reinderien / crlayout

Designing Operations layout question

Problem

A popular textbook with Part 2 "Designing Operations", chapter 9 "Layout Decisions", has a homework question 9.5 that has a lot of traffic on various tutor and homework help sites on the internet:

9.5 Roy Creasey [sometimes shown as Ted Post] Enterprises, a machine shop, is planning to move to a new, larger location. The new building will be 60 feet long by 40 feet wide. Creasey envisions the building as having six distinct production areas, roughly equal in size. He feels strongly about safety and intends to have marked pathways throughout the building to facilitate the movement of people and materials. See the following building schematic.

Building Schematic (with work areas 1-6)

1--2--3
|  |  |
4--5--6

His foreman has completed a month-long study of the number of loads of material that have moved from one process to another in the current building. This information is contained in the following flow matrix.

Flow Matrix Between Production Processes

From \ To  Materials Welding Drills Lathes Grinders Benders
Materials          0     100     50      0        0      50 
Welding           25       0      0     50        0       0 
Drills            25       0      0      0       50       0 
Lathes             0      25      0      0       20       0 
Grinders          50       0    100      0        0       0 
Benders           10       0     20      0        0       0

Finally, Creasey has developed the following matrix to indicate distances between the work areas shown in the building schematic.

Distance Between Work Areas

   1  2  3  4  5  6
1    20 40 20 40 60
2       20 40 20 40
3          60 40 20
4             20 40
5                20
6                  

What is the appropriate layout of the new building?

Analysis

This is a discrete optimization problem. Given that there are only 6! = 720 permutations of the process placement this is trivially solvable by brute force:

• Fill out the lower triangle of the distances matrix such that it's symmetrical
• Represent the design as an array of six integers, where each index is an area and each value is a process
• Generate all permutations of that array
• For each area-area source-destination pair, calculate the sum of the product of the distance and the flow. This total is the cost.
• Find the minimum cost.
• Print all solutions that have the minimum cost.

Solution

As it turns out, the four optimal solutions are all reflections of a single solution about one or both axes.

The output is:

Number of processes to locate: 6
The smallest total rate-distance product cost found is 13000
The best area arrangements at this cost are:

        1         2         3 
   Lathes  Grinders    Drills 
        4         5         6 
  Welding Materials   Benders 

        1         2         3 
   Drills  Grinders    Lathes 
        4         5         6 
  Benders Materials   Welding 

        1         2         3 
  Welding Materials   Benders 
        4         5         6 
   Lathes  Grinders    Drills 

        1         2         3 
  Benders Materials   Welding 
        4         5         6 
   Drills  Grinders    Lathes