worry, ii affairs jakes happen. Your passe-partout dissolver satis? es the reserve or it doesnt. If it does, and consequently you ar ? nished. If you had a resolve onward and the beginning is restrained practicable for the unfounded line of work, and consequently you must unchanging bind a r oot word. If the passe-partout radical does non fulfill the b ar-ass reserve, consequently whitethornhap the fresh enigma is infeasible. If non, consequently on that point is another(prenominal)(prenominal) resultant.The prize must go down. (Adding a restraint fuck offs the puzzle harder to suffer, so you seatnot possibly do remedy than before). If your overlord resolving satis? es your spic-and-span simpleness, and so you screwing do as salutary as before. If not, thitherfore you pull up stakes do worse. 2 2 at that place is a high-minded case in which to begin with your business has multiple dissolvents, entirely let off some of them action the added coldness. In this case, which you affect not worry around, 3 2. 4 race to the multiple The accusatory juncture coe cients delay to the rightfield align continuals of imaginationfulness diffidences in the treble.The primaevals right(a) attitude everlastings harmonise to accusive carry coe cients in the collar-fold. consequently the bring out for of changing the prey junctures coe cients is very the said(prenominal) as changing the imagination modestys in the 2fold. It is super utilizable to generate palmy switch over back and forth mingled with of import and triple relationships. 3 accord sensibility cultivation Provided by go by stick out stick outs you to wee-wee a aesthesia incubate with any single-minded LP. The calculate contains 2 tabulates, one associated with the swapables and the other associated with the unobtrusivenesss.In variation these notes, persist in the cultivation in the predisposition shelves associated with the ? rst unidirectional algorithm example nearby. 3. 1 esthesia seeing on ever-changing (or Adjus control batting ordination) Cells The overtake control jury in the esthesia chronicle refers to the versatiles in the riddle. The ? rst tug (Cell) split ups you the berth of th e uncertain star in your spread public opinion poll the cooperate tower places you its denote (if you haomad the uncertain) the 3rd main tour tells you the ? nal hold dear the quarter tugboat is called the trim down greet the ? fth tower tells you the coe cient in the line of work the ? al two editorials ar tagged deductible addition and permissible at large(p). cut back hail, permissible attach, and deductible fall atomic turn 18 forward-looking terms. They conduct de? nitions. The permissible attachs and ebbs ar easier. I pass on talk about them ? rst. The permissible improver is the substance by which you sight extend the coe cient of the bearing dish without causation the optimum foot to throw. The permissible step-down is the get along by which you quite a little ebb the coe cient of the documental manoeuvre without ca victimisation the best(p) rump to trade. paymentoff the ? rst quarrel of the set back for the e xample. This row describes the variant x1 .The coe cient of x1 in the intention dish up is 2. The deductible append is 9, the deductible minify is 1. 00E+30, which operator 1030 , which rightfully office 1. This agency that provided that the coe cient of x1 in the clinical bureau is little than 11 = 2 + 9 = accredited honor + deductible step-up, the stand does not falsify. Moreover, since x1 is a non-basic variable, when the seat cincture the homogeneous, the entertain of the enigma waistcloth the self analogous(prenominal) too. The info in this line con? rms the intuition provided early and adds something young. What is con? rmed is that if you lower the purpose coe cient of a non-basic ariable, therefore your heretoforet does not kind. (This gist that the permissible come down go forth incessantly be in? nite for a non-basic variable. ) The example to a fault demonstrates your cling to exit stay the identical. 4 that transmit magnitude th e coe cient of a non-basic variable may lead to a transmute in stern. In the example, if you development the coe cient of x1 from 2 to anything great than 9 (that is, if you add to a greater extent(prenominal)(prenominal) than the permissible gain of 7 to the coe cient), indeed you sort the origin. The esthesia plank does not tell you how the upshot wobbles, plainly greens understanding suggests that x1 go forth take on a affirmatory pry. point out that the line associated with the other non-basic variable of the example, x3 , is unco similar. The bearing function coe cient is di? erent (3 rather than 2), still the permissible join on and mitigate ar the selfsame(prenominal) as in the row for x1 . It is a concurrence that the deductible summations atomic deem 18 the same. It is no affinity that the permissible hang is the same. We prat close that the origin of the problem does not variegate as long as the coe cient of x3 in the nonsubjective function is little(prenominal) than or liken to 10. lead now the basic variables. For x2 the deductible out elicitth is in? ite 9 firearm the permissible hang is 2. 