Algebraic modeling in datalog

1
Algebraic modeling in datalog

• Diego Klabjan, Jun Ma, Robert Fourer
  Northwestern University

2
Datalog

• Data query languages
• SQL, Xquery
• No procedural or declarative abilities
• Procedural and declarative languages
• No data querying capabilities
• Best of both worlds

DATALOG
3
Algebraic Modeling
4
SQL

• Modeling in SQL by Linderoth, Atamturk,
  Savelsbergh
• SQL mainly about querying
• Not suited for algebraic modeling
• Everything stored in tables
• Variables
• Constraints
• Modeling non intuitive

5
Modeling in Datalog

• Powerful data capabilities
• Superset of SQL
• Data from database
• Querying and loading
• Declarative language
• Natural constructs
• Intuitive
• Hardly any development effort

6
Modeling in datalog

• Given values to decision variables
• Easy to check feasibility
• Clearly not optimality
• Underlying logic programming in Datalog
• No extra effort required
• Not the case for most other algebraic modeling
  languages

7
Basic building blocks (Diet problem)

• Index sets
• NUTR(x), NUTRname(xn) -gt string(n). FOOD(x),
  FOODname(xn) -gt string(n).
• Parameters
• Input data
• Variables
• Buyf b -gt FOOD(f), float64(b),bgt0.

amtn, f a -gt NUTR(n), FOOD(f), float64(a),
a gt 0. nutrLown nL -gt NUTR(n),
float64(nL), nL gt 0. costf c -gt FOOD(f),
float64(c), c gt 0.
8
Production/transportation model

• Multiple products (PROD), plants (ORIG) and
  destinations (DEST)
• Ship from plants to destinations
• Transportation problem for each product

9
Production/transportation model

• Limited production capacity at each plant
• Objective to minimize production and
  transportation costs

10
Datalog Model

• Sets
• Data

11
Datalog model

• Input parameters

12
Datalog model

• Variables
• How much to transport
• How much to produce
• Other variables
• Integer replace float with int
• Binary are integer with upper bound of 1

13
Datalog model

• Production availability
• sumTime predicate captures the left-hand side
• agg built-in aggregator
• Constraint negated (stratification restrictions
  of Datalog)

14
Datalog model

• Demand constraints

15
Datalog model

• Supply constraints

16
Datalog model

• Objective function
• Production cost
• Transportation cost

17
Total cost

• Sum of the two cost components

18
Enhanced modeling capabilities

• The fleeting model
• Assign fleets to flights
• Modeled as a multi-commodity network flow problem
• Network aspects
• Challenges
• Nodes at each airports sorted based on the
  arrival/departure time in a circular fashion
• Ordered and circular lists

19
Flights

• Specification of flights
• Leg(l), Legname(ln) -gt string(n).
• Legtable(s1,t1,s2,t2,l) -gtStation(s1), Time(t1),
  Station(s2), Time(t2), Leg(l).
• LegtabledStationls1 -gtStation(s1), Leg(l).
• LegtabledTimelt1 -gt Time(t1), Leg(l).
• LegtableaStationls2 -gtStation(s2), Leg(l).
• LegtableaTimelt2 -gt Time(t2), Leg(l).

20
NETWORk nodes

• For each station there is a timeline consisting
  of network nodes
• Either arrival or departure at station

node(s,t) -gt Station(s), Time(t). node(s,t) lt-
Legtable(s1,t1,_,_,_),(ss1, tt1)Legtable(_,_,
s2,t2,_),(ss2, tt2).
21
Ordered Circular lists

• Declare next in the list
• Time is an integer-like structure to capture
  times
• nodenxts,t1 t2 -gt Station(s), Time(t1),
  Time(t2).
• Order
• nodenxts,t1 t2 lt- node(s,t1), node(s,t2),
  (Timedatetimet1ltTimedatetimet2,!anythingInBe
  tween(s, t1,t2) nodefrstst2,
  nodelstst1).

22
Missing aspects

• Piecewise linear functions
• Model them explicitly
• Unfortunately OS cannot handle them explicitly
• LogicBlox needs to convert them into a linear
  mixed integer program
• Disjunctions
• Nonlinear functions
• Long term goal