# Quantitative methods for Business

18_______________________________________________________________Notes for Week 3The following terms and concepts are introduced:Why working with straight lines?•Any manufacturing process has both fixed and variable costs:-Fixed cost can be rent paid for the premises.-Variable costs can be the cost of materials and labour per each unit produced.•Given the value of the finished product, how many units are needed to break even?Definitions:•A termis a part of an expression that is connected to another term by a plus or minus sign.•A constantis a term whose value does not change.•A variable is a term that represents a quantity that may have different values.•An expressionis a combination of constantsand variables using arithmetic operations.•A coefficientis a factor by which the rest of a term is multiplied.•The degreeof expression is the highest exponent of any variable in the expression.•An equationis a statement that two expressions are equal.Solving linear equations(steps)1.Place like terms of the variable on the left side of the equation and the constant terms on the right side.2.Collect like terms and constant terms.3.Divide both sides of the equation by the coefficient of the variable. Solving simultaneous equations (steps to findingvaluesof two variables)1.Make the coefficient of either of the variables in one equation equal to its coefficient in the other equation.2.Eliminate the variable that has the same coefficient by subtracting one equation from the other equation3.Substitute the value found into one of the original equations to determine the value of the other variables.Graphs•Ordered pairs of observations canbe plotted on a graph.•Plotting ordered pairs on a graph can be used to solve many mathematical problems, for example:-simultaneous equations by linear graphing of straight line equations-problems involving non-linear graphs-problems involving break-even analysis.Break-even analysis•Helpful in manufacturing situations such as how many items must be produced and sold in order to break even?•Performed by solvingequations or using graphs•Sensitivity analysis:oHow sensitive the model is to changes, contemplated by the management (e.g. different price) or expected in the future (e.g. projected sales levels)19WEEK4–LINEARPROGRAMMINGMODELSINBUSINESSThis week we explore Linear Programming models. This includes identifying the components and formulating the model. We then consider both the graphical and the Excel Solver solution method. A worked example with screenshots of Excel Solveris available at the course website.Background reading Week 4lecture slidesLinear Programming Supplement(see Course Website, Lecture Notes page)Excel Supplement –Topic 5Review exercisesExercise 1The Crumb and Custard Bakery makes coffee cakes and Danish pastries in large pans. The main ingredients are flour and sugar. There are 12.5 kilograms of flour and 8 kilograms of sugar available, and the demand for coffee cakes is 4. Two and a half kilograms of flour and one kilogram of sugar are required to make a pan of coffee cakes, and two and a half kilograms of flour and two kilograms of sugar are required to make a pan of Danish pastries. A pan of coffee cakes has a profit of \$3, and a pan of Danish pastries has a profit of \$5. Determine the number of pans of coffee cakes and Danish pastries to produce each day so that the profit is maximised. Formulate a linear programming model for the bakery and solve this linear programming model using the graphical method.Solve this linear programming model using Excel Solver and use the Answer and Sensitivity Reports to answer the following questions:(a)If The Crumb and Custard Bakery produces the optimum number of cakes and pastries per day, will the maximum demand for coffee cakes be met? If not by how much will it be missed?(b)Will the bakery be using up all available flour and sugar each day? If not, how much will be left?(c)Suppose that the profit per coffee cake turns out to be only \$2. What effect, if any, will this have on the optimum solution?(d)What would be the effect on the optimum solution of increasing the amount of sugar available per dayto 9.5 kilograms?Exercise 2A company is planning to produce two new products that we will call A and B. These products have per unit profits of \$5 and \$8, respectively. Each unit of product must be processed on two assembly lines for required productiontimes as shown in the following table:Hours per unitABHours availableLine 112460Line 26540The manager would like to determine the quantities of each product to produce in order to maximise the profit generated by the sales of these products.Use the diagram below to determine the optimal allocation of resources and the maximum profit. 