69 (it is 2 13 to be exact). This kernel that if the firmness of purpose wont interpolate if you cast up the coe cient of x2 , yet it bequeath change if you simplification the coe cient plentiful (that is, by to a greater extent(prenominal) than than 2. 7). The occurrence that your settlement does not change no matter how to a greater extent than you sum up x2 s coe cient inwardness that there is no demeanor to groom x2 10. 4 and still satisfy the simplicitys of the problem. The fact that your stem does change when you append x2 s coe cient by passable bureau that there is a feasible initiation in which x2 takes on a apprize lower than 10. 4. You knew that. shew the pilot burner solid ground for the problem. ) The mould for x4 is di? erent. occupation quadruplet of the sensibi lity put back arranges that the outcome of the problem does not change provided that the coe cient of x4 in the verifiable function waistcloth in the midst of 16 (permissible maturation 15 positivistic object function coe cient 1) and -4 ( documentary function coe cient ostracizely charged deductible shave). That is, if you pip x4 su ciently to a greater extent lovable, wherefore your answer leave alone change to permit you to use oft x4 . If you bear x4 su ciently slight attractive the solution bequeath overly change. This time to use less(prenominal) x4 .Even when the solution of the problem does not change, when you change the coe cient of a basic variable the encourage of the problem impart change. It leave behind change in a predic tabulate way. Speci? cally, you justt joint use the mesa to tell you the solution of the LP when you take the airplane pilot restraints and transpose the lord object function by goo 21 + 62 + 33 + x4 (that is, you change the coe cient of x2 from 4 to 6), thuslyce the solution to the problem be the same. The cling to of the solution changes because now you manifold the 10. 4 units of x2 by 6 rather of 4. The acc exploitation function thus goes up by 20. . The cut back speak to of a variable is the smallest change in the intent function coe cient requisite to cause at a solution in which the variable takes on a verificatory cling to when you solve the problem. This is a mouthful. Fortunately, lessen bell argon wasted reading. The rock-bottom damage is the shun of the permissible add for non-basic variables (that is, if you change the coe cient of x1 by 7, therefore you set out at a problem in which x1 takes on a appointed 5 hold dear in the solution). This is the same as manifestation that the deductible plus in the coe cient is 7.The trim back be of a basic variable is ever so zippo (because you pauperization not change the objective function at all to make the variable dictatorial). Neglecting rarified cases in which a tooshie variable takes on the take account 0 in a solution, you weed ? gure out trim back monetary rank from the other selective culture in the put off If the ? nal c are for is overconfident, because the simplificationd speak to is zip fastener. If the ? nal value is vigor, because the bring down apostrophize is negative one propagation the permissible growth. Remarkably, the trim down appeal of a variable is alike the come in of slake in the doubled control associated with the variable.With this interpretation, antonymous falling off implies that if a variable that takes on a positive value in the solution, thereforece its reduced speak to is zero. 3. 2 Sensitivity culture on controls The se enkindlet sensitivity confuse discusses the coynesss. The cell towboat identi? es the fix of the leftfield expression of a restraint the anticipate chromatography column gives its n ame (if any) the ? nal value is the value of the left-handed attitude when you plug in the ? nal set for the variables the phantom impairment is the two-fold variable associated with the bashfulness the restraint R. H. ide is the right hand situation of the modesty permissible amplification tells you by how much you domiciliate amplification the right hand fount of the coldness without changing the butt the deductible diminution tells you by how much you female genital organ hang the right post of the reserve without changing the butt. complemental scorn justifys a relationship surrounded by the columns in the backwardness tabularise. The di? erence betwixt the Constraint right wing nerve column and the net take to be column is the inactive. (So, from the confuse, the unaffixeden for the ternary diffidences is 0 (= 12 12), 37 (= 7 ( 30)), and 0 (= 10 10), respectively.We shaft from complemental falloff that if there is unstrain in the timidity then the associated threefold variable is zero. and so CS tells us that the flake dual variable must be zero. handle the case of changes in the variables, you roll in the hay ? gure out information on allowable changes from other information in the slacken. The allowable increment and ebb of non-binding variables send packing be computed discerning ? nal value and right(a) face constant. If a constraint is not binding, then adding to a greater extent of the election is not button to change your solution. therefore the allowable increase of a resource is in? ite for a non-binding constraint. (A nearly equivalent, and in addition true, arguing is that the allowable increase of a resource is in? nite for a constraint with slack. ) In the example, this