Ryerson Polytechnic University Department of Mathematics, Physics, and Computer Science Final Examinations, April, 2003

Transcription

1 Ryerson Polytechnic University Department of Mathematics, Physics, and Computer Science Final Examinations, April, 2003 MTH Operations Research I Duration: 3 Hours. Aids allowed: Two sheets of notes are allowed, and a non-programmable calculator. Write your name and student ID number in the spaces provided below The marks for each question are indicated by [ ]. Total marks = 140. Read each question carefully and make sure you answer all parts of each question. If you use the back side of the pages to continue your answers, clearly indicate that you are doing so. There are 8 questions in this test - please make sure all the pages are here. Answer all questions Last Name: First Name: Student Number:

2 (1) [15] Consider the following LP problem: max z = 5x 1 + 3x 2 + x 3 s.t. 2x 1 + x 2 + x 3 6 x 1 + 2x 2 + x 3 7 x 1, x 2, x 3, 0 Graphically solve this dual of this LP. Then use complementary slackness to solve the LP (max) problem.

3 (2) [15] Mondo produces motorcycles at three plants. At each plant, the labour, raw material, and production costs (excluding labour costs) required to build a motorcycle are as shown in the table below. Each plant has sufficient machine capacity to produce up to 750 motorcycles per week. Each of Mondo s workers can work up to 40 hours per week and is paid $12.50 per hour worked. Mondo has a total of 525 workers and now owns 9,400 units of raw material. Each week, at least 1400 Mondos must be produced. Let x 1 = motorcycles produced at plant 1; x 2 = motorcycles produced at at plant 2; and x 3 = motorcycles produced at plant 3. The LINDO output on the following page enables Mondo to minimize the cost (labour + production) of meeting demand. Use the output to answer the following questions: (a) What would be the new optimal solution to the problem if the production cost at plant 1 were only $40? (b) How much money would Mondo save if the capacity of plant 3 were increased by 100 motorcycles? (c) By how much wold Mondo s cost increase if it had to produce one more motorcycle? Raw Material Production Plant Labour Needed Needed Cost 1 20 hours 5 units $ hours 8 units $ hours 7 units $100

4 (3) [20] Consider the following LP and its optimal tableau (shown below): maz z = 3x 1 + 4x 2 + x 3 s.t. x 1 + x 2 + x x 1 x 2 + x 3 15 x 1 + x 2 = 10 x 1, x 2, x 3, 0 z x 1 x 2 x 3 s 1 e 2 a 2 a 3 rhs M M Answer the following questions using sensitivity analysis. (a) Find the range of values of the objective function coefficient of x 1 for which the current basis remains optimal.

5 (b) Find the range of values of the objective function coefficient for x 2 for which the current basis remains optimal.

6 (c) A new variable x 4 is added to the problem; maz z = 3x 1 + 4x 2 + x 3 + 6x 4 s.t. x 1 + x 2 + x 3 + 2x x 1 x 2 + x 3 15 x 1 + x 2 + x 4 = 10 x 1, x 2, x 3, 0 Should the variable x 4 be entered into the current basis? (d) Suppose the column for x 1 changes; maz z = 3x 1 + 4x 2 + x 3 s.t. x 1 + x 2 + x 3 50 x 1 x 2 + x 3 15 x 1 + x 2 = 10 x 1, x 2, x 3, 0 Is the current basis still optimal?

7 (4) [20] Currently, City University can store 200 files on hard disc, 100 files in computer memory, and 300 files on tape. Users want to store 300 wordprocessing files, 100 packaged-program files, and 100 data files. Each month a typical word-processing file is accessed eight times, a typical packagedprogram file, four times; and a typical data file, two times. When a file is accessed, the time it takes for the file to be retrieved depends on the type of file and on the storage medium (see table below). Time (mintues): Word Packaged Processing Program Data Hard disc Memory Tape (a) If the goal is to minimize the total time per month that users spend accessing their files, formulate a balanced transportation problem that can be used to determine where files should be stored. Sketch the network and write-out the LP problem. (Hint: The sources are the hard disc, the computer memory, and the tape; while the destinations are the word-processing files, the packaged-programs, and the date files.)

8 (b) Use the minimum cost method to find a bfs. (c) Use the transportation simplex method to find an optimal solution.

9 (5) [15] Formulate the problem of finding the shortest path from node 1 to node 7 in the figure below as a Minimal Cost Network Flow Problem (MCNFP). Just formulate the problem, do not solve.

10 (6) [15] Find a non-optimal bfs for the network shown below, and perform one iteration of the network simplex method.

11 (7) [20] Let [ be the pay-off matrix of a two person zero-sum game. ] (a) Find the min-max and max-min strategies and their values. (b) Formulate the linear programming problems that find the optimal mixed strategies (but do not solve). Write down a formula that gives the value of the game in this case.

12 (c) If each player choose their moves randomly with equal probability for each choice, what would be the value of the game? (d) If one player chooses his moves randomly with equal probability for each choice, and the other player chooses her optimal mixed strategy as found in part (b), what would be the value of the game?

13 (8) [20] Use the branch-and-bound method to solve the following integer programming problem: minz = 3x 1 + x 2 s.t. 5x 1 + x x 1 + x 2 8 x 1, x 2, 0 x 1, x 2, integer