1. Introduction to Operations Research (II)
Z. Max Shen
IEOR, Berkeley
Last update September 7, 1999

2. Production Scheduling
Let all things be done decently and in order.
– I Corinthians

3. Goals of Production Scheduling
High Customer Service: on-time delivery
Low Inventory Levels: WIP and FGI
High Utilization: of machines
UTILIZATION
SERVICE
INVENTORY

4. QUICK WASH AND DRY
There are 3 jobs (dirty plates and utensils) that need to be processed (washed and dried) on two machines. These jobs need to be done first on machine 1 (wash) and then on machine 2 (dry). These jobs can be done only one at a time at each machine and at one machine at a time.
The times needed to process these 3 jobs on machines 1 and 2 are given below.

5. QUICK WASH AND DRY
Find a sequence to process these jobs on machine 1 and machine 2 so that the time at which the last job on machine 2 is finished is minimized.

6. QUICK WASH AND DRY
There are 6 jobs (dirty plates and utensils) that need to be processed (washed and dried) on two machines. These jobs need to be done first on machine 1 (wash) and then on machine 2 (dry). These jobs can be done only one at a time at each machine and at one machine at a time.
The times needed to process these 6 jobs on machines 1 and 2 are given below.

7. Classic Multi Machine Results
Minimizing “Makespan” on Two Machines: given a set of jobs that must go through a sequence of two machines, what sequence will yield the minimum makespan?
Mapespan is sequence dependent.
Simple algorithm (Johnson 1954):
1. Sort the times of the jobs on the two machines in two lists.
2. Find the shortest time in either list and remove job from both lists.
If time came from first list, place job in first available position.
If time came from second list, place job in last available position in sequence.
3. Repeat until lists are exhausted.
The resulting sequence will minimize makespan.

8. Johnson’s Algorithm Example
Data:
Iteration 1: min time is 4 (job 1 on M1); place this job first and remove from lists:

9. Johnson’s Algorithm Example (cont.)
Iteration 2: min time is 5 (job 3 on M2); place this job last and remove from lists:
Iteration 3: only job left is job 2; place in remaining position (middle).
Final Sequence: 1-2-3
Makespan: 28

10. Gantt Chart for Johnson’s Algorithm Example
Short task on M2 to
“clear out” quickly.
Short task on M1 to
“load up” quickly.