Michal Kohni

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XPRESS IVE Seminary

• Michal Koháni
• Department of Transportation Networks
• Faculty of Management Science and Informatics
• University of Zilina, Slovakia

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XPRESS IVE Basic Features

• is an advanced modeling and solving language and
  environment, where optimization problems can be
  specified and solved with the utmost precision
  and clarity
• enables you to gather the problem data from text
  files and a range of popular spreadsheets and
  databases, and gives you access to a variety of
  solvers, which can find optimal or near-optimal
  solutions to your model
• Some features easy syntax, supports dynamic
  objects,
• More information www.dashoptimization.com

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XPRESS IVE Student Version

• Maximum number of constraints (rows) 400
• Maximum number of variables (columns) 800
• Maximum number of matrix coefficients (elements)
  5000
• Maximum number of binary and integer variables,
  etc (global elements) 400

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XPRESS IVE
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XPRESS IVE
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XPRESS IVE
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XPRESS IVE
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XPRESS IVE
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XPRESS IVE
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Production Problem

• An enterpriser produces and sells chips and
  French-fries at unit profits of 80 and 50 crowns
  per kilogram of the products respectively.
• To produce 1 kg of chips, there is necessary 2 kg
  of potatoes and 0.4 kg of oils. To produce 1 kg
  of French-fries, it is necessary 1.5 kg of
  potatoes and 0.2 kg of oil.
• The enterpriser bought 100 kg of potatoes and 16
  kg of oils, when they were on stock. Which
  quantities of the particular products should the
  enterpriser produce to maximize his profit and
  not to exceed the limited quantities of potatoes
  and oil?

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Production Problem Model
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Production Problem

• Name of the model
• Options
• Parameters
• Declarations (variables, arrays)
• External Data Entry
• Objective Function
• Structural Constraints
• Output and solution

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Production Problem Model Name, Declarations

• model Production_Problem
• uses mmxprs
• declarations
• x1,x2 mpvar
• end-declarations

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Production Problem Objective Function, Contraints

• ! Objective function
• Profit 80x1 50x2
• ! constraints
• 2x1 1.5x2 lt 100
• 0.4x1 0.2x2 lt 16
• maximize(Profit)

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Production Problem Output, Solution

• ! Value of objective function - getobjval
• writeln(Total Profit , getobjval)
• ! Values of variables getsol()
• writeln(x1 ,getsol(x1))
• writeln(x2 ,getsol(x2))
• end-model

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The Transportation Problem
Demands bj tons of cement of customers j 1, 2,
3, 4 are to be satisfied as cheap as possible,
where the values of bj are 7, 8, 10 a 11
respectively. The demands can be satisfied only
from warehouses i 1, 2, 3, which disposes with
supplies ai tons of cement , where the values of
ai 10, 15, 11 respectively. The unit costs for
transportation of one ton of cement from
warehouse i to customer j are cij (see the
table).
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The Transportation Problem Model
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The Transportation Problem Model
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The Transportation Problem

• Name of the model
• Options
• Parameters
• Declarations (variables, arrays)
• Data Entry
• Objective Function
• Structural Constraints
• Output and solution

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The Transportation Problem Model Name,
Declarations

• model Transportation_Problem
• uses mmxprs
• declarations
• x array (1..3, 1..4) of mpvar !variables
• a array (1..3) of integer !supplies
• b array (1..4) of integer !demands
• c array (1..3, 1..4) of integer !Trans.unit
  cost
• end-declarations

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The Transportation Problem Objective Function,
Contraints

• a 10, 15, 11
• b 7, 8, 10, 11
• c 4, 5, 5, 3,
• 6, 6, 7, 8,
• 5, 7, 7, 5
• ! Objective function
• Cost sum (i in 1..3, j in 1..4) c(i,j)x(i,j)
• ! constraints
• forall (i in 1..3) sum (j in 1..4) x(i,j) lt a(i)
• forall (j in 1..4) sum (i in 1..3) x(i,j) b(j)
• minimize(Cost)

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The Transportation Problem Output, Solution

• ! Value of objective function - getobjval
• writeln(Total Cost , getobjval)
• ! Values of variables with loop getsol()
• forall (i in 1..3, j in 1..4)
• writeln ( x ,i,,,j, , getsol (x(i,j)))
• end-model

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Uncapacited Facility Location ProblemToy example
from Wednesday

• 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.

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Uncapacited Facility Location Problem Toy
example from Wednesday

• The prime cost e0 is 2000 and e1 is 1000 per km.
• Handling cost gi 0 and bj1.

