Can Synthetic Biology Solve Our Problems - 1161 Words

Can synthetic biology solve our problems in biofuel production? In recent years the increased interest in biofuel production has been sparked by the environmental damage, economic impact and dwindling reserves of petroleum-based fuels and chemicals [1] causing a huge influx in investment in improving biofuel production processes. Synthetic biology has allowed biofuel production to advance by providing new methods in which they can create optimal biocatalysts for sustainable and efficient production of biofuels [2]. Synthetic biology itself is an interdisciplinary branch of biology concerned with designing, engineering and synthesising novel biological systems that are not found in nature, as well as redesigning existing ones. [3] It has also been said that is has enabled scientists to â€Å"create life from scratch† and has allowed scientists to not only better understand the underlying mechanisms involved in such processes but the basic principles of biology as a whole [4]. One particular popular definition states it is theâ€Å"designing and constructing of biological devices, biological systems, and biological machines for useful purposes.†[5]. A biofuel, by definition, is a fuel that’s made from living things or their waste [6] and the most commonly extracted and used is ethanol and diesel and are usually produced from crops such as wheat, soybeans, corn and sugarcane [7]. Obviously, as prices of crude oil are soaring biofuel is becoming more and more in demand and it does haveShow MoreRelatedBioengeneering: Improving Health and Lifestyle for Humans874 Words   |  4 Pagesimprove the health of humans by applying biology in engineering. Imagine a world without sick people, or people with deformity. This may be hard to imagine, but with the remarkable inventions and solutions developed and produced by bioengineers, this scenario we can currently only imagine in our heads will hopefully eventually become an ordinary norm. It is crucial to help people who were born with a body that restricts them from doing something everyone else can. They deserve to be able to move andRead MoreHistory of Engineering1060 Words   |   5 Pagesfor ingenium which means innate quality, especially mental power. Engineer dates back to 1325 when an engine’er, someone who operates an engine, was referred to by a conductor as an engineer. (Ford) Engineers go through much education before they can regularly get a job. Most engineering jobs require 4 years at a university. A degree in the appropriate field is highly recommended. Courses offering hands on training are important. Engineers usually must be registered and/or licensed. EngineersRead MoreThe Problem Of Global Poverty1609 Words   |  7 PagesPoverty is one of the largest dilemmas plaguing the world today. Solutions to solving the global issue of poverty are constantly debated, as world leaders try to find the best possible approach. However, in order to work towards solving the problem of global poverty, we must first identify the key cause. The main source of global poverty is the inaccessibility to food as a necessary biological need, otherwise known as world hunger. Food is necessary in order to live. It provides fuel for basic biologicalRead MoreBio Hackers : Saving The World !899 Words   |  4 PagesSaving the World! A chip that slips in under the skin to track a person’s critical information such as heart rate, pulse, temperature and other signs is currently being created and tested by Bio-hackers. Another Bio-hacker is developing crops that can resist cold temperatures such as orange crops. Biohackers are making a difference today with their eyes on tomorrow. Bio-hackers are tinkering with the DNA of existing organisms to create new ones and will lead to innovations of a biological natureRead MoreWhat Is Nature Or What It?1491 Words   |  6 Pagesespecially when discussing the topic of what is nature or what is natural. We all have different connotations attached to the word nature. None of these connotations are wrong they are just different. They are a product of our religion, our biology, and the experiences of our life. There have been many interesting pieces of writing that have ether directly or indirectly defined nature to us what nature is. Whether it is an acenet story passed down though oral tradition through out generations orRead MoreDevelopment And Growth Of The Global Agricultural Biotechnology Industry1171 Words   |  5 Pagesto of worthy of amassing 29.3 billion US dollars by the end of the year 2020 growing at a CAGR rate of 9.5% during the predicted period from the year 2013 to 2020. The global as the market has been separated into transgenic crops, tools, and synthetic biology-enabled items based upon its functions and uses Demand for food manufacture has grown owing to large population resulting in the requirement of more food to fulfill the food needs of the worlds people. This is a critical factor contributingRead MoreWhat Is Nature Or What It?1480 Words   |  6 Pagesnature or what is natural because it is not a question that has a clear answer. We all have different connotations attached to the word nature. None of these connotations are wrong. They are just different. They are a product of our religion, our