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Homework answers / question archive / Question 1 [6 points] Each week, at 12am Monday morning, Pfizer vaccines for COVID-19 arrive by air to one of two airports in Toronto: Billy Bishop ‘Toronto City Airport

Question 1 [6 points] Each week, at 12am Monday morning, Pfizer vaccines for COVID-19 arrive by air to one of two airports in Toronto: Billy Bishop ‘Toronto City Airport

Computer Science

Question 1 [6 points]

Each week, at 12am Monday morning, Pfizer vaccines for COVID-19 arrive by air to one of two airports in Toronto: Billy Bishop ‘Toronto City Airport. (100,000 doses) or Toronto Pearson Airport (250,000 doses). From there, they are immediately transported to the 7 hospital immunization clinics and 22 city run mass vaccination sites. Unfortunately, the Pfizer vaccine can only be kept at fridge temperatures for five days. Otherwise, the vaccine must be stored at temperatures between —80°C to —60°C and the hospital immunization clinics are the only sites that have this capability. As a result, at, 12am Saturday morning, all the hospitals send a replenishment order of two days worth of demand to the city run clinics to ensure that the sites have enough vaccines for the weekend.

Hospital Site Number

1

2

3

4

5

6

7

Weekly Storage Capacity (thousands)

30

37

44

51

58

65

72

Weekly Fixed Storage Cost (per dose)

1.05

1.10

1.15

1.20

1.25

1.30

1.35

 

Table 1: Weekly storage capacity and weekly storage costs of the hospital immunization clinics.

The 29 locations administer exactly 50,000 vaccinations per day, seven days per week. The seven hospital immunization clinics administer four times as many vaccinations per day as the city run clinics. The weekly storage capacity of hospital immunization clinics and the weekly fixed costs associated with storing vaccines at that facility are given in Table 1. City run vaccination sites can store no more than 5000 doses. In addition, due to operational constraints within the logistic company, the following two restrictions must be adhered to when developing the transhipment plan:

1. The number of vaccine doses stored at hospital site 5 must be less than or equal to 80% of the number of doses stored at hospital site 7.

2. The number of vaccine doses stored at hospital site 6 must be greater than or equal to 75% of the number of doses stored at hospital sites 2, 3, and 4 combined.

Formulate and solve a linear program to determine how many doses of vaccine should be initially sent from the airports to each of the 29 vaccination locations, and to be transshipped later in the week, to minimize storage costs while adhering to all constraints. Then, answer the following 10 questions:

(a) How many vaccinations can a hospital administer per week?

(b) How many doses should initially be sent to each city run vaccination site?

(c) How many decision variables are in the formulation?

(d) Write down the constraint associated with the second restriction.

(c) Write down the constraint associated with ensuring that the weekly storage capacity of each hospital is adhered to.

(f) Write down the constraint associated with ensuring that the city run vaccination sites have enough vaccines for the weekend.

(g) What is the optimal storage cost for the week?

(h) How many doses are being stored at Hospital 7 in the optimal solution?

(i) What is preventing the solution from storing all of the vaccines at the least costly hospitals?

(j) Can you recommend a way to reduce total costs without changing the weekly fixed storage cost?

Question 2 [6 points]

The 50 million tree program is a tree planting charity whose mandate is to increase forest cover in Ontario. As of 2020, more than 30 million trees have been planted. This year, the organization made a big push to acquire funding so that they could plant 10 million trees in 2022. There are a total of 36 potential planting locations in Ontario but. it remains to determine which sites should be chosen. For each site i = 1,..., 36, there is a cost ¢; for each tree that is planted where

                           ci = 0.05 + 1/20 (i mod 10),

and mod represents the modulo operation. Unfortunately, due to the Conservation Authorities Act of Ontario, many intricate laws that must be adhered to. In particular:

  • At least 13 planting locations must be chosen in Ontario.
  • Between 33,111 and 668,457 trees can be planted at any location if selected.
  • At most one planting location can be chosen amongst the sites 1, 10, and 20.
  • Exactly three planting locations must be chosen amongst the sites 3, 9, 15, 21, 27, and 33.
  • No more than 4 planting locations must be chosen amongst the sites 2, 4, 6, 8, 12, 14, and 16.
  • If planting location 30 is chosen then the sites 31 and 32 cannot be chosen.
  • If planting location 21 is chosen then the sites 22 and 23 must. be chosen.
  • The sum of all trees planted at sites 1-18 must equal the sum of all trees planted at sites 19-36.

Formulate a MILP model to minimize the sum of costs related to planting the 10 million trees while respecting the legal requirements in the Conservation Act. Then, answer the following 10 questions.

(a) If there were no restrictions, what three planting locations would be in the optimal solution?

(b) Without solving the MILP, what is the minimum number of planting locations given the maximum number of trees that can be planted at any location?

(c) Are there any redundant constraints? If so, which one?

(d) Does the objective contain fixed costs only, variable costs only, or both?

(e) Write down the constraints associated with linking the integer and binary decision variables,

(f) Write down the constraints associated with planting locations 21, 22, and 23.

(g) How many decision variables are in the formulation?

(h) What is the optimal planting cost?

(i) How many trees are planted at site number 4?

(j) What are the most trees planted at any location?

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