Question

Formulate the followings into optimization problems. While you could use the integer constraints, the linear structure should be maintained. However, you shouldn’t use the integer constraints if you can formulate the problem without them. Describe the decision variables, objective function and constraints carefully. You don’t need to solve this problem.

(b) (10 pts) Three workers are available to perform two jobs. The timei takes each worker to perform each job is given in the following table. Formulate the problem to determine an assignment of workers to jobs that minimizes the total time required to perform the four jobs. The solution should be integers Worker1 Worker2 Worker3 Job 1 T11 13 12 Job 2 T21 T 22 T 23

Answer 1

Let the decision variables be X11, X12, X13, X21, X22, and X23 corresponding to the time taken for job to the worker.

The optimization objective function can be written as

Minimize Z where

Z = summation Xij*Tij

Subject to constraints

X11 + X12 + X13 = 1          (job 1 should be assigned to 1 person only)

X21 + X22 + X23 = 1          (job 2 should be assigned to 1 person only)

X11 + X21 <= 1                   (worker 1 should be assigned 1 or no jobs)

X12 + X22 <= 1                   (worker 2 should be assigned 1 or no jobs)

X13 + X23 <= 1                   (worker 3 should be assigned 1 or no jobs)

Xij = {0,1}