Topic 4 Questions and Suggested Solutions
Transportation problems
Question 1
Tom would like 3 pints of home brew today and an additional 4 pints of home brew tomorrow.
Dick is willing to sell a maximum of 5 pints total at a price of $3.00 per pint today and $2.70 per
print tomorrow. Harry is willing to sell a maximum of 4 pints total at a price of $2.90 per pint
today and $2.80 per pint tomorrow.
Tom wishes to know what his purchases should be to minimize his cost while satisfying his thirst
requirements. Formulate and solve a linear program for this problem.
Formulation
Total supply = 5+4 = 9
> Total demand = 3+4 = 7
Let Xij = number of pints of brew from person i (= 1 Dick, 2 Harry)
to day j (= 1 today, 2 tomorrow)
Min z = 3.0X11 + 2.7X12
+ 2.9X21 + 2.8X22
s.t.
X11 + X12 5
X21 + X22 4
X11 + X21 = 3
X12 + X22 = 4
with Xij 0
Solution
A B C D E F G
1 Unit Cost Destination
2 Today Tomorrow
3 Source Dick $3.00 $2.70
4 Harry $2.90 $2.80
5
6
7 Purchases Destination
8 Today Tomorrow Total Supply
9 Source Dick 0 4 4 ² 5
10 Harry 3 0 3 ² 4
11 Total Received 3 4
12 = = Total Cost
13 Demand 3 4 $19.50
N.B.: "2" in Cells 9F and 10F should be "".
1
, Question 2
The Versatech Corporation has decided to produce three new products. Five branch plants now
have excess product capacity. The unit manufacturing cost of the first product would be $31, $29,
$32, $28, and $29 in plants 1, 2, 3, 4, and 5, respectively. The unit manufacturing cost of the
second product would be $45, $41, $46, $42, and $43 in plants 1, 2, 3, 4, and 5, respectively. The
unit manufacturing cost of the third product would be $38, $35, and $40 in plants 1, 2, and 3, re-
spectively, whereas plants 4 and 5 do not save the capability for producing this product. Sales
forecasts indicate that 600, 1000, and 800 units of products 1, 2, and 3, respectively, should be
produced per day. Plants 1, 2, 3, 4, and 5 have the capacity to produce 400, 600, 400, 600, and
1000 units daily, respectively, regardless of the product or combinations of products involved.
Assume that any plant having the capability and capacity to produce them can produce any
combination of the products in any quantity.
Management wishes to know how to allocate the new products to the plants to minimize total
manufacturing cost. Formulate and solve a linear program for this problem.
Formulation
Total supply = 400+600+400+600+1000 = 3000
> Total demand = 600+1000+800 = 2400
Let Xij = number of units of product j (= 1,2,3) to be produced from plant i (= 1,2,3,4,5)
Min z = 31X11 + 45X12 + 38X13
+ 29X21 + 41X22 + 35X23
+ 32X31 + 46X32 + 40X33
+ 28X41 + 42X42 + -X43
+ 29X51 + 43X52 + -X53
s.t.
X11 + X12 + X13 400
X21 + X22 + X23 600
X31 + X32 + X33 400
X41 + X42 + X43 600
X51 + X52 + X53 1000
X11 + X21 + X31 + X41 + X51 = 600
X12 + X22 + X32 + X42 + X52 = 1000
X13 + X23 + X33 + X43 + X53 = 800
with Xij 0
Solution
2
Transportation problems
Question 1
Tom would like 3 pints of home brew today and an additional 4 pints of home brew tomorrow.
Dick is willing to sell a maximum of 5 pints total at a price of $3.00 per pint today and $2.70 per
print tomorrow. Harry is willing to sell a maximum of 4 pints total at a price of $2.90 per pint
today and $2.80 per pint tomorrow.
Tom wishes to know what his purchases should be to minimize his cost while satisfying his thirst
requirements. Formulate and solve a linear program for this problem.
Formulation
Total supply = 5+4 = 9
> Total demand = 3+4 = 7
Let Xij = number of pints of brew from person i (= 1 Dick, 2 Harry)
to day j (= 1 today, 2 tomorrow)
Min z = 3.0X11 + 2.7X12
+ 2.9X21 + 2.8X22
s.t.
X11 + X12 5
X21 + X22 4
X11 + X21 = 3
X12 + X22 = 4
with Xij 0
Solution
A B C D E F G
1 Unit Cost Destination
2 Today Tomorrow
3 Source Dick $3.00 $2.70
4 Harry $2.90 $2.80
5
6
7 Purchases Destination
8 Today Tomorrow Total Supply
9 Source Dick 0 4 4 ² 5
10 Harry 3 0 3 ² 4
11 Total Received 3 4
12 = = Total Cost
13 Demand 3 4 $19.50
N.B.: "2" in Cells 9F and 10F should be "".
1
, Question 2
The Versatech Corporation has decided to produce three new products. Five branch plants now
have excess product capacity. The unit manufacturing cost of the first product would be $31, $29,
$32, $28, and $29 in plants 1, 2, 3, 4, and 5, respectively. The unit manufacturing cost of the
second product would be $45, $41, $46, $42, and $43 in plants 1, 2, 3, 4, and 5, respectively. The
unit manufacturing cost of the third product would be $38, $35, and $40 in plants 1, 2, and 3, re-
spectively, whereas plants 4 and 5 do not save the capability for producing this product. Sales
forecasts indicate that 600, 1000, and 800 units of products 1, 2, and 3, respectively, should be
produced per day. Plants 1, 2, 3, 4, and 5 have the capacity to produce 400, 600, 400, 600, and
1000 units daily, respectively, regardless of the product or combinations of products involved.
Assume that any plant having the capability and capacity to produce them can produce any
combination of the products in any quantity.
Management wishes to know how to allocate the new products to the plants to minimize total
manufacturing cost. Formulate and solve a linear program for this problem.
Formulation
Total supply = 400+600+400+600+1000 = 3000
> Total demand = 600+1000+800 = 2400
Let Xij = number of units of product j (= 1,2,3) to be produced from plant i (= 1,2,3,4,5)
Min z = 31X11 + 45X12 + 38X13
+ 29X21 + 41X22 + 35X23
+ 32X31 + 46X32 + 40X33
+ 28X41 + 42X42 + -X43
+ 29X51 + 43X52 + -X53
s.t.
X11 + X12 + X13 400
X21 + X22 + X23 600
X31 + X32 + X33 400
X41 + X42 + X43 600
X51 + X52 + X53 1000
X11 + X21 + X31 + X41 + X51 = 600
X12 + X22 + X32 + X42 + X52 = 1000
X13 + X23 + X33 + X43 + X53 = 800
with Xij 0
Solution
2