20_____________________________________________________________Assessment exercise for Assignment 22.Dean runs The Creamy Barwhich specialises in artisan ice cream sold at a local farmer’s market. Prevailing prices in the local market are \$8 for a take-home tub of Classic Vanilla and \$15 for a tub of Chocolate Almond Fudge. The local dairy farmer delivers 48 litres of milk every Friday in preparation for market day. Classic Vanilla will need 0.5 litres per tub and Chocolate Almond Fudge requires 3 times as much. Both flavours require 500g of sugar to enhance the taste. There is a total of 20kg of sugar available per market day. For the signature velvety mouthfeel, Dean adds 0.5 litres of heavy cream to Classic Vanilla and double the amount for Chocolate Almond Fudge. He ordered 50 litres of heavy cream from the supplier. Task 1Construct a mathematical model for this problem.In doing so, consider the following:(a)What are the decision variables for this problem?(b)Using decision variables identified in part (a), formulate the objective function for this problem. Is the quantity of interest to be maximised or minimised? (c)What constraints are relevant to this problem? Using the decision variables from part (a), formulate those constraints.Task 2Use Excel Solver to obtain a solution to the mathematical problem from Task 1. Your submission should include:•your Excel spreadsheet•the Sensitivity Report•the Answer ReportTask 3Use your Excel output to answer the following questions:Profit line (Profit = 40)21(a)Describe the linear programming solution to the Deanof The Creamy Barin terms of: •The optimumnumber of take-hometubsof Classic Vanilla andChocolate Almond Fudgeto prepare each market day. •The maximum revenue per market day.•Whether all the milk purchasedwill be fully utilised. •Whether all the sugarallocatedwill be fully utilised. •Whether all the heavy creamordered will be fully used. Which of the Solver reports helps you answer these questions? (b)What is the maximum profit per market day if Dean paid \$1.2 per litre for milk and cream and \$45 for sugar? Note that Dean also draws a \$100 salary per market day. Which Solver reportallows you to answer this question?(c)Due to the popularity of the Chocolate Almond Fudgeflavour, Dean is hoping to increase the price to \$20per take-home tub. Would the solution obtained in Task 2 still be optimal? Which of the EXCEL reports helps you answerthis question? Justify your answer carefully. How would the solution and The Creamy Bars’ revenuechange, if at all? (d)In preparation for the scorching heat in summer, Dean would like to purchase an extra 10 litres of milk to increase ice cream production. Would the solution obtained in Task 2 still be optimal? Which of the EXCEL reports helps you answer this question? Justify your answer carefully. How would the solution and The Creamy Bars’ revenue change, if at all? Attach the new Answer Report ONLY, for the scenario in which Dean purchases 58 litres of milk, verifying your calculated maximum revenue per market day.Task 4 Write a report outlining the solution and discussing your findings from Task 3 (at most two pages, double-spaced, at least 2cm margins, 12pt Times New Roman font or equivalent).Here are a few points to consider while working through this assignment question:1.The first step is always to work out the mathematical set up for the problem. This means identifying decision variables, formulating the objective function and then formulating constraints. At this stage, we are not trying to solve the problem or work out interactions among constraints. We simply list all conditions that must be satisfied. When you complete Task 1, you should have two decision variables, the objective function written in terms of those decision variables, and five constraints, also written in terms of decision variables (some using both decision variables, others just one of them).2.The second step isto find a solution. Task 2 tells you specifically to use Excel Solver to find this solution. The key here is to translate all mathematical expressions from Task 1 into Excel format. Instructions for doing so can be found under Topic 5 in the Excel booklet, as well as in the Linear Programming supplement. In addition, the Lecture notes page in this website gives you access to Excel spreadsheets used to generate Excel output shown in lecture slides for Week 5. It may be worthwhile examining thembefore attempting Task 2.3.The final step is interpreting the solution that has been found, which is Task 3.4.The report in Task 4 is a summary of the results from linear programming and sensitivity analysis in Tasks 2 and 3.22_____________________________________________________________________Notes for Week 4Linear Programming is used to obtain optimal solutions to problems that involve restrictions or limitations, such as: Materials, Budgets, Labour & Machine time.Linear Programming components•Objective: the goal of an LP model is maximisation or minimisation•Decision variables: amounts of either inputs or outputs•Constraints: limitations that restrict the available alternatives•Parameters: numerical values such as the price per unit•Feasible solution space: the set of all feasible combinations of decision variables as defined by the constraints Linear Programming assumptions•Linearity: the impact of decision variables is linear in constraints and objective function –there are no powers higher than one.