Let us consider one producer P and four
customers, which are supplied each day
with one item of product each.
Producer P
Customers can be supplied only by trucks
and each truck can carry exactly one item
1
of the product at transportation cost
2000 crowns per unit distance. But, there
is a railway, which starts from P and goes
1
near to the customers through two
places, where transshipment places may
1
be constituted (each for 6000 crown per
1
1
day) . This transportation means is able
to transports one item at 1000 crowns
1
Customers per distance unit.
1
1
We expect that the average fixed
charges may move from 2000 to
8000.
The demand satisfaction costs may
vary from 70% to 150% of they
declared value.
1
Producer P (1)
The fixed charges
and costs were only
roughly estimated.
1
1
C1
C3
1
1
1 (2)
1
1
(3)
1
C2
Customers
C4
fi
0
6
I
1
2
6
3
J 1
cij C1
P 8
P1 5
P2 8
2
3
4
C2
C3
C4
8
5
10 10
7 7
8
6
6
We expect that the average
fixed charges may move
from 2 to 8 (thousands).
The
demand
satisfaction
costs may vary from 70% to
150% of they declared value.
2
fi =  fi ; cij = cij.
<0.33,1.33>
 <0.7, 1.5>
fi
0
6
6
I
1
2
3
J 1
cij C1
P
P1
P2
8
5
8
2
3
4
C2
C3
C4
8
5
8
 10
7
6
 10
7
6
We expect that the average fixed
charges may move from 2 to 8.
The demand satisfaction costs
may vary from 70% to 150% of
they declared value.
3
<0.33, 1.33>
 <0.7, 1.5>
fi
0
6
6


I
1
2
3
J 1
cij C1
P
P1
P2
8
5
8
2
3
4
C2
C3
C4
8
5
8
 10
7
6
 10
7
6
Let the parameters  and  independently
increment by 0.33 from their lower limits to upper
limits.
On the whole, we get 16 instances to solve.
4

In declaration part:

Declare ‘names’ for all set of constraints using data type ‘linctr’
constr_1 : array (1..4) of linctr
! name for first set of constraints
constr_2 : array (1..3,1..4) of linctr ! name for second set of constraints

In model part:

Name all constraints
forall (j in 1..4) constr_1(j) := sum(i in 1..3) z(i,j)=1
forall (i in 1..3, j in 1..4) constr_2(i,j) := z(i,j)<=y(i)
Run the model repeteadly in the loop for different alpha and beta
5
One salt producer P supplies five customers, which are supplied each day.
Each customer demands 2 tons of salt every day.
Customers can be supplied only by trucks and each truck can carry exactly
two tons of the salt. Unit transport cost for the truck is 50 NOK per
transported unit and km.
There is a also ferry, which starts from producer P and goes near to the
customers through two places P1 and P2, where transshipment places
may be constituted. If the transshipment will be at the transshipment place,
we must pay the rent for 600 NOK per day. Unit transport cost for the ferry
is 20 NOK per transported unit and km. The distance matrix is given.
Find a solution of this problem with minimal total cost. Write the
mathematical model and solve it in Xpress-IVE.
6
Local authorities want to build at most 2 fire brigades at some places from the set 1, 2, 3
and 4 so that the size of the part of population from set {1, 2, …, 10}, which is out of the
time limit Tmax =10 minutes from the nearest fire brigade, should be minimized. Local
authority has a budget of 110000 NOK, which can be used to build fire brigades. Building of
fire brigade costs:
at location 1: 55000 NOK
at location 2: 45000 NOK
at location 3: 45000 NOK
at location 4: 55000 NOK
Population bj in nodes {1, 2, …, 10} is: 2000, 1500, 100, 1200, 1150, 400, 100, 800, 350, 250.
Transportation network with distances (in km) is on the picture below. Average speed of the
fire brigade’s vehicle is 60 km per hour. To calculate travel times use the shortest distances
matrix dij
Find a solution of this problem. Write the description of decision variables and all constants
used in the model on this sheet or on its opposite site, write the mathematical model and
solve it in Xpress-IVE. Put down the answer: How many people will not be served (will be out
of the time limit) and at which locations will be the fire brigades located?
16
6
10
10
10
12
1
10
8
26
2
3
7
12
14
4
10
12
5
10
12
20
22
26
10
9
7
Let us consider that computer vendor want to locate 1 shop at some
places from the set 1, 2, 3 and 4 . The number p=1 of facilities is
fixed, but not each customer must be served. Service of customer j
brings profit, which is proportional to its demand bj , but only when
its distance from some located facility is less or equal than 25 km.
Each customer’s demand is equal to 10 (bj=10)
16
6
10
10
10
12
1
10
8
26
2
3
7
12
14
4
10
12
26
10
12
20
22
5
10
9
8
10
Maximize
b x
j 1
j
j
4
Subject to  a ij yi  x j
for j  1,..,10
i 1
4
y
i 1
i
p
yi  {0,1}
for i  1,..,4
x j  {0,1}
for j  1,..,10
9