# A little programming language for integer linear programming

Integer linear programming (ILP) is a good fit for making rotas. Expressing rota constraints as mathematical expressions is concise and easy to read. When translated into code, however, it’s not so nice. The PuLP library does a pretty good job, but it’s more verbose.

Here are a pair of constraints from the Solving Scheduling Problems with Integer Linear Programming memo about fairly distributing rota assignments amongst the available people:

$\forall t \in \mathcal T \text{, } \forall p \in \mathcal P \text{, } \forall r \in \mathcal R \text{, } X_p \geqslant A_{tpr}$ $\forall p \in \mathcal P \text{, } X_p \leqslant \sum_{t \in \mathcal T} \sum_{r \in \mathcal R} A_{tpr}$

It’s a bit dense, but it only has the necessary information. Now here’s the corresponding Python:

for slot in range(slots):
for person in people:
for role in roles:
problem += is_assigned[person] >= assignments[slot, person, role]

for person in people:
problem += is_assigned[person] <= pulp.lpSum(assignments[slot, person, role] for slot in range(slots) for role in roles)


That’s a bit more verbose, takes up more space, we’ve got this pulp.lpSum thing, this mysterious problem variable.. I’d prefer to be able to write an ASCII equivalent of the mathematical form, and have the Python generated for me.

The source file for this memo is Literate Haskell, you can load it directly into GHCi. So here’s the necessary ceremony:

{-# LANGUAGE LambdaCase #-}
{-# LANGUAGE GADTs #-}

import Data.Foldable (for_)
import Data.List     (intercalate, sort)
import Data.Maybe    (listToMaybe)


Let’s begin!

## Concrete syntax

My main motivation when coming up with the concrete syntax was “it should look like the maths, but be ASCII, and not be LaTeX because that would be a pain”. It should also be concise, in particular it shouldn’t be necessary to specify what type quantifiers range over (that should be inferred).

I’m going to use the basic rota generator described in the other memo as a running example.

An ILP language is, necessarily, not very expressive. So I decided on the following types:

• User-defined types (sets of values)
• Predicates
• Integers
• N-dimensional binary arrays
• N-dimensional integer arraysI’ve not implemented these because the example didn’t need them.

Predicates take parameters and arrays take indices, the types of these will also be modelled. A predicate which takes two integers is a different type to a predicate which takes an integer and a set-value.

We also want to distinguish between “parameter” variables, which the user of the model will supply, and “model” variables, which the model will solve for.

Here’s an example with some comments:

-- Define three new types
type TimeSlot, Person, Role

-- M is an input of type integer
param integer M

-- is_leave is an input of type (TimeSlot, Person) -> bool
param predicate is_leave(TimeSlot, Person)

-- A is a 3D binary array the solver will try to produce
model binary A[TimeSlot, Person, Role]

-- X is a 1D binary array (or "vector" if you like special-case
-- terminology...)  the solver will try to produce
model binary X[Person]

Now we have our constraints and objective function:

-- In every time slot, each role is assigned to exactly one person
forall t, r; sum{p} A[t,p,r] = 1

-- Nobody is assigned multiple roles in the same time slot
forall t, p; sum{r} A[t,p,r] <= 1

-- Nobody is assigned a role in a slot they are on leave for
forall p, t if is_leave(t,p), r; A[t,p,r] = 0

-- Nobody works too many shifts
forall p; sum{t, r} A[t,p,r] <= M

-- Assignments are fairly distributed
forall t, p, r; X[p] >= A[t,p,r]
forall p; X[p] <= sum{t} r A[t,p,r]

maximise sum{p} X[p]

Look how concise they are! They don’t reference any types either!

Even though forall and sum are conceptually similar (bring a new variable into scope and do some sort of quantification) I picked different syntax for them because forall introduces multiple actual constraints: one for each value of the user-defined type being quantified over. The forall quantifier is part of the meta-language, the sum quantifier is part of the ILP language.