biology, and the experiences of our life. There have been many provocative thoughts expressed about what is natural that have either directly or indirectly defined nature to us. Whether it is an ancient story passed down through oral tradition throughout generationsRead MoreImpacts of Applications of Chemistry on Society and the Environment3915 Words   |  16 Pagestimes, Chemistry has played a pivotal role in the advancement and enrichment of civilization, although sometimes it has also caused harmful and occasional long-reaching catastrophic effects on the environment. The importance of this sphere of science can be demonstrated by the fact that entire periods in history were named the Iron Age and the Bronze Age, according to the level of chemical endeavor of that time. The content in this report will comment on the various implications of science on societyRead MoreUnit 1 KEY QUESTIONS Essay2265 Words   |  10 Pagesdiscover more about biology and to gain insight on plant and animal species. The stated intent of the voyage was to obtain evidence that supported the biblical theory of creation as well as chart poorly known parts of the South American coastline. 2. Why does the Antibiotic resistance problem represent an example of evolution? The antibiotic problem is a perfect example of evolution because it shows how species have adapted and evolved based on their environment. It shows how bacteria can adjust and eitherRead MoreTechnological Singularity1755 Words   |  8 PagesRaymond Kurzweil, one of the leading inventors of our time, in his most recent futurist manifesto: â€Å"The Singularity Is Near: When Humans Transcend Biology† (2005). Kurzweil estimates that machines will inevitably be able to surpass our thinking capabilities within a few decades. Kurzweils speculative reasoning has been heavily debated and challenged. In Aamodt and Wangs article they point out that there are fundamental differences between our brains and computers that makes Kurzweils predictions

Sensitivity Analysis Free Essays

string(30) " solution remains the same\)\." Linear Programming Notes VII Sensitivity Analysis 1 Introduction When you use a mathematical model to describe reality you must make approximations. The world is more complicated than the kinds of optimization problems that we are able to solve. Linearity assumptions usually are signi? cant approximations. We will write a custom essay sample on Sensitivity Analysis or any similar topic only for you Order Now Another important approximation comes because you cannot be sure of the data that you put into the model. Your knowledge of the relevant technology may be imprecise, forcing you to approximate values in A, b, or c. Moreover, information may change. Sensitivity analysis is a systematic study of how sensitive (duh) solutions are to (small) changes in the data. The basic idea is to be able to give answers to questions of the form: 1. If the objective function changes, how does the solution change? 2. If resources available change, how does the solution change? 3. If a constraint is added to the problem, how does the solution change? One approach to these questions is to solve lots of linear programming problems. For example, if you think that the price of your primary output will be between $100 and $120 per unit, you can solve twenty di? rent problems (one for each whole number between $100 and $120). 1 This method would work, but it is inelegant and (for large problems) would involve a large amount of computation time. (In fact, the computation time is cheap, and computing solutions to similar problems is a standard technique for studying sensitivity in practice. ) The approach that I will describe in these notes takes full adva ntage of the structure of LP programming problems and their solution. It turns out that you can often ? gure out what happens in â€Å"nearby† linear programming problems just by thinking and by examining the information provided by the simplex algorithm. In this section, I will describe the sensitivity analysis information provided in Excel computations. I will also try to give an intuition for the results. 2 Intuition and Overview Throughout these notes you should imagine that you must solve a linear programming problem, but then you want to see how the answer changes if the problem is changed. In every case, the results assume that only one thing about the problem changes. That is, in sensitivity analysis you evaluate what happens when only one parameter of the problem changes. 1 OK, there are really 21 problems, but who is counting? 1 To ? x ideas, you may think about a particular LP, say the familiar example: max 2Ãâ€"1 subject to 3Ãâ€"1 x1 2x 1 + + + 4Ãâ€"2 x2 3Ãâ€"2 x2 + + + + 3x 3 x3 2x 3 3x 3 + + + x4 4x 4 3x 4 x4 x ? ? ? 12 7 10 0 We know that the solution to this problem is x0 = 42, x1 = 0; x2 = 10. 4; x3 = 0; x4 = . 4. 2. 