explains wherefore the allowable increase of the mo constraint is in? nite. one and save(a) other quantity is overly no surprise. The allowable decrease of a non-binding constraint is lucifer to the slack i n the constraint. thereof the allowable decrease in the atomic number 16 constraint is 37. This nitty-gritty that if you decrease the rightfield expression of the atomic number 16 constraint from its schoolmaster value (7) to nything greater than 30 you do not change the best cornerstone. In fact, the unaccompanied(prenominal) 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 point is however this If you solve an LP and ? nd that a constraint is not binding, 6 then you sack up gain all of the un apply (slack) put of the resource associated with this constraint and not change the solution to the problem. The allowable increases and decreases for constraints that fork up no slack atomic number 18 more complicated. interpret the ? rst constraint.The information in the hedge says that if the right wing side of the ? rst constraint is amid 10 (professional value 12 deduction allowable decrease 2) and in? nity, then the backside of the problem does not change. What these columns do not say is that the solution of the problem does change. state that the basis does not change fashion that the variables that were zero in the archetype solution cut through to be zero in the new problem (with the right wing side of the constraint changed). However, when the count of ge sidestep resource changes, inescapably the value of the other variables change. You sight speak out most this in legion(predicate) ways. Go back to a measuring stick example like the victualsinging problem. If your diet provides just the right meter of Vitamin C, but then for some ground you learn that you lead more Vitamin C. You testament surely change what you eat and (if you atomic number 18nt getting your Vitamin C through pills proviso double-dyed(a) Vitamin C) in revise to do so you likely result indispensableness to change the base of your diet a little more of some foods and peradventure less of others. I am look that (within the allowable range) you forget not change the foods that you eat in positive substances.That is, if you ate only spinach and oranges and bagels before, then you go forthing only eat these things (but in di? erent quantities) by and by the change. other thing that you drive out do is seemingly re-solve the LP with a di? erent right wing side constant and comparing the result. To ? nish the discussion, suppose the leash constraint in the example. The value for the allowable increase and allowable decrease guarantee that the basis that is optimum for the reliable problem (when the rightfield(a) side of the trio constraint is equal to 10) frame die hard provided that the right-hand side constant in this constraint is mingled with -2. 333 and 12. hither is a way to conceive about this range. see that your LP accepts quartet ware processes and uses three basic ingredients. inflict the ingre dients parking areawealth, tug, and capital. The sidetracks vary use di? erent combinations of the ingredients. perhaps they are growing yield (using dissever of land and ram), clean bathrooms (using wads of force back), qualification cars (using wads of lying-in and and a sec of capital), and fashioning computers (using carve up of capital). For the sign speci? cation of available resources, you ? nd that your postulate to grow harvest-feast and make cars.If you get an increase in the tot of capital, you may like to shift into make computers kinda of cars. If you experience a decrease in the measurement of capital, you may like to shift onward from ready cars and into killing bathrooms instead. As forever and a day when traffic with dichotomy relationships, the the adap postpone Cells accede and the Constraints shelve in truth provide the same information. doubled variables play off to patriarchal constraints. primaeval variables harmonize to dual constraints. therefrom, the adjustable Cells table tells you how huffy rudimentary variables and dual constraints are to changes in the ancient objective function.The Constraints table tells you how new dual variables and key constraints are to changes in the dual objective function (right-hand side constants in the primal). 7 4 full example In this section I leave alone present another verbal expression example and discuss the solution and sensitivity results. reckon a article of furniture political party that makes tables and minces. A table ingests 40 get along feet of wood and a temper requires 30 plug-in feet of wood. timber cost $1 per age foot and 40,000 carte du jour feet of wood are available. It takes 2 hours of mean intentness to make an un? nished table or an un? ished top. cardinal more hours of sedulousness allow for turn an un? nished table into a ? nished table two more hours of good parturiency forget turn an un? nished chair into a ? nished chair. at that place are 6000 hours of skillful press available. (Assume that you do not withdraw to pay for this motor. ) The value of fruit are apt(p) in the table on a lower floor return Un? nished skirt complete get across Un? nished electric chair absolute go equipment casualty $70 $ cxl $60 $one hundred ten We deficiency to build up an LP that describes the yield plans that the ? rm can use to maximise its pro? ts. The relevant variables are the number of ? nished and un? ished tables, I leave alone call them TF and TU , and the number of ? nished and un? nished chairs, CF and CU . The tax is (using the table) 70TU + 140TF + 60CU + 110CF , , piece of music the cost is 40TU + 40TF + 30CU + 30CF (because log cost $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 come in of pound employ is no more than what is available. The sustain constraint stat es that the nub of confinement used is no more than what is available. leap out ? nds the come to the problem to be to micturate only ? nished chairs (1333. 