## Abstract syntax

Let’s talk abstract syntax. I didn’t want to write a parser, so we’ll skip over that. We’re now getting to our first real bit of Haskell code.

In the concrete syntax there’s only one forall and it’s followed by a list of variables to quantify over, and then the constraint. To simplify the implementation, the abstract-syntax-CForall has exactly one variable, and can either be followed by another CForall or a CCheck (the bit like sum{r} A[t,p,r] <= 1).

A CForall also contains an optional predicate restriction, which is expressed as the predicate’s name followed by the list of arguments.

type Name = String

data Constraint a
= CForall (TypedName a) (Maybe (Name, [Name])) (Constraint a)
| CCheck Op (Expression a) (Expression a)
deriving Eq

data Op = OEq | OLt | OGt | OLEq | OGEq
deriving Eq


We’ll talk about the TypedName bit in the next section, but it’s essentially the name of the quantifier variable. Here are some examples, where E1 and E2 are placeholders for expressions:

• concrete: E1 <= E2

abstract: CCheck OEq E1 E2

• concrete: forall x; E1 = E2

abstract: CForall (Untyped "x") Nothing (CCheck OEq E1 E2)

• concrete: forall x, y if p(x, y); E1 < E2

abstract: CForall (Untyped "x") Nothing (CForall (Untyped "y") (Just ("p", ["x", "y"])) (CCheck OLt E1 E2))

The expression language is a bit richer, there are more forms of expressions than there are constraints:

data Expression a
= ESum (TypedName a) (Expression a)
| EIndex Name [Name]
| EVar Name
| EConst Integer
| EMul Integer (Expression a)
| EAdd (Expression a) (Expression a)
deriving Eq


Here are some more examples:

• concrete: 1 + 2

abstract: EAdd (EConst 1) (EConst 2)

• concrete: X[y,z]

abstract: EIndex "X" ["y", "z"]

• concrete: 3 * sum{i} X[i]

abstract: EMul 3 (ESum (Untyped "i") (EIndex "X" ["i"]))

Reading terms expressed in this abstract syntax would be a bit of a pain, so here’s some pretty-printing:

instance Show (Constraint a) where
show (CForall tyname (Just (rname, rargs)) c) =
"forall " ++ show tyname ++ " if " ++ rname ++ "(" ++ strings rargs ++ "); " ++ show c
show (CForall tyname _ c) = "forall " ++ show tyname ++ "; " ++ show c
show (CCheck op expr1 expr2) = show expr1 ++ " " ++ show op ++ " " ++ show expr2

instance Show (Expression a) where
show (ESum tyname expr) = "sum{" ++ show tyname ++ "} " ++ show expr
show (EIndex name args) = name ++ "[" ++ strings args ++ "]"
show (EVar name) = name
show (EConst i) = show i
show (EMul i expr) = show i ++ " * " ++ show expr
show (EAdd expr1 expr2) = show expr1 ++ " + " ++ show expr2

instance Show Op where
show OEq = "="
show OLt = "<"
show OGt = ">"
show OLEq = "<="
show OGEq = ">="


It looks like this:

λ> CForall (Untyped "x") Nothing
(CForall (Untyped "y") (Just ("p", ["x", "y"]))
(CCheck OLt
(EMul 3 (ESum (Untyped "i") (EIndex "X" ["i"])))
(EConst 10)))
forall x; forall y if p(x, y); 3 * sum{i} X[i] < 10

The strings helper function used in CForall and EIndex just comma-separates a list of strings:

strings :: [String] -> String
strings = intercalate ", "


## Type system

Previously, I said we would have these types:

• User-defined types
• Predicates
• Integers
• N-dimensional binary arrays
• N-dimensional integer arrays (not actually implemented)

And we also need to distinguish between “parameter” variables and “model” variables. ILP solvers only operate on matrices, so actually what we have are three parameter types:

• User-defined types
• Predicates
• Integers

And two model types:

• N-dimensional binary arrays
• N-dimensional integer arrays
data Ty
= ParamCustom Name | ParamInteger | ParamPredicate [Ty]
| ModelBinary [Ty]
deriving Eq


