M5_Lab_Problem_Set_without_solutions

DAT 500N – Prescriptive Analytics Fall 2023 Dong / Farahat / Zeng Module 5: Mixed-Integer Optimization Lab Problem Set Required Problems: 1-3 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? Machine Fixed Cost Variable Cost Capacity 1 $1000 $21 500 2 $ 950 $23 600 3 $ 875 $25 750 4 $ 850 $24 400 5 $ 800 $20 600 6 $ 700 $26 800 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?

Lab Problem Set (Module 5) – DAT 500N Prescriptive Analytics – Fall 2023 – Dong / Farahat / Zeng 2/5 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: Possible Building Sites Region Sanford Altamonte Apopka Casselberry Maitland 1 X X 2 X X X X 3 X X 4 X X 5 X X 6 X X 7 X X Cost ($1,000s) $450 $650 $550 $500 $525 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?

Lab Problem Set (Module 5) – DAT 500N Prescriptive Analytics – Fall 2023 – Dong / Farahat / Zeng 4/5 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: Distances (in Miles) Between Groves and Plants Processing Plant Grove Ocala Orlando Leesburg Mt. Dora 21 50 40 Eustis 35 30 22 Clermont 55 20 25 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?

Lab Problem Set (Module 5) – DAT 500N Prescriptive Analytics – Fall 2023 – Dong / Farahat / Zeng 5/5 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 summ arized as “ all- units” price discounts in the table below.) Vendor A Vendor B Product Requirement Unit Price Volume Required Unit Price Volume Required 1 500 $225 0 - 250 $224 0 - 300 $220 251 - 500 $214 301- 500 2 1000 $124 0 - 600 $120 0 - 1000 $115 601 - 1000 (no discount) 3 2500 $60 0 - 1000 $54 0 - 1500 $56 1001 - 2000 $52 1501 - 2500 $51 2001 - 2500 Total Capacity 2500 2000 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))?