# Chapter 19: Chapter 19

Problem 478

Suppose that two costume companies each make clown, skeleton, and space costumes. They all sell for the same amount and use the same machinery and workmanship. Furthermore, the market for costumes is fixed; a certain given number of total costumes will be sold this Halloween. But each company has its own individual styles which affect how the costumes sell. On the basis of past experience, the following matrix has been inferred. It indicates, for example, that if both make clown outfits, then for every 20 that are sold, company I will lose 2 sales to company II. Similarly, if company I makes clown outfits and company II makes space suits, then for each 20 sold, company I will sell 4 more than company II.

Problem 479

Consider a buyer who at the beginning of each month decides whether to buy brand 1 or brand 2 that month. Each month the buyer will select either brand 1 or 2 . The selection he makes at the beginning of any one month depends at most on the selection he made in the immediately preceding month and not on any other previous selections. Let \(\mathrm{p}_{\mathrm{ij}}\) denote the probability that at the beginning of any month the buyer selects brand \(j\) given that he bought brand i the preceding month. Suppose we know that \(p_{11}=3 / 4, p_{12}=1 / 4\), \(\mathrm{p}_{21}=1 / 2\) and $\mathrm{p}_{22}=1 / 2$ 1) First draw the transition diagram. 2) Assuming that the first month under consideration is January, determine the distribution of probabilities for April.

Problem 480

Consider a market that is controlled by two brands \(B_{1}\) and $\mathrm{B}_{2} .\( Suppose it is known that \)10 \%$ of the buyers of each brand switch to the other brand during any given month. At the beginning of a given month 500 buyers are divided so that 300 purchase \(\mathrm{B}_{1}\) and 200 purchase \(\mathrm{B}_{2} .\) What will be the number of buyers who will purchase \(B_{1}\) and \(B_{2}\) during the given month?

Problem 482

A certain textile mill finishes cotton cloth obtained from weaving mills. The mill turns out two styles of .cloth, a lightly printed style and a heavily printed one. The mill's output during a week is limited only by the capacity of its equipment for two of the finishing operations - - printing and bleaching .. and not by demand considerations. The maximum weekly output of the printing machinery is 800 thousand yards of cloth if the light pattern is printed exclusively, 400 thousand yards if the heavy pattern is printed exclusively, or any combination on the printing line $\mathrm{L}+2 \mathrm{H}=800\( (where \)\mathrm{L}$ represents light pattern and H heavy pattern). In a week, the maximum the bleaching equipment can handle is 500 thousand yards of the light-patterned cloth exclusively, 550 thousand yards of the heavy-patterned cloth exclusively, or any combination on the bleaching line, \(1.1 \mathrm{~L}+\mathrm{H}=550\). The mill gained \(\$ 300\) and \(\$ 290\) per thousand yards of the light . . and heavy-patterned cloths, respectively. 1) Draw the graph of the two lines described above. 2) Solve the linear programming problem of maximizing the gain from a week's production.

Problem 485

Suppose that we have 2 factories and 3 warehouses. Factory I makes 40 widgets. Factory II makes 50 widgets. Warehouse A stores 15 widgets. Warehouse B stores 45 widgets. Warehouse \(\mathrm{C}\) stores 30 widgets. It costs \(\$ 80\) to ship one widget from Factory I to warehouse \(A, \$ 75\) to ship one widget from Factory I to warehouse \(\mathrm{B}, \$ 60\) to ship one widget from Factory \(\mathrm{I}\) to warehouse \(\mathrm{C}, \$ 65\) per widget to ship from Factory II to warehouse \(\mathrm{A}, \$ 70\) per widget to ship from Factory II to warehouse \(B\), and \(\$ 75\) per widget to ship from Factory II to warehouse \(\mathrm{C}\). 1) Set up the linear programming problem to find the shipping pattern which minimizes the total cost. 2)Find a feasible (but not necessarily optimal) solution to the problem of finding a shipping pattern using the Northwest Corner Algorithm. 3) Use the Minimum Cell Method to find a feasible solution to the shipping problem.

Problem 486

A small-trailer manufacturer wishes to determine how many camper units and how many house trailers he should produce in order to make optimal use of his available resources. Suppose he has available 11 units of aluminum, 40 units of wood, and 52 person-weeks of work. (The preceding data are expressed in convenient units. We assume that all other needed resources are available and have no effect on his decision.) The table below gives the amount of each resource needed to manufactur each camper and each trailer. Suppose further that based on his previous year's sales record the manufacturer has decided to make no more than 5 campers. If the manufacturer realized a profit of \(\$ 300\) on a camper and \(\$ 400\) on a trailer, what should be his production in order to maximize his profit?

Problem 487

1) Define sample space, random variable and expected value. 2) The probability that an American man in his early fifties will survive for another year is about \(.99\). How much should such a man pay for \(\$ 20,000\) worth of term insurance, exclusive of administrative costs and profit for the company? 3) Suppose that the probability of finding 0,1,2, and 3 people in line ahead of you at the Apex Supermarket is .2,4, .25, and .15, respectively. If the probability of finding 0,1,2 and 3 people in line at the B\&Q Supermarket across the street is 3,3, .2, and .2, respectively, at which supermarket would you find shorter lines in the long run? (Assume that both markets have the practice of opening up a new line as soon as one exceeds 3 people in length.)

Problem 488

1) Show that the standard deviation of \(\mathrm{X}\) and \(\mathrm{Y}\) corresponds to vector addition (according to the parallelogram law). 2) Solve the following problems: (a) Suppose that 3 resistors are placed in series. The standard deviations of the given resistances are 20,20 and 10 ohms, respectively. What is the standard deviation of the resistance of the series combination? (b) Gears \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}^{\prime}\) s widths have standard deviations of \(0.001,0.004\), and \(.002\) in., respectively. If these are assembled side by side on a single shaft what is the standard deviation of this total width?

Problem 493

The technique used to estimate a relationship of the form \(\mathrm{Y}=\beta_{0}+\beta \mathrm{X}\) is known as simple regression. Generalize this technique to estimate a relationship of the form, $$ \mathrm{Y}=\beta_{0}+\beta_{1} \mathrm{X}_{1}+\beta_{2} \mathrm{X}_{2}+\ldots+\beta_{\mathrm{K}} \mathrm{X}_{\mathrm{K}} $$ where \(\mathrm{K}\) is some positive integer. This new technique is known as multiple regression.

Problem 494

A highly specialized industry builds one device each month. The total monthly demand is a random variable with the following distribution. $$ \begin{array}{lllll} \text { Demand } & 0 & 1 & 2 & 3 \\ \mathrm{P}(\mathrm{D}) & 1 / 9 & 6 / 9 & 1 / 9 & 1 / 9 \end{array} $$ When the inventory level reaches 3, production is stopped until the inventory drops to \(2 .\) Let the states of the system be the inventory level. The transition matrix is found to be Assuming the industry starts with zero inventory find the transition matrix as \(\mathrm{n} \rightarrow \infty\).