Problems_Integer Programming - BU-275-OC1 - Business Decision Models

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Wilfrid Laurier University *

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275

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Industrial Engineering

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Apr 3, 2024

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BU275: Practice problems for “Integer Programming”' Problem 1 Dagma Chemicals Inc., produces a special oil-base material that is currently in short supply. Dagma’s management faces the problem of deci how many units it should supply to each customer. Since the 4 customers are in different industries, different prices can be charged due to the various industry pricing structures. However, slightly different production costs at the 2 plants and varying transportation costs between the plants and customers make a “sell to the highest bidder” strategy unacceptable. After considering price, production costs, and transportation costs, Dagma established the following profit per unit for each plant-customer alternative. Customer ($ per unit) Plant D1 D2 D3 D4 Sarnia 32 34 32 40 Calgary 34 30 28 38 The plant capacities and customer orders are shown in the following table. Plant capacity (units) Distributor orders (units) Sarnia 5000 D1 2000 Calgary 4000 D2 3000 D3 5000 D4 4000 Each Distributor’s order must be met in full or not at all. Partial order fulfillment is not permitted. Because of the high demand management has decided that only a maximum of 3 distributors will be served, and that either the demand for distributor 3 or distributor 4 will be satisfied but not both. However, because of its history of good relationships, Dagma has promised to distributor 2 that its demand will be completely satisfied. The company has also decided that if it ships any amount from Sarnia to distributor 1 then Calgary should ship at least the same amount to distributor 3. Formulate Dagma'’s problem as a mixed integer programming problem. Problem 2 A hiker is deciding which and how many items he can carry in his knapsack when going on a hike. The table below gives the weight of each item in kilograms and the value of each item to the hiker(which is a score). Determine which items he can carry if he wants to maximize the value without exceeding the weight capacity of his knapsack which is 14 kilograms. Item 1 2 3 4 Weight (in kg) 5 7 4 3 Value 8 11 6 4 1 Most questions are based on “Introduction to Management Science”, Hillier and Hillier, 5th edition, McGraw Hill, and on “Introduction to Management Science”, Taylor, 12th edition, Pearson.

Problem 3 A toy manufacturer is planning to produce new toys. The setup cost of the production facilities and the unit profit for each toy are given below: Toy | Setup cost ($) | Profit ($) 1 45000 12 2 76000 16 The company has two factories that are capable of producing these toys. In order to avoid doubling the setup cost only one factory should be used for each toy. The production rates of each toy are given below (in units/hour): Toy1l | Toy2 Factory 1 52 38 Factory2 | 42 23 Factories 1 and 2, respectively, have 480 and 720 hours of production time available for the production of these toys. The manufacturer wants to know which of the new toys to produce, where and how many of each (if any) should be produced so as to maximize the total profit. Using appropriate decision variables (including 0-1 type) formulate the above problem as a mixed integer programming problem. Problem 4 There are 6 possible projects to choose from. Model each of the following constraints independent of the other constraints. a. Project 2 must be done At least two of the projects must be done. Project 3 is done if and only if Project 5 is done. Project 1 or 4 but not both must be done Project 4 can be done only if both project 1 and project 3 are done If projects 1 and 5 are done, project 6 must be done. If project 1 is done, project 2 should not be done. Project 3 should only be done if project 4 is not done. Smeoo0 o

Problem 5 Brenda Last, an undergraduate business major at State University, is attempting to determine her course schedule for the fall semester. She is considering seven 3-credit-hour courses, which are shown in the following table. Also included are the average number of hours she expects to have to devote to each course each week (based on information from other students) and her minimum expected grade in each course, based on an analysis of the grading records of the teachers in each course: Course Average Hours per Week ~ Minimum Grade Management I 5 B Principles of Accounting 10 c Corporate Finance 8 c Quantitative Methods 12 D Marketing Management 7 c Tava Programming 10 D English Literature 8 B An Ain a course earns 4 quality credits per hour, a B earns 3 quality credits, a C earns 2 quality credits, a D earns 1 quality credit, and an F earns no quality credits per hour. Brenda wants to select a schedule that will provide at least a 2.0 grade point average. In order to remain a full-time student, which she must do to continue receiving financial aid, she must take at least 12 credit hours. Principles of Accounting, Corporate Finance, Quantitative Methods, and Java Programming all require a lot of computing and mathematics, and Brenda would like to take no more than two of these courses. To remain on schedule and meet prerequisites, she needs to take at least three of the following courses: Management |, Principles of Accounting, Java Programming, and English Literature. Brenda wants to develop a course schedule that will minimize the number of hours she has to work each week. Formulate a 0-1 integer programming model for this problem.

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