Chapter 7
Transportation, Assignment, and
Transshipment Problems
12. a.
b.
Min 10x11 + 16x12 + 32x13 + 14x21 + 22x22 + 40x23 + 22x31 + 24x32 +
s.t.
x11 + x12 + x13
x21 +
x22 + x23
x31 + x32 +
x11
+ x21
+ x31
x12
+
x22
+ x32
x13
+ x23
+
34x33
x33
x33



=
=
=
1
1
1
1
1
1
xij  0 for all i, j
Solution x12 = 1, x21 = 1, x33 = 1
13. a.
Total completion time = 64
Optimal assignment: Jackson to 1, Smith to 3, and Burton to 2. Time requirement is 62 days.
b.
Considering Burton has saved 2 days.
c.
Ellis.
7-1
Chapter 7
14. a.
b.
Max 100x11 + 95x12 + 85x21 + 80x22 + 90x31 + 75x32
s.t.
x11 + x12
x21 + x22
x31 + x32
x11
+ x21
+ x31
x12
+ x22
+ x32



=
=
1
1
1
1
1
xij  0, i = 1,2,3; j = 1,2
Optimal Solution: x12 = 1, x31 = 1 Value: 185
Assign Bostock to the Southwest territory and Miller to the Northwest territory.
16. a.
7-2
Transportation, Assignment, and Transshipment Problems
1
1
10
Shoe
6
12
8
Location
1
1
15
1
2
18
Toy
5
11
Location
2
17
10
3
1
1
13
Auto Parts
16
Location
14
1
3
12
4
1 Housewares
13
10
14
1
5
Video
Location
4
16
6
12
7-3
1
Chapter 7
b.
1 if department i is assigned location j
Let xij  
 0 otherwise
Max
+
+
10x11 +
17x31 +
14x51 +
6x12 +
10x32 +
16x52 +
12x13 + 8x14 + 15x21 + 18x22
13x33 + 16x34 + 14x41 + 12x42
6x53 + 12x54
x11
x12 +
x13
+
+
5x23 + 11x24
13x43 + 10x44
s.t.
x11
+
x31
+
x51
+
+ x52
x21
+ x22
+
x14
x12
x13
x32 +
+
+
+
x14
x33
+
x34
x53
x31
+
+
x54
+
x23
+ x24
x32
+ x33
+
x21
+
x22
+
x23
+ x24
x41
+
x42
+
x43
+
x41
+ x42
+
x34
+ x51
+
x43
+ x44
x52
+ x53
+
x44
x54





=
=
=
=
1
1
1
1
1
1
1
1
1
xij  0 for all i, j
Optimal Solution:
Toy:
Auto Parts:
Housewares:
Video:
Profit: 61
18. a.
Location 2
Location 4
Location 3
Location 1
This is the variation of the assignment problem in which multiple assignments are possible.
Each distribution center may be assigned up to 3 customer zones.
The linear programming model of this problem has 40 variables (one for each combination of
distribution center and customer zone). It has 13 constraints. There are 5 supply ( 3)
constraints and 8 demand (= 1) constraints.
The problem can also be solved using the Transportation module of The Management Scientist.
The optimal solution is given below.
Plano:
Flagstaff:
Springfield:
Boulder:
19.
Assignments
Kansas City, Dallas
Los Angeles
Chicago, Columbus, Atlanta
Newark, Denver
Total Cost -
Cost ($1000s)
34
15
70
97
$216
b.
the Nashville distribution center is not used.
c.
All the distribution centers are used. Columbus is switched from Springfield to Nashville.
Total cost increases by $11,000 to $227,000.
A linear programming formulation and the optimal solution are given. For the variables, we let
the first letter of the sales representatives name be the first subscript and the sales territory be
the second subscript.
7-4
Transportation, Assignment, and Transshipment Problems
Max
44xWA + 80xWB + 52xWC + 60xWD + 60xBA + 56xBB + 40xBC
+ 72xBD + 36xFA + 60xFB + 48xFC + 48xFD + 52xHA
+ 76xHB + 36xHC + 40xHD
s.t.
xWA +
xWB +
xWC +
xWD
xBA +
xWA
+
xBB +
xBA
xWB
xBC +
xBD
xFA +
xFB +
xFA
+
xBB
+
xWC
+
+
xWD
xHA +
xHB +
xHC +
xHD
xHA
+
+
+
xFD
xFB
xBC
+
xFC +
xHB
xFC
xBD
+
+
xHC
xFD
+
xHD

1

1

1

1
=
1
=
1
=
1
=
1
xij  0 for all i, j
Optimal Solution
Sales
Washington - B
Benson - D
Fredricks - C
Hodson - A
80
72
48
52
252
Total
20.
A linear programming formulation of this problem can be developed as follows. Let the first
letter of each variable name represent the professor and the second two the course. Note that a
DPH variable is not created because the assignment is unacceptable.
Max 2.8AUN + 2.2AMB +
3.3AMS +
3.0APH + 3.2BUN +
··
·
+ 2.5DMS
s.t.
AUN +
AUN +
AMB
BUN
+
+
AMS +
BMB +
CUN +
BUN
AMB
+
+
CUN +
BMB +
AMS +
All Variables  0
Optimal Solution:
A to MS course
B to Ph.D. course
C to MBA course
D to Undergraduate course
Max Total Rating
Rating
3.3
3.6
3.2
3.2
13.3
7-5
APH
BMS
CMB
DUN
DUN
CMB
BMS
APH
+
+
+
BPH
CMS + CPH
DMB + DMS
+
+
+
DMB
CMS + DMS
BPH + CPH




=
=
=
=
1
1
1
1
1
1
1
1
Chapter 7
21. a.
Min 150x11
+ 210x12
+ 170x21
+ 270x13
+ 230x22
+ 180x31
+ 220x23
+ 230x32
+ 160x41
+ 225x33
+ 240x42 +
230x43
s.t.
x11
+
x11
x12
x21
+
+
+x21
x12
x13
x22
x31
+
+
+x31
+x22
x13
x23
x32
x41
+x41
+x32
+x23
+
+
x33
x42 +
+x42
+x33
+x43
x43




=
=
=
1
1
1
1
1
1
1
xij  for all i, j
Optimal Solution: x12 = 1, x23 = 1, x41 = 1
Total hours required: 590
Note: statistician 3 is not assigned.
b.
The solution will not change, but the total hours required will increase by 5. This is the extra
time required for statistician 4 to complete the job for client A.
c.
The solution will not change, but the total time required will decrease by 20 hours.
d.
The solution will not change; statistician 3 will not be assigned. Note that this occurs because
increasing the time for statistician 3 makes statistician 3 an even less attractive candidate for
assignment.
7-6