## Statistics- ADM2302 A, B, C, D, E Fall 2015 Assignment # 3

## Statistics- ADM2302 A, B, C, D, E Fall 2015 Assignment # 3

ADM2302 A, B, C, D, EFall 2015Assignment # 3Assignment #3Transportation Problems, Integer and Binary Programming and Goal programming,ADM2302 students are reminded that submitted assignments must be neat, readable, and well-organized.Assignment marks will be adjusted for sloppiness, poor grammar and spelling, as well as for technical errors.Please note that:While working together is encouraged, plagiarism on assignments will not be accepted.Each student must sign the individual statement of integrity to be included with the submission.Each student must provide an individual original submission of completed Assignment #3.Students should submit a PDF of their type-written (i.e., not handwritten) assignment via blackboardlearn by the due-date.Assignments must be stapled or they will not be marked.Problem1:This problem is a combination of binary integer programming and transportation problemA company has 6 sales centers in Ontario and has decided to open new depots to deliver its products from thedepots to the sales centers. There are two types of costs associated with the delivery: set-up costs (fixedcosts) are capital costs which may usually be written off over several years, and transportation costs whichdepend on the distance covered. We assume that they have been put on some comparable basis, by takingthe costs over a year.There are 6 sites available for the construction of new depots to deliver products to the sales centers.The following table (Table 1) gives the transportation costs (in thousand dollars) of delivering the entiredemand of each sales center from a depot (not the unit costs). Certain deliveries that are impossible aremarked with the infinity symbol (∞).Sales CentersDepot12345611008050506010021209060706510031401108080751304160125100100801505190150130∞∞∞6200180150∞∞∞Table 1: Transportation costs for satisfying entire demand of each sales centerThe construction costs (fixed cost) for each depot as well as the capacity of each depot are listed in Table 2.Fall 2015Page 1ADM2302 A, B, C, D, EFall 2015Assignment # 3Depot123456Cost(1000$)3500900010000400030009000Capacity(tons)300250100180275300Table 2: Fixed costs and capacity limits of the depot locationsThere are estimations for demand of each sales center which are shown in the following table.Sales centerDemand (tons)12341208075100Table 3: Demand data51106100Considering that the demand of a sales center needs to be satisfied and a sales center may be delivered tofrom several depots, which depots should be opened to minimize the total cost of construction and ofdelivery, while satisfying all demands? Formulate algebraically the corresponding model for this problem, butDo NOT solve the problem.Problem2:A car rental company has decided to provide roadside services for 6 zones in a large city. The company wantsto determine where (zone) to locate the roadside services. To reduce the companyâs costs the manager wantsto locate a minimum number of roadside services and ensure that at least one service is within 15 minutes ofeach zone.The times (in minutes) required to drive between zones are:ToFrom123456101020302020210025352010320250153020430351501525520203015014620102025140Formulate an integer/binary programming model that will select the minimum number of roadside servicesthat the company will need to achieve its policy objective. Do NOT solve the problem.Fall 2015Page 2ADM2302 A, B, C, D, EFall 2015Assignment # 3Problem3:Capital budgetingA firm has 6 projects that it would like to undertake over the next 5 years but because of budget limitationsnot all can be selected. The total budget that the firm has considered to invest in the projects is $12,400,000.The following table displays the expected revenue (NPV) of each project after 5 years and the required yearlycapital for each investment.Table 1: Investment DetailsCapital (in $000) required per yearInvestment/Project123456ExpectedNPV ($000)$2700$3330$7010$5770$2900$4870Year 1$ 975$1200$2500$1550$1400$1900Year 2$ 350$ 200$1200$1350$ 350$1900Year 3$ 200$ 200$ 850$ 675$87.5$ 350Year 4$ 100$ 200$ 400$ 337.5$ 21.875$ 350Year 5$50$ 200$ 400$168.75$0$ 350The capital available for the time period of each year is shown in the following table:(The $12,400,000 in investment capital is spread over the 5 years)Year 1$ 5800Capital (in $000) allocated per yearYear 2Year 3Year 4$ 3500$ 1300$ 900Year 5$ 900In addition, the firm must follow a few federal and state laws regarding these projects:1.2.3.Surplus capital funds in any year cannot be carried over from year to year.If the firm decides to invest in the second investment/project, it must also invest in the fourth.If the firm decides to invest in the first investment/project, it cannot invest in the thirdinvestment/project.Considering the budget limitations and the laws, which of the investments/projects should be chosen tomaximize potential NPV? Formulate the Integer Linear Programming model in algebraic form, and usingExcelâs Solver to find a solution. Provide a printout of your answer report (Attach the excel file to yoursubmission)Fall 2015Page 3ADM2302 A, B, C, D, EFall 2015Assignment # 3Problem4:Goal ProgrammingA large bookstore in Ottawa operates 7 days per week. This bookstore needs the following number of fulltime employees working each day of the week:DayNumber ofemployeesSunday47Monday22Tuesday28Wednesday35Thursday34Friday43Saturday53Each employee must work 5 consecutive days each week and then have 2 days off. For example, anyemployee who works Sunday through Thursday has Friday and Saturday off. This bookstore currently has atotal of 60 employees available to work. The director of the department has developed the following set ofprioritized goals for employee scheduling:(1) The store wants to avoid hiring any additional employees. Weight=40(2) The most important days for the department to be fully staffed are Saturday and Sunday. Weight=31(3) The next most important day to be fully staffed is Friday. Weight=25(4) The department would like to be fully staffed the remaining 4 days in the week. Weight=20Formulate a goal programming model to determine the number of employees who should begin their 5-dayworkweek each day of the week to achieve the department’s objectives. Do NOT solve the problem.Fall 2015Page 4

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