DEPARTMENTS OF ENGINEERING SCIENCE AND BIOENGINEERING AND STATISTICS

ENGSCI / STATS 255 SC

ASSIGNMENT 3

DUE: 11:59PM, 20 MAY 2025

Instructions for handing in. Please:

1.   Attempt ALL questions.

2.   Assignments may be typed in a word processor of your choice, or handwritten neatly.

3.   Set answers out in order of the questions. Do NOT jump between questions.

4.   There is no need to copy the questions out in your submission.

5.   Problems entered into Excel must include your student ID in the Excel spreadsheet. Submit only the sections of computer output that are asked for.

6.   You can generate your Excel output either by printing the required reports directly to paper (print in landscape mode where appropriate), or by using the Snipping Tool (Windows) or Command - Shift -

4 (Mac). You can display formulae in Excel by using Control + ` or Command + ` or by using

Formulas → Formula Auditing →Show Formulas (see image below, left). Excel output showing

formulae should include row and column headings; use Page Layout → Sheet Options → Headings → Print (see image below, right) to enable this in printouts.

Submission: Assignments must be submitted using Canvas as a single PDF file, before the due date and time. Handwritten assignments will need to be scanned. Prepare your assignments well in advance of the deadline in case of technical issues, as no extensions will be provided in this case.

Notes:

•   Summarising, analysing and communicating information is an important part of Operations

Research. For this reason you will be expected to write answers which clearly communicate your thoughts. The mark you receive will be based on your written English as well as your technical  work.

• We encourage working together. Discussing assignments and methods of solution with other

students or getting help in understanding from staff and students is acceptable and encouraged. You must write up your final assignment individually, in your own words.

•   By submitting this assignment, you confirm that you understand the University's policies on

cheating, plagiarism and group work; that your submission is entirely your own work and you have not allowed access to any part of the assignment to any other person. See the appropriate sections in the Course Outline for more details.

•   This assignment makes up 7.5% of the final assessment for this course.

Question 1: Planning a Broadway Show (16 marks)

Broadway Productions is a company that puts together musicals for Broadway. A new musical has just been signed, and the producer has identified the following tasks that need to be accomplished before the show can open:

Three of these tasks don’t require any other tasks to be completed before they are begun: preparing each part, scoring the music and hiring the cast. The choreography is done after the music is scored. The dance rehearsal cannot begin until each part is prepared, the cast is hired, and the choreography completed.   The scenery is designed and built after the dance rehearsal.  Costumes are designed and made once the cast is hired.  The dress rehearsal is done after the dance rehearsal and after the costumes are ready.   The dress rehearsal is followed by the full rehearsal, which also requires the scenery.  The final rehearsal follows the full rehearsal.

a)    Identify the immediate predecessors of each task and present it in a table.

b)   Develop an Activity-on-Arc project network for this project by hand, in the manner  shown in the workbook.  Label all arcs with their associated activity. Introduce and clearly label a D1, D2, etc any dummy arcs you need. Note that drawing an activity-on-node diagram will score zero marks. (Your network should be drawn by hand, not using the spreadsheets. But you can use the Project Management Simulation spreadsheet to check your diagram if you wish.)

c)    Enter this network and duration data into the 255 “Project Management CPM & PERT” spreadsheet to do a PERT analysis for this project. Because our expected durations are fractional, you should change the formatting on the EST, EFT, LST, LFT, and slack calculated values to show 3 decimal places. Hand in a screenshot showing your data entry and the resulting table of EST, EFT, …, variance calculations, and also the expected project completion time, variance of project completion time, and std deviation of project completion time values. Be sure to include your ID number.

d)   Using the spreadsheet results, list the tasks on the expected critical path.

e)    The dance company needs to publish a date for the opening night that they are 99% confident they will meet. How many days should they allow for the project duration to give this level of confidence?

f)    The director has just learnt that the costume design and making is having major staffing problems, and is likely to have its duration extended by 12 days. Should the director be worried? Carefully justify your answer by referring to the PERT spreadsheet results table; you must identify and report in your answer the specific value(s) in this table that support your conclusion. (You should not make any changes to the spreadsheet; just use the output produced above to answer this question.)

Question 2: Crashing (13 marks)

Consider the following project network & data that we wish to crash. (Note that this table shows, for example, that task A can be shortened to 15-6=9 days at a cost of $1/day*6 days=$6.) D1 and D2 are dummy arcs.

a)    Construct a linear programme in Excel to crash this network.  Be sure to follow the  standard  255 spreadsheet layout for linear programmes. You should hand in a screenshot showing your model and an optimal solution for a required completion time of 24 days. (No other screenshots are required.) Be sure to show your ID number on the spreadsheet.

b)   For your solution in a), list all the tasks that have to be crashed and give their new durations.

c)    By repeatedly solving variations of your LP model using Excel, complete the following table to show, for the specified project duration, the total cost to crash the project to achieve this duration, and also the marginal cost. The marginal cost (in $/day) is the additional cost for reducing the project duration by one more day. So, for example, the marginal cost for a duration of 28 days is the difference between the total crash cost for 28 days and the total crash cost for 29 days.

d)  What trend do you notice in the marginal costs?

Question 3: V and W Drinks (21 marks)

A new startup company wishes to make its workers feel valued by providing them with complimentary V drinks. They think that the workers will collectively consumer 25 drinks per day on average. The company has the drinks delivered by a supplier at $3 per drink, and each delivery costs $15. These orders need to be placed 3 days before they are delivered. The drinks are stored in a fridge on site which costs $0.03 per drink per day. The company is able to order more drinks at any point. The company is interested in minimising the amount they pay for purchasing and storing the drinks.

Note: This company works 7 days a week, so you can ignore weekends as a complicating factor.

a)  What is the appropriate inventory model for the company’s situation? Write down the parameters of the problem and their values; express units of time in terms of days.

b)  Calculate the optimal order quantity. Show your working. (Simply round your answer to get an integer value.)

c)   Suppose that the company can get quantity discounts based on the number of drinks they order. For up to 150 drinks the cost is $3 per drink, for 151 to 200 the cost is $2.70 per drink, and for 201 drinks or more the cost is $2.60 per drink. Calculate the optimal order quantity and total cost for each cost level and determine the overall optimal order quantity. Show your working (including your Q* values). (Hint: This is a little different to the examples we did in class.)

The company now has now set up a production line producing W drinks (with the slogan “twice a V”). They wish to test these drinks out on their employees by switching from V to W. They can produce W drinks at a cost of $2.50 per drink at a rate of 45 drinks per day. Requests for a new production run have to be given 6 days before production can start. Each new production run costs $50 to set up.

d)  What is the appropriate inventory model for the company’s new situation?

e)  Assuming demand does not change, determine the optimal production quantity. (Simply rounding the answer is OK.)

f)   Sketch the inventory levels over time for one cycle, labelling:

(i) the end of the production period (in days),

(ii) the cycle length (in days), and

(iii) the maximum inventory level (in drinks).

g)  Determine the “re-order” point for this system, clearly stating whether this occurs during or after the production run.