Let us consider one producer P and four customers, which are supplied each day with one item of product each. Customers can be supplied only by trucks and each truck can carry exactly one item of the product at transportation cost 2000 crowns per unit distance. But, there is a railway, which starts from P and goes near to the customers through two places, where transshipment places may be constituted (each for 6000 crown per day). This transportation means is able to transports one item at 1000 crowns per distance unit. We expect that the average fixed charges may move from 2000 to The demand satisfaction costs may vary from 70% to 150% of they declared value. 1 Producer P Customers

2 J1234 fifi Ic ij C1C1 C2C2 C3C3 C4C4 01 P P1P P2P The fixed charges and costs were only roughly estimated. Producer P (1) Customers (2) (3) C1C1 C2C2 C3C3 C4C4 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.

3 J1234 fifi Ic ij C1C1 C2C2 C3C3 C4C4 01 P  8 8  8 8  10  6 2 P1P1  5 5  5 5  7 7  7 7 3 P2P2  8 8  8 8  6 6  6 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.    f i =  f i ; c ij =  c ij.

4 J1234 fifi Ic ij C1C1 C2C2 C3C3 C4C4 01 P  8 8  8 8  10  6 2 P1P1  5 5  5 5  7 7  7 7 3 P2P2  8 8  8 8  6 6  6 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.   

5  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

6 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.

7 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 T max =10 minutes from the nearest fire brigade, should be minimized. Local authority has a budget of NOK, which can be used to build fire brigades. Building of fire brigade costs: at location 1: NOK at location 2: NOK at location 3: NOK at location 4: NOK Population b j 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 d ij 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?

8 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 b j, but only when its distance from some located facility is less or equal than 25 km. Each customer’s demand is equal to 10 (b j =10)

9