Mathematics 23 - Applied Matrix Algebra Supplement 1. Application: Production Planning

Transcription

1 Mathematics - Applied Matrix Algebra Supplement Application: Production Planning A manufacturer makes three different types of chemical products: A, B, and C. Each product must go through two processing machines, X and Y. The products require the following times in machines X and Y : () One ton of A requires hours in machine X and hours in machine Y. () One ton of B requires hours in machine X and hours in machine Y. () One ton of C requires hours in machine X and hours in machine Y. Machine X is available 8 hours per week and machine Y is available hours per week. Since management does not want to keep the expensive machines X and Y idle, it would like to know how many tons of each product to make so that the machines are fully utilized. It is assumed that the manufacturer can sell as much of each product as is made. To solve this problem, let x, x, x denote the number of tons of products A, B, and C, respectively, to be made. The number of hours that machine X will be used is which must equal 8, so we get the equation x + x + x, x + x + x = 8. Similarly, the number of hours machine Y will be used is x + x + x =. Mathematically, the problem is to find a solution set for the system of equations x + x + x = 8 x + x + x = Since we have variables and only equations, this system will have many (in fact, infinitely many) possible solutions. We need to find a parametric representation for the solution set of this system. Suppose we let x = t, so that t is the number of tons of C that is produced. First, notice that it is impossible (even nonsensical!) to make a negative amount of anything. This indicates that the smallest amount of C that can be made is. Now, since even when x, x are both, x cannot be over, it is clear that we must have t. For x =, we obtain the system which we can solve to get x + t = 8 x + t = x = t. Substituting this into one of the equations of the previous system, we obtain x + ( t) + t = 8

2 Math - Applied Matrix Algebra which simplifies to x + t = or x = t Thus we see that all solutions are given by x = t x = t x = t where t. For example, when t =, the original system is satisfied by x = 5, x =, x = and when t =, the original system is satisfied by x = 7, x =, x =. Any such solution is as good as any other, and there are infinitely many of them. There is no best solution unless additional information or restrictions are given. For each of the following problems, you should note whether the system has a solution set consisting of exactly one solution no solution infinitely many solutions. Problem. An oil refinery produces low-sulphur and high-sulphur fuel. Each ton of lowsulfur fuel requires 5 minutes in the blending plant and minutes in the refining plant; each ton of high-sulfur fuel requires minutes in the blending plant and minutes in the refining plant. If the blending plant is available for hours and the refining plant is available for hours, how many tons of each type of fuel should be manufactured so that the plants are fully utilized? Problem. A dietician is preparing a meal consisting of foods A, B, and C. The nutritional information for these types of food is (in units per ounce of food): A B C protein fats carbohydrates If the meal should provide exactly 5 units of protein, units of fat, and units of carbohydrate, how many ounces of each type of food should be used? Problem. An inheritance of $, is to be divided among three trusts, with the second trust receiving twice as much as the first trust. The three trusts pay interest at the rates of 9%, %, and % annually, respectively, and return a total in interest of $ at the end of the first year. How much was invested in each trust?

3 Mathematics - Applied Matrix Algebra Supplement Application: Economic Models for Exchange of Goods. Suppose that in a primitive society, the members of a tribe are engaged in three occupations: farming, manufacturing of tools and utensils, and the weaving and sewing of clothing. Assume that initially the tribe has no monetary system and that all goods and services are bartered. Let the three groups be denoted by F, M, and C, respectively. Suppose that this directed graph indicates how the bartering system works in practice: / F / M / C This figure indicates that the farmers keep half of their produce and give one quarter of their produce to the manufacturers and on quarter to the clothing producers. The manufacturers divide the goods evenly among three groups, one third goes to each (including themselves). The group producing clothes gives half of the clothes too the farmers and divides the other half evenly between themselves and the manufacturers. The results are summarized in this table: F M C F M C The first column of the table indicates the distribution of the goods produced by the farmers, the second column indicates the distribution of the manufactured goods, and the third column indicates the distribution of the clothing. As the size of the tribe grows, the system of bartering becomes too cumbersome and, consequently, the tribe decides to institute a monetary system of exchange. For this simple economic system, we assume that there will be no accumulation of capital or debt and that the prices for each of the three types of goods will reflect the values of the existing bartering system. The question is how to assign values to the three types of goods that fairly represent the current bartering system. This problem can be turned into a linear system using a model that was originally developed by the Nobel prize winning economist Wassily Leontief, a model that we will study more extensively later. For now, let x be the monetary value of the goods produced by the farmers, x be the value of the manufactured goods, and x be the value of the clothing produced. According to the first row of the table, the value of the goods received by the farmers amounts to half the value of the farm goods produced, plus one-third the value of