1 Changing Objective Function Suppose that you solve an LP and then wish to solve another problem with the same constraints but a slightly di? erent objective function. (I will always make only one change in the problem at a time. So if I change the objective function, not only will I hold the constraints ? ed, but I will change only one coe cient in the objective function. ) When you change the objective function it turns out that there are two cases to consider. The ? rst case is the change in a non-basic variable (a variable that takes on the value zero in the solution). In the example, the relevant non-basic variables are x1 and x3 . What happens to your solution if the coe cient of a non-basic variable decreases? For example, suppose that the coe cient of x1 in the objective function above was reduced from 2 to 1 (so that the objective function is: max x1 + 4Ãâ€"2 + 3Ãâ€"3 + x4 ). What has happened is this: You have taken a variable that you didn’t want to use in the ? rst place (you set x1 = 0) and then made it less pro? table (lowered its coe cient in the objective function). You are still not going to use it. The solution does not change. Observation If you lower the objective function coe cient of a non-basic variable, then the solution does not change. What if you raise the coe cient? Intuitively, raising it just a little bit should not matter, but raising the coe cient a lot might induce you to change the value of x in a way that makes x1 0. So, for a non-basic variable, you should expect a solution to continue to be valid for a range of values for coe cients of nonbasic variables. The range should include all lower values for the coe cient and some higher values. If the coe cient increases enough (and putting the variable into the basis is feasible), then the solution changes. What happens to your solution if the coe cient of a basic variable (like x2 or x4 in the example) decreases? This situation di? ers from the previous one in that you are using the basis variable in the ? rst place. The change makes the variable contribute less to pro? . You should expect that a su ciently large reduction makes you want to change your solution (and lower the value 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 2Ãâ€"1 +2Ãâ€"2 +3Ãâ€"3 + x4 ), 2 maybe you would want to set x2 = 0 instead of x2 = 10. 4. On the other hand, a sma ll reduction in x2 ’s objective function coe cient would typically not cause you to change your solution. In contrast to the case of the non-basic variable, such a change will change the value of your objective function. You compute the value by plugging in x into the objective function, if x2 = 10. 4 and the coe cient of x2 goes down from 4 to 2, then the contribution of the x2 term to the value goes down from 41. 6 to 20. 8 (assuming that the solution remains the same). You read "Sensitivity Analysis" in category "Essay examples" If the coe cient of a basic variable goes up, then your value goes up and you still want to use the variable, but if it goes up enough, you may want to adjust x so that it x2 is even possible. In many cases, this is possible by ? nding another basis (and therefore another solution). So, intuitively, there should be a range of values of the coe cient of the objective function (a range that includes the original value) in which the solution of the problem does not change. Outside of this range, the solution will change (to lower the value of the basic variable for reductions and increase its value of increases 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 discussed this topic when we talked about duality. I argued that dual prices capture the e? ct of a change in the amounts of available resources. When you changed the amount of resource in a non-binding constraint, then increases never changed your solution. Small decreases also did not change anything, but if you decreased the amount of resource enough to make the constraint binding, your solution could change. (Note the similarity between this analysis and the case of changing the coe c ient of a non-basic variable in the objective function. Changes in the right-hand side of binding constraints always change the solution (the value of x must adjust to the new constraints). We saw earlier that the dual variable associated with the constraint measures how much the objective function will be in? uenced by the change. 2. 3 Adding a Constraint If you add a constraint to a problem, two things can happen. Your original 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 have a solution. If the original solution does not satisfy the new constraint, then possibly 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 well as before. If not, then you will do worse. 2 2 There is a rare case in which originally your problem has multiple solutions, but only some of them satisfy the added constraint. In this case, which you need not worry about, 3 2. 4 Relationship to the Dual The objective function coe cients correspond to the right-hand side constants of resource constraints in the dual. The primal’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 really the same as changing the resource constraints in the dual. It is extremely useful to become comfortable switching back and forth between primal and dual relationships. 3 Understanding Sensitivity Information Provided by Excel Excel permits you to create a sensitivity report with any solved LP. The report contains two tables, one associated with the variables and the other associated with the constraints. In reading these notes, keep the information in the sensitivity tables associated with the ? rst simplex algorithm example nearby. 3. 