OMIS 4000 Assignment 1
Due: October 6, 11:59 PM EST
Question 1. [23 Points] – Binary Program
Luke’s diner is a family-owned restaurant with 6 employees; i =1,…,6. Sally and Henry are managers of the restaurant while George, Harmony, Mike, and Maria are regular employees. The restaurant opens five days per week (Wednesday through Sunday) at 7 AM and closes at 11 PM. Shifts are broken down into two 8-hour time blocks per day, i.e., 7 AM – 3 PM and 3 PM – 11 PM. As a result, there are 𝑗=1,…,10 possible shifts per week with odd numbers representing morning shifts and even numbers representing evening shifts (e.g., 𝑗=1 represents Wednesday morning). As demonstrated in the table below, the cost of scheduling employee 𝑖 to shift 𝑗 depends on whether it is a weekday or weekend.
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Employee Name
Weekday Cost Per Shift ($)
Weekend Cost Per Shift ($)
For the upcoming week, formulate and solve (with Python) a binary linear programming model to minimize the total cost of scheduling employees to shifts such that the following restrictions are adhered to:
1. At least 3, but no more than 5, employees can be assigned to the same shift.
2. Each employee must work between 32 and 48 hours per week.
3. Each employee must be assigned to at least one weekend shift (i.e., Saturday or Sunday). 4. At least one manager must be assigned to each one of the ten weekly shifts.
5. Employees cannot work back-to-back shifts (i.e., 7 AM – 11 PM) on the same day of the week.
Question 2. [43 Points] – Mixed-Integer Linear Program
The UNICEF Supply Community is at the forefront of UNICEF’s humanitarian work. Currently, the community is in charge of preparing a loading plan for a cargo ship destined for Ghana. The following four products need to be transported aboard the ship:
Nutrition Supplies Medical Supplies Pharmaceut ica ls Sanitatio n Supplies
Amount Available (tons)
4,800 2,500 1,200 1,700
Volume per Ton (cubic feet)
40 25 60 55
Profit per Ton ($)
70 50 60 80
Any and/or all of the available commodities can be loaded. However, the ship has three cargo holds with the following capacity restrictions:
Cargo Hold
Forward Center
Weight Capacity (tons)
3,000 6,000 4,000
Volume Capacity (cubic feet)
145,000 180,000 155,000
Only one type of commodity can be placed into any cargo hold. However, because of balance considerations, the weight in the forward cargo hold must be within 10% of the weight in the rear cargo hold. Further, the weight in the center cargo hold must be between 40% and 60% of the total weight on board.
a. Formulate a Mixed-Integer Linear programming (MILP) model for this problem. b. Implement your model in Python and solve it.
c. What is the optimal solution?
Question 3. [24 Points] – Nonlinear Program
The year is 2024 and the U.S. presidential campaign is well underway. It has become evident that the race is going to be close. As such, one of the presidential candidates has hired , a prescriptive analytics consultant, to estimate the percentage of votes the campaign will receive in each of the states based on the amount of money the campaign spends and the number of times the candidate visits each state during the final two weeks before the election. The results of Adrian’s analysis provided the following function:
Percentage of votes in state k = (akV2k —bkD2k)
Vk = the number of times the candidate visits state k in the last two weeks of the campaign, and Dk = the amount of money (in $1,000,000s) the campaign spends on advertising in state k in the
last two weeks of the campaign.
The following table summarizes Adrian’s estimates of the parameters a and b for each state,
along with the number
Florida Georgia California Texas Illino is n
of electoral votes at stake in each state:
0.145 0.31 0.117 0.27 0.123 0.21 0.125 0.28 0.128 0.26 0.121 0.22 0.134 0.24 0.121 0.38
Electoral Votes
The candidate believes that he can make 21 campaign stops in the next two weeks, and there is $15 million left in the campaign budget available for advertising. He wants to spend at least $500,000 in each of these states in the next two weeks. He also wants to make at least one, but no more than five, campaign stops in each of these states. Within these constraints, Adrian wants to allocate the available resources to maximize the number of electoral votes the candidate can receive. Assume the candidate needs 51% of the vote to win in each state.
a. Formulate a Nonlinear Programming (NLP) model for this problem to determine i) how much money the candidate should spend on advertising in each state; ii) how many campaign stops the candidate should make in each state.
b. Implement your model in Python and solve it. What is the expected number of electoral votes generated by this solution?
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