DAT5567 – Prescriptive Analytics

Fall 2025

Module 5: Linear Optimization with Integer Variables

Lab Problem Set

Required Problems: 1-3

Please upload to Canvas both the .ipynbfile as well as an .html or .pdf rendering of that file with the solution output displayed. You can solve the problems in a single notebook or in separate notebooks (but please label them clearly). You are not required to provide the mathematical formulation (but are strongly encouraged to have it handy before you start writing your code).

Additional optional problems: 4-5

Problems labelled with an asterisk (*) are more challenging

Problem 1

(Fixed Cost) Radford Castings can produce brake shoes on six different machines. The following table summarizes the manufacturing costs associated with producing the brake shoes on each machine along with the available capacity on each machine. If the company has received an order for 1,800 brake shoes, how should it schedule these machines?

a. Formulate a (mixed-) integer linear optimization model for this problem.

b. Construct a PuLP model for this problem and solve it.

c. What is the optimal solution?

Problem 2

(Set-covering) Health Care Systems of Florida (HCSF) is planning to build a number of new emergency-care clinics in central Florida. HCSF management has divided a map of the area into seven regions. They want to locate the emergency centers so that all seven regions will be conveniently served by at least one facility.  Five possible sites are available for constructing the new facilities. The regions that can be served conveniently by each site are indicated by X in the following table:

a. Formulate a (mixed-) integer linear optimization model to determine which sites should be selected so as to provide convenient service to all locations in the least costly manner.

b. Construct a PuLP model for this problem and solve it.

c. What is the optimal solution?

Problem 3

(Minimum Order) Clampett Oil purchases crude oil products from suppliers in Texas (TX), Oklahoma (OK), Pennsylvania (PA), and Alabama (AL), from which it refines four end-products: gasoline, kerosene, heating oil, and asphalt. Because of differences in the quality and chemical characteristics of the oil from the different suppliers, the amount of each end product that can be refined from a barrel of crude oil varies depending on the source of the crude. Additionally, the amount of crude available from each source varies, as does the cost of a barrel of crude from each supplier. These values are summarized below. For example, the first line of this table indicates that a barrel of crude oil from Texas can be refined into 2 barrels of gasoline, 2.8 barrels of kerosene, 1.7 barrels of heating oil, or 2.4 barrels of asphalt. Each supplier requires a minimum purchase of at least 500 barrels.

The company owns a tanker truck that picks up whatever crude oil it purchases. This truck can hold 2,000 barrels of crude. The cost of sending the truck to pick up oil from the various locations is shown in the column labeled  “Trucking  Cost.”  The company s plans for its next production cycle specify 750 barrels of gasoline, 800 barrels of kerosene, 1,000 barrels of heating oil, and 300 barrels of asphalt to be produced.

a. Formulate a (mixed-) integer linear optimization model that can be solved to determine the purchasing plan that will allow the company to implement its production plan at the least cost.

b. Construct a PuLP model for this problem and solve it.

c. What is the optimal solution?

Problem 4

(Trasportation) Tropicsun is a leading grower and distributor of fresh citrus products with three large citrus groves scattered around central Florida in the cities of Mt. Dora, Eustis, and Clermont. Tropicsun currently has 275,000 bushels of citrus at the grove in Mt. Dora, 400,000 bushels at the grove in Eustis, and 300,000 at the grove in Clermont. Tropicsun has citrus processing plants in Ocala, Orlando, and Leesburg with

processing capacities to handle 200,000, 600,000, and 225,000 bushels, respectively. Tropicsun contracts with a local trucking company to transport its fruit from the groves to the processing plants. The trucking company charges a flat rate of $8 per mile regardless of how many bushels of fruit are transported. The  following table summarizes the distances (in miles) between each grove and processing plant:

Tropicsun wants to determine how many bushels to ship from each grove to each processing plant to minimize the total transportation cost.

a. Formulate a (mixed-) integer linear optimization model for this problem

b. Create a PuLP model for this problem and solve it.

c. What is the optimal solution?

Problem 5*

Universal Technologies, Inc. has identified two qualified vendors with the capability to supply some of its electronic components. For the coming year, Universal has estimated its volume requirements for these components and obtained price-break schedules from each vendor. (These are summarized as “all-units” price discounts in the table below.)

All-units price discounts work as follows. Take Vendor A and Product 3 for example. If 1,400 units are purchased from Vendor A, each of the 1,400 units costs $56 each, and the total cost is $56*1,400=$78,400.

Universal s engineers have also estimated each vendor s maximum capacity for producing these components, based on available information about equipment in use and labor policies in effect.

Finally, because of its limited history with Vendor A, Universal has adopted a policy that permits no more than 60% of its total unit purchases on these components to come from Vendor A.

a. Formulate a (mixed-) integer linear optimization model that finds the minimum-cost plan for Universal.

b. Construct a PuLP model for this problem and solve it.

c. What is the optimal solution?

d. Suppose that Vendor A provides a new price-discount schedule for component 3. This one is an “incremental” discount, as opposed to an “all-units” discount, as follows.

Unit price = $60 on all units up to 1000

Unit price = $56 on the next 1000 units

Unit price = $51 on the next 500 units

With the change in pricing at Vendor A, what is the minimum purchasing cost for Universal, and what is the impact on the optimal purchase plan (compared to Part (c))?