1 Sensitivity Information on Changing (or Adjustable) Cells The top table in the sensitivity report refers to the variables in the problem. The ? rst column (Cell) tells you the location of the variable in your spreadsheet; the second column tells you its name (if you named the variable); the third column tells you the ? nal value; the fourth column is called the reduced cost; the ? fth column tells you the coe cient in the problem; the ? al two columns are labeled â€Å"allowable increase† and â€Å"allowable decrease. † Reduced cost, allowable increase, and allowable decrease are new terms. They need 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 causing the optimal basis to change. Th e 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 means 1030 , which really means 1. This means 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 stays the same, the value of the problem stays 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 means 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 increasing the coe cient of a non-basic variable may lead to a change in basis. In the example, if you increase the coe cient of x1 from 2 to anything greater than 9 (that is, if you add more 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 positive value. Notice that the line associated with the other non-basic variable of the example, x3 , is remarkably 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. Consider 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 means that if the solution won’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 minus allowable decrease). That is, if you make x4 su ciently more attractive, 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 2Ãâ€"1 + 6Ãâ€"2 + 3Ãâ€"3 + x4 (that is, you change the coe cient of x2 from 4 to 6), then the solution to the problem remains the same. The value of the solution changes because now you multiply the 10. 4 units of x2 by 6 instead of 4. The objective function therefore goes up by 20. . The reduced cost of a variable is the smallest change in the objective function coe cient needed to arrive at a solution in which the variable takes on a positive value when you solve the problem. This is a mouthful. Fortunately, reduced costs are redundant information. The reduced cost is the negative of the allowable increase for non-basic variables (that is, if y ou 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 saying that the allowable increase in the coe cient is 7. The reduced cost of a basic variable is always zero (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 costs 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 slack in the dual constraint associated with the variable. With this interpretation, complementary slackness 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 Constraints The second sensitivity table discusses the constraints. The cell column identi? es the location of the left-hand 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 price 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 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 guarantees a relationship between the columns in the constraint table. The di? erence between the â€Å" Constraint Right-Hand Side† column and the â€Å"Final Value† column is the slack. (So, from the table, the slack for the three constraints is 0 (= 12 12), 37 (= 7 ( 30)), and 0 (= 10 10), respectively. We know from Complementary Slackness that if there is slack in the constraint then the associated dual variable is zero. Hence CS tells us that the second 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 example, this explains why the allowable increase of the second constraint is in? nite. One other quantity is also no surprise. The allowable decrease of a non-binding constraint is equal to the slack in the constraint. Hence t he 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 point is only this: If you solve an LP and ? nd that a constraint is not binding, 6 then you can remove all of the unused (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 diet problem. If your diet provides exactly the right amount of Vitamin C, but then for some reason you learn that you need more Vitamin C. You will certainly change what you eat and (if you aren’t getting your Vitamin C through pills supplying pure Vitamin C) in order to do so you probably will need to change the comp osition of your diet – a little more of some foods and perhaps less of others. I am saying 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. Here is a way to think about this range. Suppose that your LP involves four production processes and uses three basic ingredients. Call the ingredients land, labor, and capital. The outputs vary use di? erent combinations of the ingredients. Maybe they are growing fruit (using lots of land and labor), cleaning bathrooms (using lots of labor), making cars (u sing lots of labor and and a bit 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 building 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 duality 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 how 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 present another formulation example and discuss the solution and sensitivity results. Imagine a furniture company that makes tables and chairs. A table