4 Math - Applied Matrix Algebra Supplement the manufactured products, and half the value of the clothing goods. Thus the total value of the goods received by the farmers is x + x + x. If the system is fair, the total value of goods received by the farmers should equal x, the total value of the farm goods produced. Thus we have the linear equation x + x + x = x. Using the second row of the table and equating the value of the goods produced and received by the manufacturers, we obtain a second equation: x + x + x = x. Finally, the third row of the table yields: x + x + x = x. These equations can be rewritten as a homogeneous system: x + x + x = x x + x = x + x x = The reduced row-echelon form of the augmented matrix for this system is 5. We can find a parametric representation for the solution set of this system by letting x = t. Then we immediately get x = t and x = 5 by back-substitution. It follows that the variables x, x, x should be assigned values in the ratio x : x : x = 5 : : Consider the parallel with modern currency systems; it doesn t matter what the actual units of money are, just that one can find a fair system of exchange. This simple system is an example of the closed Leontief input-output model. Leontief s models are fundamental to our understanding of economic systems. Problem. Determine the relative values of x, x, x if the distribution of goods is as described in this table: F M C F M C

5 Mathematics - Applied Matrix Algebra Supplement Application: Production Costs. A company manufactures three products. Its production expenses are divided into three categories. In each category, an estimate is given for the cost of producing a single item of each product. An estimate is also made of the amount of each product to be produced per quarter. These estimates are given in Tables and : Product Expenses A B C Raw materials...5 Labor...5 Overhead...5 Table. Production Costs Per Item (dollars) Season Product Summer Fall Winter Spring A 5 5 B C 58 Table. Amount Produced Per Quarter The company would like to present at their stockholders meeting a single table showing the total costs for each quarter in each of the three categories: raw materials, labor, and overhead. Solution. Consider the tables as represented by the matrices M =...5 and P = If we form the product MP, the first column will represent the costs for the summer quarter: Raw materials: (.)() + (.)() + (.5)(58) = 87 Labor: (.)() + (.)() + (.5)(58) = 5 Overhead: (.)() + (.)() + (.5)(58) = 7

6 Math - Applied Matrix Algebra Supplement The second column will represent the costs for the fall quarter: Raw materials: (.)(5) + (.)() + (.5)() = Labor: (.)(5) + (.)() + (.5)() = 9 Overhead: (.)(5) + (.)() + (.5)() = 9 Columns and of MP represent the costs for the winter and spring quarters MP = The entries in the first row of MP represent the total cost of raw materials for each of the four quarters. The entries in rows and represent the total cost for labor and overhead, respectively, for each of the four quarters. The yearly expenses in each of the columns may be added to obtain the total production costs for each quarter. Table summarizes the total production costs. Season Summer Fall Winter Spring Year Raw materials Labor Overhead Total costs Table. Total Production Costs (dollars) Problem. A toy manufacturer makes toy airplanes, boats, and cars. Each toy is fabricated in a factory F in Taiwan and then assembled in factory F in the U.S. The total cost of each product consists of the manufacturing cost and the shipping cost. Then the costs at each factory (in $US) can be described as and F = Manufacturing Costs Shipping Costs Manufacturing Costs Shipping Costs Airplanes Boats Cars.. F =.5.5 Airplanes Boats.. Cars Find a matrix that gives the total manufacturing and shipping costs for each product. Warning: you need to use some practical reasoning here; do not mimic the previous example exactly!

