ENGSCI 313 – Assignment MFO

发布时间：2024-06-01

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ENGSCI 313 – Assignment MFO

Due date: May 31. Submit report + excel file online via canvas by 11:59pm. Total marks: [20]

Bespoke Solutions Ltd (BSL) are about to start a large project tailoring and installing a warehouse stock management system for a large retailer. The project includes the design of a database, migration of historical data, integration of forecasting tools, barcoding and sensor hardware, and training of the retailer’s team on the use of the new computerised system.

The project planning team identify: 11 project tasks, which tasks need to be completed before a given task can start (predecessors), how long each task will take if all aspects of the task are conducted “in house” – i.e. not using subcontractors (duration), the shortest time in which a task could be completed if extra subcontractors were employed (min), and the cost per day of the subcontractors for each task (i.e. how expensive it is to “crash” tasks within the project). This information is included in the table below.

Task	Description	Predecessor(s)	Duration (days)	min	Crashing cost ($1000 per day)
A	Draw up plan	-	1	1	0
B	Assign tasks	A	4	3	1
C	Obtain hardware	A	12	8	5
D	Programming	B	50	30	7
E	Hardware installation	C	9	6	4
F	Software testing	D	25	18	8
G	Produce user manual	E	15	8	3
H	Migrate databases	E	12	10	5
I	System testing	F	20	15	3
J	User training	G,H	15	15	0
K	User testing	I,J	22	18	2

The task precedence information can be summarized using a network diagram. Note the dummy arcs included (the arcs from nodes 7 to 10, 8 to 10, and 9 to 11) to ensure the precedence of multiple tasks is observed. These dummy arcs take 0 duration.

One question of interest to BSL is how quickly can they complete this project “in-house” (without paying subcontractors to expedite, or “crash”, any of the tasks)? One way to do this is to identify the longest path in the network (running from node 0 to node 12). We saw in class how to find the shortest path in a network (using a transhipment model with a single unit of demand and supply, and costing the arcs with their distances, or durations). The shortest path is the least cost solution in this model. To find the longest path you can use the negative of the task duration as each arc cost.

The solution to the corresponding transhipment problem will identify the longest path (known as the critical path), which determines how long the project will take to complete.

1. Download “forMFOAssignment313.xltx” from canvas, [DO NOT Copy someone else’s

download of the canvas file]. Save the file as a “ .xlsx” file. Add an additional worksheet, and set up the worksheet with the generalised net-flow out formulation of the transhipment problem described above. Solve it using the solver add-in to determine the minimum project completion time for BSL working in-house. [2m]

a. List the decision variables. [1m]

b. What is the minimum project completion time? [1m]

c. What is the sequence of tasks on the critical path? [1m]

The above model has included variables on arcs (determining the amount of flow – hint for 1a!).

Another way to approach this problem is to use variables on nodes, where these take on the value of the start and end times of the tasks. So x0 is the project start time (t = 0), x12 is the project completion time, and (for example) x2 is the finish time of task B and the start time of task D. Each arc now determines a constraint in a linear programme: e.g., because task B takes 4 days it yields the constraint: x2 − x1 ≥ 4. If we include a constraint x0 ≥ 0 (i.e. choose for all variables to be non-negative), finding the project completion time can be done by minimising the value of x12 .

2. Add another worksheet to your workbook and set up the variable-on-node linear

programme described above. Solve this model using the solver add-in (you should get the same answer as in question 1), and choose to keep the answer and sensitivity report. [3m]

a. Include a copy of the answer report as your answer to 2a. [2m]

b. Each shadow price in the constraint section of the sensitivity reports corresponds to one of the arcs in the network. What do you notice about all the arcs with a shadow price of 0 (hint – look again at your answer to 1c)? Does it make sense that these arcs have a shadow price of 0 – explain? [2m]

To support a planned promotional campaign, the retailer wants to transition to the new system quickly, and so offers a bonus of $200,000 if the project can be completed in 90 days. The model introduced in question 2 can be extended to determine if using subcontractors to meet the 90-day target is worth doing (if it costs less than $200,000 it would be worth it). The model extension requires additional “crashing” variables that determine by how many days each task should be reduced. For example, for task B we can introduce a variable YB that is bounded below by 0 and above by 1 (as the maximum reduction for task B is 4 − 3 = 1). This variable is then included in the constraint associated with task B in the formulation from question 2: x2 − x1 + YB ≥ 4. Thus, if task B starts after 1 day (x1 = 1) and we choose to crash the task by half a day (YB = 0.5) then task B will be completed after 4.5 days (x2 = 4.5). Note that actually task B can finish any time after 4.5 days, but if we set a project completion deadline of 90 days (x12 ≤ 90) and change the objective function to the crashing cost (i.e. the product sum of the reductions for each task multiplied by the per day cost) then a minimise objective function will push the project completion time to 90 (despite the “ ≤” constraint) and the constraints with any non-zero reduction variables to being “ =” constraints.

3. Add another worksheet to your workbook and extend the variable-on-node linear

programme to model the project crashing problem described above. Solve this model using the solver add-in, and choose to keep the answer report. [4m]

a. Include a copy of the answer report as your answer to 3a. [2m]

b. Is it worth crashing the project to secure the $200,000 bonus? [2m]