Departments A, B, C, and D need to be assigned to four rooms 1, 2, 3, and 4. These rooms are arranged in a row, in that order, with 20 meters between each. The departmental work flows are contained in the table below

(a) What is the material handling total of assigning A-1, B-2, C-3, D-4?
(b) What is the material handling total of assigning A-1, B-3, C-4, D-2?

Flow Matrix
Dept. A Dept. B Dept. C Dept. D
Dept. A 0 30 5 20
Dept. B 5 0 40 20
Dept. C 0 10 0 40
Dept. D 10 5 0 0

(a) The material handling total of A-1, B-2, C-3, D-4 is 5500. Details: 1-2 (A-B): 20 × 30 + 20 × 5 = 700; 1-3 (A-C): 40 × 5 + 40 × 0 = 200; 1-4 (A-D): 60 × 20 + 60 × 10 = 1800; 2-3 (B-C): 20 × 40 + 20 × 10 = 1000; 2-4 (B-D): 40 × 20 + 40 × 5 = 1000; and 3-4 (C-D): 20 × 40 + 20 × 0 = 800. The sum of these six elements is 5500.

(b) The material handling total of A-1, B-3, C-4, D-2 is 5400 (which is also the optimal solution). Details: 1-2 (A-D): 20 × 20 + 20 × 10 = 600; 1-3 (A-B): 40 × 30 + 40 × 5 = 1400; 1-4 (A-C): 60 × 5 + 60 × 0 = 300; 2-3 (B-D): 20 × 20 + 20 × 5 = 500; 2-4 (C-D):40 × 40 + 40 × 0 = 1600; and 3-4 (B-C): 20 × 40 + 20 × 10 = 1000. These six elements sum to 5400.