requires 40 board feet of wood and a chair requires 30 board feet of wood. Wood costs $1 per board foot and 40,000 board feet of wood are available. It takes 2 hours of skilled labor 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 skilled 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 for this labor. ) The prices of output are given in the table below: Product Un? nished Table Finished Table Un? nished Chair 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 revenue is (using the table): 70TU + 140TF + 60CU + 110CF , , while the cost is 40TU + 40TF + 30CU + 30CF (because lumber 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 lumber 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 sell 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 sell for $60. Because the allowable increase in the coe cient is $50, it would not be pro? table to produce 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 apparently 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 best 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 worth 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 point 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 additional carpenter? Skilled labor is not worth 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 hour s 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, producing 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 hand-crafted cabinet. 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 worthwhile? 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. 72 7891 and increasing the output of cabinets to 8. 163265306. (Again, please tolerate the fractions. ) You could not have guessed these ? gures in advance, but you could ? gure 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 do es not lead to cabinet production. 10 How to cite Sensitivity Analysis, Essay examples

Pak Railway Thesis free essay sample

Railways,  lifeline  of the country, is a  national  state-run transport service. It is under the administration of federal government and its head quarter is in Lahore. It is an important source of  transportation  throughout Pakistan. It carries millions of passengers throughout the country. It used to carry huge freight in Pakistan. This cheap and safe mode for passengers is now facing a number of issues. A number of services of Pakistan Railways have been cancelled, suspended or terminated and many more will be suspended in near future because of mismanagement and shortage of locomotives, fuel and money. The chapter of all major services, from Lahore to Karachi, has been closed. It is pertinent to mention that all  AC serviceshave been stopped. The incompetent administration has failed to attain locomotives from any quarter of the world. Passengers are suffering due to mismanagement of administration. Pakistan Railways decision to suspend goods  train  service due to severe shortage of locomotives and fuel is another blow to this organization. It is now basically financially bankrupt organization. In other words it is on the verge of financial collapse. The political interference, nepotism, corruption, poor maintenance of tracks bridges and mismanagement in almost every field are the major causes of failure of Pakistan Railways. Pakistan Railways purchased 69 completely built locomotive units from China under 2003 agreement. These are about 37% cheaper than the European locomotives but considered to be faulty. It is stated that 32 of these have already been scraped. Dong Fang  Electric  Corporation  has been severely criticized for producing low quality locomotives. The other viewpoint is that misuse of the machinery was the major cause of the failure of Chinese locomotives. According to Sheikh Rashid, the former railway minister, crankshafts of locomotives worth Rs10 million were damaged because of the use of substandard lubrication oil. It may be mentioned here that normally a locomotive consists of six traction motors while the Pakistan Railways is operating them with only three or four motors. This is the major cause of mid-way breakdown of trains. The passengers, in such a case, have to wait for a long time till repair or replacement of faulty  engine  takes place. Naturally trains are too late and passengers can be seen sitting at platforms with their  luggage. A number of trains lack facility of light at nights because of the failure of the  generators  and ill attitude of management. Another reason that prevents people to go by  train  is increase in corruption by the ticketing officials. The reservation of birth is an uphill task. There are  complaints  that reservation is confirmed after receiving bribe of hundred or more rupees by passengers. Pakistan Railways is no more the best choice to travel for the passengers. Haji Ghulam Ahmed Bilour, federal minister for Pakistan Railways, is a very controversial figure. He is considered to be somewhat responsible for the deteriorating situation of Pakistan Railways. His viewpoint is that the whole railway system is obsolete. He complains that half of the total locomotives are out of order. Almost 86 % bridges are more than 100 years old. The trains, tracks and machinery are outdated or faulty. He says that Pakistan needs 25 to 30  engines  annually. He is now making a plan to