Co370 Midterm
UNIVERSITY OF WATERLOO CO370/CM443: Deterministic OR Models Midterm Examination – FALL TERM 2011 Thursday, October 20, 2011, 7-9 PM (conflict time 4:45-6:45 PM) CLOSED BOOK
Surname: First Name: Signature: ID#: INSTRUCTIONS: 1. Write your name and Student ID# in the blanks above. 2. There are five questions. Some questions may be longer or more difficult than others. Read all the questions first and budget your time appropriately for each question. 3. Answer each question in your solution booklet. Problem 1 2 3 4 5 Total Number of Exam Pages: Value 20 15 30 20 15 100 Mark Awarded
11+2+1 pages (including AMPL appendix and cover sheet)
1
Problem 1: Sensitivity Analysis using Tableaus
(20 marks)
WireCo produces wire cloth, insect screens, and meshes. The production of each type of product requires aluminum, wiring hours, and weaving hours. Currently, 50 lb. of aluminum, 24 wiring hours, and 60 weaving hours are available. The resource requirements and sales price for one item of each type of product are listed below. Aluminum (lb.) Wiring (hours) Weaving (hours) Price ($) Wire cloth 1 0.5 1 6 Insect screen 1 2 2 14 Mesh 2 1 4 13
WireCo solves the following LP to find an optimal production plan. max s.t. 6xC + 14xS + 13xM xC + xS + 2xM + s1 0.5xC + 2xS + xM + s2 xC + 2xS + 4xM + s3 xC , xS , xM , s1 , s2 , s3
= = = ≥
50 24 60 0.
(aluminum) (wiring) (weaving)
Here xC , xS , and xM denote respectively the number of wire cloths, insect screens, and meshes manufactured, and s1 , s2 , and s3 are the slack variables for the aluminum, wiring, and weaving constraints respectively. Running the simplex method on this LP yields the following final tableau. z xC + 9xS − 3xS + s1 + 6xS − xS + xM + 11s2 + 0.5s3 − 2s2 + 4s2 − s3 − s2 + 0.5s3 = = = = 294 2 36 6
(a) What is the optimal solution? What are the shadow prices of the resource constraints? (4 marks) Solution: The optimal solution is to produce 36 wire cloths and 6 meshes to earn revenue of $294.
Surname: First Name: Signature: ID#: INSTRUCTIONS: 1. Write your name and Student ID# in the blanks above. 2. There are five questions. Some questions may be longer or more difficult than others. Read all the questions first and budget your time appropriately for each question. 3. Answer each question in your solution booklet. Problem 1 2 3 4 5 Total Number of Exam Pages: Value 20 15 30 20 15 100 Mark Awarded
11+2+1 pages (including AMPL appendix and cover sheet)
1
Problem 1: Sensitivity Analysis using Tableaus
(20 marks)
WireCo produces wire cloth, insect screens, and meshes. The production of each type of product requires aluminum, wiring hours, and weaving hours. Currently, 50 lb. of aluminum, 24 wiring hours, and 60 weaving hours are available. The resource requirements and sales price for one item of each type of product are listed below. Aluminum (lb.) Wiring (hours) Weaving (hours) Price ($) Wire cloth 1 0.5 1 6 Insect screen 1 2 2 14 Mesh 2 1 4 13
WireCo solves the following LP to find an optimal production plan. max s.t. 6xC + 14xS + 13xM xC + xS + 2xM + s1 0.5xC + 2xS + xM + s2 xC + 2xS + 4xM + s3 xC , xS , xM , s1 , s2 , s3
= = = ≥
50 24 60 0.
(aluminum) (wiring) (weaving)
Here xC , xS , and xM denote respectively the number of wire cloths, insect screens, and meshes manufactured, and s1 , s2 , and s3 are the slack variables for the aluminum, wiring, and weaving constraints respectively. Running the simplex method on this LP yields the following final tableau. z xC + 9xS − 3xS + s1 + 6xS − xS + xM + 11s2 + 0.5s3 − 2s2 + 4s2 − s3 − s2 + 0.5s3 = = = = 294 2 36 6
(a) What is the optimal solution? What are the shadow prices of the resource constraints? (4 marks) Solution: The optimal solution is to produce 36 wire cloths and 6 meshes to earn revenue of $294.