7 Mathematics - Applied Matrix Algebra Supplement Application: A Simple Model for Marital Status. In a certain town, percent of the married women get divorced each year and percent of the single women get married each year. There are 8 married women and single women. Assuming that the total population of women remains constant, how many married women and how many single women will there be after year? Solution. Form a matrix A as follows: The entries in the first row of A will be the percent of married and single women, respectively, that are married after year. The entries in the second row will be the percent of women who are single after year. Thus we obtain the matrix A = If we let 8 x =, the number of married and single women after year can be computed by multiplying A times x Ax = =..8 After year there will be married women and single women.. Problem. Find the number of married and single women after years. Problem. Find the number of married and single women after years. Problem. Write a formula (that is, an equation or expression) for how to calculate the number of single and married women after n years. (Do not attempt an explicit calculation, just write down the formula.) This sort of population dynamics problem is very common and we will return to this example later in a more advanced setting.

8 Mathematics - Applied Matrix Algebra Supplement 5 Application: Markov Processes. Which of the following can be transition matrices of a Markov process? a) b) c) d) Which of the following are probability vectors? a) b) c) d) 5 5. Determine the value of each missing entry (denoted by ) so that the matrix will be a transition matrix of a Markov process. There may be more than one right answer a)..5 b)..5.. Consider the transition matrix a) If x () = P =.7..., compute x (), x (), and x () to three decimal places. b) Explain why P is regular and find its steady-state vector. 5. Consider the transition matrix a) If x () = P = , compute x (), x (), and x () to three decimal places.

9 Math - Applied Matrix Algebra Supplement b) Explain why P is regular (yes, it really is regular) and find its steady-state vector.. Which of the following transition matrices are regular? a) b) c) d) Show that each of the following transition matrices reaches a state of equilibrium a) b) c) d) Let P =. a) Show that P is not regular (compare this to a). b) Show that P k x k. 9. Find the steady-state vector of each of the following regular matrices a) b) c) d)

10 Mathematics - Applied Matrix Algebra Supplement Application: Markov Processes. Every day, a behavioral psychologist places a rat in a cage with two doors, A and B. The rat can go through door A, where it receives an electric shock, or through door B, where it receives some food. A record is kept of the door through which the rat passes. At the start of the experiment, the rat is equally likely to go through either door. After going through door A and receiving a shock, the probability of going through the same door on the next day is.. After going through door B and receiving food, the probability of going through the same door on the next day is.. a) Write the transition matrix for the Markov process. b) What is the probability of the rat going through door A on the third day after starting the experiment? c) What is the steady-state vector?. The subscription department of a magazine sends out a letter to a large mailing list inviting subscriptions for the magazine. Some of the people receiving this letter already subscribe to the magazine, while others do not. From this mailing list, percent of those who already subscribe will subscribe again, while 5 percent of those who do not now subscribe, will subscribe. a) Write the transition matrix for the Markov process. b) Last time the letter was sent out, it was found that % of the people receiving it ordered a subscription. What percentage of those receiving the current letter can be expected to order a subscription?. A new mass transit system has just gone into operation. The transit authority has made studies that predict the percentage of commuters who will change to mass transit (M) or continue driving their automobile (A). The following transition matrix has been obtained: This year Next year M A. a) If x () =.7 system after year? After years? Find x () to answer this. M A.7...8, what percentage of commuters will be using the mass transit

11 Math - Applied Matrix Algebra Supplement b) What percentage of the commuters will be using the mass transit system in the long run?. A study has determined that the occupation of a boy, as an adult, depends upon the occupation of his father and is given by the following transition matrix, where P=professional, F=farmer, and L=laborer. Father s occupation P F L Son s occupation P F L a) What is the probability that the grandchild of a professional will also be a professional? b) In the long run, what proportion of the population will be farmers? 5. Suppose Alpha and Broadwave are the two cable companies which service a certain town. Cylink Communications wants to expand to provide service to the same town, and compete with Alpha and Broadwave. Using extensive research data and analytical techniques, Cylink has determined that if it does so, the transition matrix giving customer preferences will be This year A B C Next year A B C for each year, where A=Alpha, B=Broadwave, and C=Cylink. a) Will the market attains equilibrium? b) If the market attains equilibrium, what is the market share in the long run? If the market does not attain equilibrium, what state vectors does the market oscillate between?