•Divisibility: values of decision variables which are not whole numbers are acceptable•Certainty: values of parameters are known and constant•Non-negativity: negative values of decision variables are unacceptableGraphical Linear Programming method for finding optimal solutions when there are only two decision variables•Set up objective function and constraints in mathematical format•Plot the constraints•Identify the feasible solution space•Plot the objective function•Determine the optimum solutionLinear Programming in Excel(any number of variables)Sensitivity Analysis•Effects of changes to the objective function parameters•Effects of changes to right-hand sides of constraints.23WEEK5–INTRODUCTIONTOSTATISTICSANDPROBABILITYThis week we investigate quantitative research principles and statistics in business. This will involve defining the different types of data and variables, discussing methods of collecting data and identifying theappropriate visual display. We also investigate elementary probability, giving definitions of events, and contingency tables to calculate probabilities of interest and check statistical independence.Background reading Week 5lecture slidesFrom the textbook: Topic 6Sampling and Visual Presentation of Data(Ch.2, Ch.7 Section 7.4 and 7.5 only)and all material in Topic 9 Elementary probability and independence.Excel Supplement –Topic 6Review exercisesfrom the textbookTopic 6 Sampling and Visual Presentation of Data:Use Excelfor thesequestions. The data can be downloaded from the course website.•Section 2.3: p. 174, exercise 2.17(Electricity Costs). •Section 2.4: reproduce the results in Figure 2.13on page 178. •Section 7.4: p. 207, exercise 7.27(Sydney council) (a), (b), (c).Additional exercises:1.Suppose you study family income in a random sample of 300 families. You find that the mean family income is \$41,000; the median is \$35,000; and the highest and lowest incomes are \$250,000 and \$2,400, respectively.(a)Is the distribution of income symmetric, right-skewed or left-skewed? How many families in the sample earned less than \$35,000? Explain how you know.(b)Based on the given information, can you determine how many families earned more than \$41,000? Why or why not?2.What data types are these:a.Sales for last quarter of \$4,920,115.b.Length of a TV commercial.c.Did you file an insurance claim last month? Yes/No.d.A recruiter’s rating of job candidates (outstanding, good, adequate, weak, unsatisfactory).e.Dividend per share paid by Microsoft stock.3.Would you believe this study? Based solely on the information given in each case, decide whether you have any reason to doubt the results of the study. Explain your reasoning.a.A magazine tested its new format by including a postcard that readers could return to say whether or not they preferred the new format over the old format.b.A study financed by a major pharmaceutical company finds that its new drug is no more effective against high blood pressure than older, less expensive drugs.c.A survey was conducted that asked people whether they cheat on their income taxes. The survey required respondents to provide their names and addresses, but the letter that came with the survey promised full confidentiality.4.In an ad for the Club, a device used to discourage car thefts, it was stated that “The Club reduces your odds of car theft by 400%”. What is wrong with this statement?245.This graph advises job hunters which sort of start-up company they should approach:Source: Visual.ly/WTFViz.netElementary probability and independence:6.Section 4.2: p. 295, exercise 4.16 (Pay TV). Construct a contingency table from the data to include row and column totals. You can construct the table manually or in Excel. In addition, also determine:oThe probability a household is medium income and will not subscribeoThe probability a household is medium income or will not subscribeoGiven a household is low income, the probability it will subscribe.Additional exercise:A small grocery store would like to know if the number of items purchased by a customer is independent of the type of payment method the customer chooses to use. Having this information can help the store manager determine how to set up various checkout lanes. The manager collected a random sample of 368 customer transactions. Results are shown below.Number of items purchasedPayment methodCashBank/credit cardTotal1 to 54048886 to 10617413511 to 2040498921 plus253156Total166202368a.What is the probability that a customer pays cash?b.What is the probability that a customer pays by bank or credit card?c.What is the probability that a customer purchases 6 to 10 items?d.Are payment methodand number of items purchasedindependent? What calculations allow you to answer this question?e.Does it make sense for the manager to offer a cash-only checkout lane that is not restricted to the number of items purchased?25_______________________________________________________________Assessment exercise for Assignment 23.A simple random sample of 220 university students were asked what pasta they usually order and with which sauce. The preferences of these respondents are summarised below:(a)Use Excel to obtain a 100% stacked column chartfor the data from the table. What does your chart suggest between the relationship between Pastaand Sauce?EXCEL Instructions: Refer to Topic 6 in the Excel Bookletfor instructions on how to obtain 100% stacked column chart.