Remember the TypedName in the constraint and expression abstract syntax? It was used wherever a new name was brought into scope: CForall and ESum. A TypedName is either a Name by itself or a Name associated with a Ty:

data IsTyped
data IsUntyped

data TypedName a where
Untyped :: Name       -> TypedName IsUntyped
Typed   :: Name -> Ty -> TypedName IsTyped

instance Eq (TypedName a) where
Untyped n1 == Untyped n2 = n1 == n2
Typed n1 ty1 == Typed n2 ty2 = n1 == n2 && ty1 == ty2


When generating code, we’ll need to know which types are being quantified over. So the type checker will fill in the types as it goes, turning our untyped expressions and constraints into typed expressions and constraints.

type UntypedConstraint = Constraint IsUntyped
type UntypedExpression = Expression IsUntyped
type TypedConstraint = Constraint IsTyped
type TypedExpression = Expression IsTyped


And let’s add some pretty-printing for types too:

instance Show Ty where
show (ParamCustom name) = "param<" ++ name ++ ">"
show ParamInteger = "param<integer>"
show (ParamPredicate args) = "param<predicate(" ++ strings (map show args) ++ ")>"
show (ModelBinary args) = "model<binary[" ++ strings (map show args) ++ "]>"

instance Show (TypedName a) where
show (Untyped name) = name
show (Typed name ty) = show ty ++ " " ++ name


## Running example

Our running example is the set of basic rota constraints from the other memo. We’ve already seen the concrete syntax, here’s the abstract syntax:

type Binding = (Name, Ty)

globals :: [Binding]
globals =
[ ("M", ParamInteger)
, ("is_leave", ParamPredicate [ParamCustom "TimeSlot", ParamCustom "Person"])
, ("A", ModelBinary [ParamCustom "TimeSlot", ParamCustom "Person", ParamCustom "Role"])
, ("X", ModelBinary [ParamCustom "Person"])
]

constraints :: [UntypedConstraint]
constraints =
[ -- In every time slot, each role is assigned to exactly one person
CForall (Untyped "t") Nothing
(CForall (Untyped "r") Nothing
(CCheck OEq
(ESum (Untyped "p") (EIndex "A" ["t", "p", "r"]))
(EConst 1)))
-- Nobody is assigned multiple roles in the same time slot
, CForall (Untyped "t") Nothing
(CForall (Untyped "p") Nothing
(CCheck OLEq
(ESum (Untyped "r") (EIndex "A" ["t", "p", "r"]))
(EConst 1)))
-- Nobody is assigned a role in a slot they are on leave for
, CForall (Untyped "p") Nothing
(CForall (Untyped "t") (Just ("is_leave", ["t", "p"]))
(CForall (Untyped "r") Nothing
(CCheck OEq
(EIndex "A" ["t", "p", "r"])
(EConst 0))))
-- Nobody works too many shifts
, CForall (Untyped "p") Nothing
(CCheck OLEq
(ESum (Untyped "t") (ESum (Untyped "r") (EIndex "A" ["t", "p", "r"])))
(EVar "M"))
-- Assignments are fairly distributed
, CForall (Untyped "t") Nothing
(CForall (Untyped "p") Nothing
(CForall (Untyped "r") Nothing
(CCheck OGEq
(EIndex "X" ["p"])
(EIndex "A" ["t", "p", "r"]))))
, CForall (Untyped "p") Nothing
(CCheck OLEq
(EIndex "X" ["p"])
(ESum (Untyped "t") (ESum (Untyped "r") (EIndex "A" ["t", "p", "r"]))))
]


That’s pretty verbose, more than the Python! Good thing that I’d write a parser for this if I were doing it for real.

## Type checking and inference

This is the hairy bit of the memo. I’ve not gone for any particular type inference algorithm, I just went for the straightforward way to do it for the syntax and types I had.