1
1
1
C1
C2
1
1
P1
1
Customers
1
1
P2
C3
C4
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Uncapacited Facility Location Problem Toy
example from Wednesday
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1
1
C1
C2
1
1
P1
1
Customers
1
1
P2
C3
C4
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Uncapacited Facility Location Problem Toy
example from Wednesday
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1
1
C1
C2
1
1
P1
1
Customers
1
1
P2
C3
C4
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XPRESS IVE

• Name of the model
• Options
• Parameters
• Declarations (variables, arrays)
• External Data Entry
• Objective Function
• Structural Constraints
• Output and solution

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Writing a model in Mosel Name of the model
options

• model ToyExample
• uses mmxprs
• !other sections
• end-model

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Writing a model in Mosel Declarations of
decision variables

• declarations
• y1,y2,y3 mpvar
• z11,z12,z13,z14,z21,z22,z23,z24 mpvar
• z31,z31,z33,z34 mpvar
• end-declarations
• y1 is_binary
• y2 is_binary
• y3 is_binary
• z11 is_binary
• z12 is_binary
• z34 is_binary

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Writing a model in Mosel Objective function
constraints

• ! Objective function
• Cost6y26y38z115z218z318z125z228z3
  2
• 10z137z236z3310z147z246z34
• ! constraints
• z11z21z311
• z12z22z321
• z13z23z331
• z14z24z341
• z11lty1
• z34lty3
• minimize(Cost) !you dont need to declare Cost

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Writing a model in Mosel Output results

• ! Value of objective function - getobjval
• writeln(Total cost , getobjval)
• ! Value of decision variable
• writeln(y1 ,getsol(y1))
• writeln(y2 ,getsol(y2))
• writeln(y3 ,getsol(y3))
• writeln(z11 ,getsol(z11))
• writeln(z21 ,getsol(z21))
• writeln(z31 ,getsol(z31))
• writeln(z34 ,getsol(z34))

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Writing a model in Mosel Results

• Total cost 30
• y1 1
• y2 1
• y3 0
• z11 0
• z12 0
• z13 0
• z14 0
• z21 1
• z22 1
• z23 1
• z24 1
• z31 0
• z32 0
• z33 0
• z34 0

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Writing a model in Mosel Loops, sums data
from file
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Writing a model in Mosel Declarations of
variables using arrays loops

• declarations
• y array (1..3) of mpvar
• z array (1..3,1..4) of mpvar
• f array (1..3) of integer
• c array (1..3,1..4) of integer
• end-declarations
• forall (i in 1..3) y(i) is_binary
• forall (i in 1..3,j in 1..4) z(i,j) is_binary

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Writing a model in Mosel Data input

• initializations from ToyData.txt
• fc
• end-initializations
• Structure of the input file ToyData.txt
• f0,6,6
• c8, 8,10,10,
• 5, 5, 7, 7,
• 8, 8, 6, 6

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Writing a model in Mosel Objective function
constraints

• ! Objective function
• Costsum(i in 1..3) f(i)y(i) sum(i in 1..3,j
  in 1..4)
• c(i,j)z(i,j)
• ! Constraints
• forall (j in 1..4) sum(i in 1..3) z(i,j)1
• forall (i in 1..3, j in 1..4) z(i,j)lty(i)
• minimize(Cost) !you dont need to declare Cost

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Writing a model in Mosel Output results

• ! Value of objective function
• writeln(Total cost , getobjval)
• ! Values of decision variable
• forall(i in 1..3) writeln(y(,i,)
  ,getsol(y(i)))
• forall(i in 1..3, j in 1..4)
• writeln(z(,i,,,j,) ,getsol(z(i,j)))

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Maximum distance problem Toy example
Let us consider that local authorities want to
locate ambulance vehicles at some places from the
set 1, 2, 3 and 4 so that a distance from the
worst located dwelling place from set 1, 2, ,
10 to an ambulance be at most 25 km.
i
i
aij1
aij0
customers
j
D
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Maximum distance problem Toy example
Distance matrix
Incidental matrix
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Maximum distance problem Toy example
Mathematic model of the Maximum distance problem
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p-Median Problem
Let us consider that local authorities want to
locate p2 facilities at some places from the set
1, 2, 3 and 4 so that an average distance
between customer and the nearest facility should
be minimized.
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p-Median Problem
Distance matrix
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p-Median ProblemToy example from Tuesday
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p-Centre ProblemToy example
Let us consider that local authorities want to
locate p2 fire brigades at some places from the
set 1, 2, 3 and 4 so that a distance from the
worst located dwelling place from set 1, 2, ,
10 to a fire brigade be minimal.
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p-Centre ProblemToy example

• The 2-Centre Problem consists in minimizing the
  maximum distance between customer and the nearest
  located facility

Distance matrix
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p-Centre ProblemToy example
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Maximum covering location problem Toy example
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 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 customers demand is equal to 10
(bj10)
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Maximum covering location problem Toy example
Distance matrix
Incidental matrix
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Maximum covering location problem Toy example
from Tuesday