(b)Use the contingency table to calculate the following probabilities; include an appropriate probability statement for each case:i.What is the probability that a respondent prefers Spaghetti?ii.What is the probability that a respondent prefers GnocchiandCarbonarasauce? iii.What is the probability that a respondent prefers Raviolior Pestosauce? iv.Suppose the preferred sauce is Bolognese. What then is the probability the pasta ordered is Spaghetti?v.Suppose that the preferred Pasta isRavioli. What then is the probability the sauceordered isCarbonara?(c)Are the events Spaghettiand Saucestatistically independentor dependent?How do you know? Show all calculations that support your answer.SaucePastaBologneseCarbonaraPestoTotalSpaghetti32251168Gnocchi38262488Ravioli18232364Total88745822026________________________________________________________________Notes for Week 5The following terms and concepts are introduced:What is statistics?•Statistics is the scientific method that enables us to make decisions as responsibly as possible. It involves collecting, presenting, and analysing data, as well as drawing inferences from themQuantitative research principles•Collecting pertinent information that is as reliable as possible.•Selecting the parts of the available information that are most helpful to make rational decisions.•Making the actual decisions as sensibly as possible on the basis of the available evidence.•Perceiving the risks entailed in the particular decision made andevaluating the corresponding risks of alternative actions.Twocategories of statistics•Descriptive statistics:-The meaningful presentation of data such that its characteristics can be effectively observed.-Graphical or pictorial display.-Condensation of large masses of data into a form such as tables.-Preparation of summary measures (e.g. average).•Statistical inference:-Relates to decision making; leads to future action rather than an inspection of the past.-Can we distinguish between genuine structural changes and the results of random fluctuation?-Values of numerical quantities were estimated. How reliable are those estimates?Sample & population•Most studies result in sample data which is then used to draw conclusions about a population.-A population is the complete collection of items to be studied.-A sample is the portion of the population that is selected for analysisTypes of data•Measurementsusing an instrument –the resulting data is called numerical•Counts–the resulting dataare called numerical•Classification datainvolves placing observations into categoriesoResulting data is calledcategoricaloWhen individuals or objects are ranked according to a criterion,rank data is obtained.•Primarydata is information collected by the person or organisation that will be using the information.•Secondarydata is information already collected by someone elseSampling•Examines some portion of a population. It may not be practical to analyse all the given data:-It may not be physically possible to collect all possible data.-It may be too expensive to collect or process all possible data.•Data may be processed more quickly by using a sample. When selecting any sample you first must answer:-What is the population from which the sample is to be chosen?-What should the size of the sample be?-What is the required accuracy of results?-How should the members be selected for inclusion?•A sample should be unbiased:-It should be genuinely representative of the population.-If there is any bias the result will be valueless.27Rules for avoiding bias in sampling•Do not only use people who volunteer to be in a sample.•Do not choose a sample using a method that omits segments of the population.•Do not use people in the sample only because they are handy or available.•The person selecting the sample should not have a vested interest in the results.Random sampling•Lottery method:-Population members are assigned a number.-Sample is then chosen according to randomly chosen numbers.-Use coins, dice, birthdates or other “random” devices.•Random numbers:-Numbers selected from a table of random numbers.-Numbers generated by a computer.•Random sample means that each member of the population has the same chance of being selected, with no one member being favoured.Other sampling methods•Census–complete enumeration•Systematic sampling•Quota sampling–e.g. same number of men and women •Multistage sampling•Importance sampling•Convenience–convenient•Judgement–by expert•Self-selected–e.g. phone-in, mail-back•Cluster sampling-Population is divided into groups (clusters).-A simple random sample of clusters is taken.-All individuals in selected clusters are studied.•Stratified sampling-Population is divided into groups (strata).-A simple random sample of equal size is taken from each group.Questionnaire and survey design•Commonly used for gathering data and market research.•Methods:-telephone interviews and cold calling.-person-to-person interviews.-questionnaires (multiple choice, open-ended questions).•Quality of results depends on quality of questions.-Wording of questions is very important!-Art as much as a science.Data displays•Count data-Tables of counts or percentages-Pie charts-Bar or column charts

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