12 Mathematics - Applied Matrix Algebra Supplement 7 Application: The Leontief Closed Model. Which of the following matrices are exchange matrices? a) b) c) d) 5. Find a nonnegative vector p satisfying (I E) p =, for the given exchange matrix: a) b) c) 5. Consider a simple economy of Farmers, Carpenters, and Tailors, akin to the one discussed in class and in the homework. Suppose: the Farmers consume of the food, of the housing (produced by the carpenters) 5 and the clothes. the Carpenters consume of the food, of the housing (produced by the carpenters) and the clothes. 5 the Tailors consume of the food, of the housing (produced by the carpenters) 5 and no clothes. Find the exchange matrix E for this problem and find a vector p with at least one positive nonzero entry satisfying (I E) p =.. Consider the international trade model consisting of three countries, c, c, and c. Suppose that: c spends its income on domestic goods, its income on imports from c, and its income on imports from c. c spends its income on imports from c 5, its income on domestic goods, and 5 its income on imports from c 5. c spends its income on imports from c, its income on imports from c, and its income on domestic goods. Find the incomes of the three countries.

13 Mathematics - Applied Matrix Algebra Supplement 8 Application: The Leontief Open Model. Which of the following matrices are productive? a) b) c) d). Suppose that the consumption matrix for the linear production model is C = Find the production vector for the following demand vectors:. a) d = b) d =. A small town has three primary industries: a copper mine, a railroad, and an electric power plant. To produce $ of copper, the copper mine uses $. of copper, $. of rail transportation, and $. of electric power. To provide $ of transportation, the railroad uses $. of copper, $. of rail transportation, and $. of electric power. To provide $ of electric power, the electric utility uses $. of copper, $. of rail transportation, and $. of electric power. Suppose that during the year there is an outside demand of $. million for copper, $.8 million for rail transportation, and $.5 million for electric power. How much should each industry produce to satisfy the demands?. A small economic system consists of two industries: coal and steel. Suppose that to produce $. worth of coal, the coal industry uses $. of coal and $.8 of steel. Suppose also that to produce $. worth of steel, the steel industry uses $. of its own product and $. of coal. Find the consumption matrix C for this system. For the demand vector d =,, coal steel solve the equation x C x = d for x. That is, find a production vector x such that the outside demand d is met, without any surplus.

14 Math - Applied Matrix Algebra Supplement 8 5. An industrial system has only a Fuel industry and a Machines industry. To produce $. worth of fuel, the fuel industry uses $. of its own product and $. worth of machines. To produce $. worth of machines, the machine industry consumes $. of its own product, and $. of fuel. Find the consumption matrix C for this system. If the external demand for fuel and machines is given by d = 5,, fuel machines use (I C) to find a production vector x such that the outside demand d is met, without any surplus.. A small community includes a farmer, a baker, and a grocer with the consumption matrix and external demand vector given by E = F B G Farmer Baker Grocer and d = Find the optimal production vector x for this model.. 7. An remote city has an industrial system consisting of Farmers, Carpenters, and Tailors. The tailors consume $. worth of the clothes and $. worth of food, for every $. of clothes they produce, but they do all their work by hand, so they use no tools. The farmers eat $. of every $. worth of food that they produce. In addition, they require $. worth of tools for working the fields, and $. worth of housing. For every $. worth of tools built by the carpenters, they eat $. of food, use $. worth of clothes, and require $. of tools for further production. What is the consumption matrix for this system? How much of each good needs to be produced in order to meet an external demand of $5, in clothes, $, in food, and $8, in tools. The tools are produced by the carpenters