repair, hire or lease locomotives in collaboration with the private sector. Moreover, he believes that a bailout package can be helpful to overpower the crisis. According to him delay in funds to Railways is the root cause of the crisis. He accuses the federal  government for  not releasing remarkable funds for the betterment of Pakistan Railways. All his plans may end in failure because of corruption in the management, financial problems, his ill-advised attitude and lack of vision. The efficiency of the railway minister is evident from its almost nil performance. He believes that two mafia gangs are very strong in Pakistan Railways but finds himself incapable to take any action against them. He seems to be too weak to solve the issues and problems faced by his ministry. The Chief Justice has expressed his disapproval for the high-ups of Railways in the following comments: â€Å"Ninety metric tons of  silver  worth millions was sold for mere Rs28,000 as scrap, while  a light bulb  worth Rs60 is being purchased at Rs400, whereas absence of maintenance turned expensive locomotives into junk one by one, besides a Grade-18 officer, a blue-eyed boy of the railways minister, is promoted to hold a Grade-20 post of secretary purchase. In the words of Chief Justice â€Å" the electricity wires meant for  electric  trains from Lahore to Khanewal have been stolen. † Moreover, he observed that â€Å"tickets were sold in advance outside  ticket  counters and tickets were not available at railway stations. † It is also in the notice of the apex court that land mafia has grabbed Pakistan Railways land in different areas of the country. He has already order ed the railways to approach the Sindh administration in this connection. Nonetheless, everybody knows the efficiency of the present Sindh government. Railways needed Rs2. 2 billion to pay the salaries and pension to its protesting employees but the government has not released enough money to overpower the deep financial crisis. The economy of the country is under severe pressure and the poor administration seems to be unable to solve the problem. The apex court has already remarked â€Å"Railways should take steps to make it functional as early as possible. † Sheikh Rashid Ahmad, former railway minister, blames corrupt officers of the department for the present situation in Pakistan Railways. According to him only 156 locomotives out of 500 are in normal working condition. According to his statement 15,000 freight  wagons  are not plying in the country and the business has gone in the hands of the private sector. He claims that 200  coaches  and several locomotives, imported from China, could have been manufactured in the local Carriage Factory in Islamabad. Due to imported  coaches  and machinery several labourers have lost their  jobs  or sent on forced leave. A limited number of loaders have been appointed by the administration on the platforms to carry the  luggage  of the passengers. It is stated that a particular amount of money is received by the administration as a bribe from these poor loaders. In return they are allowed to receive hundred rupees from the passenger to carry the  luggage. A private firm has been working under the contract to transport goods including medicines,carpets,  furniture and  electronic appliances through Pakistan Railways. The monopoly of this firm is an obstacle in the free trade of different goods. Pakistan Railways should create competitive atmosphere to provide the customers with more facilities. Pakistan Railways losses have reached billions of rupees. It seems difficult to bring trains back on tracks in the present circumstances. The situation may not change unless corrupt high-ups are removed, suspended and dismissed from the services. To restructure and modernise Pakistan Railways under the present administration seems to be the dream of a mad man. Pakistan Railways is sinking in the sea of corruption as no serious efforts are being made to eradicate it from this department. PAKISTAN RAILWAY CRISIS AND ITS SOLUTIONS Among the means of transportation railways are the cheapest and safest mode for passengers and goods. It also helps in growth of economy for the country. A plan for a rail system in  Pakistan  was first proposed in 1858. A survey for railway line was initiated by Commissioner of Sindh ,Sir Henry Edward Fere in 1858. It was proposed that a railway line from  Karachi  City  to Kotri, steam navigation from the Indus /Chenab up to  Multan  and from there another railway to  Lahore  and beyond be constructed. Thus, it was on 13th May, 1861 that first railway line was opened for public traffic between  Ã‚  Ã‚  Karachi  City  and Kotri, the distance of 105 miles. The line between  Karachi  Cityand Kemari was opened in 1889   and   by 1897 the line from Kemari to Kotri was doubled. Since 1861 when the first railway line was laid down between  Karachi  and Kotri, the expansion of the railway network by the British came at a rapid pace up until 1947. The driving factors for this growth were strategic and economic in nature. For instance to stop the invasion of the Russians from the West, the British built the Khojak tunnel, the fourth largest at that time, in seemingly inaccessible areas of Balochistan to reach Chaman railway station.