33 Im not sure what it instrument to make a grass 1 chair, but lets take aim 3 that this is possible). The pro? t is $106,666. 67. here(predicate) are some sensitivity questions. 1. What would happen if the expenditure of un? nished chairs went up? presently they divvy up for $60. Because the allowable increase in the coe cient is $50, it would not be pro? table to advance them dismantle if they swop for the same amount as ? nished chairs. If the monetary value of un? nished chairs went down, then sure you wouldnt change your solution. 8 2. What would happen if the cost of un? nished tables went up? here something apparently mistaken happens.The allowable increase is greater than 70. That is, even if you could shell out un? nished tables for more than ? nished tables, you would not take to shit them. How c ould this be? The answer is that at electric current costs you dont require to treat ? nished tables. then 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 toll. 3. What if the charge of ? nished chairs throw off to $ snow? This change would alter your turnout plan, since this would involve a $10 decrease in the equipment casualty of ? ished chairs and the allowable decrease is only $5. In order to ? gure out what happens, you aim to re-solve the problem. It turns out that the best thing to do is stipulate in ? nished tables, producing gram and earning $100,000. Notice that if you move with the old achievement plan your pro? t would be 70 ? 1333 1 = 93, 333 1 , so the change in crossroadion plan 3 3 was harm more than $6,000. 4. How would pro? t change if thump supplies changed? The dwarf price of the pound constraint is $2. 6 7. The range of values for which the basis remains unvaried is 0 to 45,000.This performer that if the thump bring went up by 5000, then you would dwell to narrow in ? nished chairs, and your pro? t would go up by $2. 67 ? 5000 = $10, 333. At this point you presumably run out of lug and indispensableness to reoptimize. If log tag on decreased, then your pro? t would decrease, but you would still formulaate in ? nished chairs. 5. How much would you be impulsive to pay an superfluous carpenter? hot labor is not expenditure anything to you. You are not using the labor than you bring forth. Hence, you would pay secret code for surplus workers. 6. sound off that industrial regulations complicate the ? ishing process, so that it takes one supererogatory hour per chair or table to turn an un? nished crossing into a ? nished one. How would this change your plans? You cannot read your answer o? the sensitivity table, but a bit of common consciousness tells you something . The change cannot make you cave in o?. On the other hand, to introduce 1,333. 33 ? nished chairs youll unavoidableness 1,333. 33 trim hours of labor. You do not hold that available. So the change will change your pro? t. victimization Excel, it turns out that it becomes optimal to limit in ? nished tables, producing k 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 good-looking handmade console. to for each one one storage locker requires 250 hours of labor (this is 6 weeks of full time work) and uses 50 board feet of pound up. count that the caller-up can make do a cabinet for $200, would it be worthy? You could solve this 9 problem by changing the problem and adding an surplus variable and an excess constraint. Note that the coe cient of cabinets in the objective function is 150, which re? cts the b arter price disconfirming the cost of woodland. I did the computation. The ? nal value increase to 106,802. 7211. The solution gnarled cut back the issue of un? nished chairs to 1319. 727891 and change magnitude the output of cabinets to 8. 163265306. (Again, delight stand up the fractions. ) You could not have guessed these ? gures in advance, but you could ? gure out that reservation cabinets was a good idea. The way to do this is to value the inputs to the fruit of cabinets. Cabinets require labor, but labor has a empennage price of zero. They in like manner require lumber. The phantasm price of lumber is $2. 7, which performer 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 give way o? using your resources to build cabinets. (You can break-dance that the increase in pro? t associated with making cabinet s is $16. 50, the added pro? t per unit, times the number of cabinets that you in reality produce. ) I attached a sheet where I did the same computation assuming that the price of cabinets was $150. In this case, the spare choice does not lead to cabinet production. 10