We’ll use a monad stack for the type checker:

type TcFun = ReaderT [Binding] (StateT [Binding] (Except String))
-- environment       ^^^^^^^^^
-- unresolved free variables           ^^^^^^^^^
-- error message                                         ^^^^^^


To get a feel for how TcFun is useful, let’s go through some utility functions.

Type errors:

typeError :: String -> TcFun a
typeError = lift . lift . throwE

eExpected :: String -> Name -> Maybe Ty -> TcFun a
eExpected eTy name (Just aTy) = typeError $"Expected " ++ eTy ++ " variable, but '" ++ name ++ "' is " ++ show aTy ++ " variable." eExpected eTy name Nothing = typeError$
"Expected " ++ eTy ++ " variable, but could not infer a type for '" ++ name ++ "'."


Throwing a type error is pretty important, so we’ll need a function for that, and for one of the more common errors.

Looking up the type of a name (if it’s bound):

getTy :: Name -> TcFun (Maybe Ty)
getTy name = lookup name <$> ask  The bindings in the state are just to keep track of free variables, and are not used when checking something’s type. Running a subcomputation with a name removed from the environment: withoutBinding :: Name -> TcFun a -> TcFun a withoutBinding name = withReaderT (remove name)  For example, if we have a global x and a constraint forall x; A[x], the x in A[x] is not the global x; it’s the x bound by the forall. Don’t worry, it’s only removed while typechecking the body of the CForall or ESum which introduced the new binding. Asserting a variable has a type: assertType :: Name -> Ty -> TcFun () assertType name eTy = getTy name >>= \case Just aTy | eTy == aTy -> pure () | otherwise -> eExpected (show eTy) name (Just aTy) Nothing -> lift$ modify ((name, eTy):)


Takes a name and an expected type, and checks that any pre-existing binding matches. If there is no pre-existing binding, the name is introduced as a free variable.

Removing a free variable from the state:

delFree :: Name -> TcFun Ty
delFree name = lookup name <$> lift get >>= \case Just ty -> do lift$ modify (remove name)
pure ty
Nothing -> typeError $"Could not infer a type for '" ++ name ++ "'."  Looks up the type of a free variable, removes the variable from the state, and returns the type. If we fail to find a type for the variable, it’s unused, which is a type error (as we can’t infer a concrete type). The basic idea is to walk through the abstract syntax: unify types when they arise; and for CForall and ESum check that the inner constraint (or expression) has a free variable with the right name and type. Typechecking argument lists: Let’s start with the simplest case: type checking an argument list, which arises in quantifier predicate constraints and EIndex. The function takes the names of the argument variables and their expected types, and checks that the variables do have those types. typecheckArgList :: [Name] -> [Ty] -> TcFun ()  The recursive case takes the name of the current argument and its expected type. It then looks up the actual type of the name. If it has a type, check that it’s the same as the expected type and either move onto the next parameter or throw an error. If there is no binding, assertType records it as a free variable. typecheckArgList (name:ns) (expectedTy:ts) = do assertType name expectedTy typecheckArgList ns ts  Ultimately the typecheckExpression and typecheckConstraint functions we’ll get to later will make sure all these free variables are bound by a forall or a sum. The base case is when we run out of argument names or types; there should be the same number of each: typecheckArgList [] [] = pure () typecheckArgList ns [] = typeError$
"Expected " ++ show (length ns) ++ " fewer arguments."
typecheckArgList [] ts = typeError $"Expected " ++ show (length ts) ++ " more arguments."  Typechecking expressions: Expressions have a few different parts, so let’s go through them one at a time. typecheckExpression :: UntypedExpression -> TcFun TypedExpression typecheckExpression e0 = decorate e0 (go e0) where  The decorate function, defined further below, appends the pretty-printed expression to any error message. So by decorateing every recursive call, we get an increasingly wide view of the error. Like this: Found variable 'x' at incompatible types param<integer> and param<index>. in x in A[x] = x in forall x; A[x] = x ESum introduces a new binding. The way I’ve handled this is by unbinding the name (in case there was something with the same name from a wider scope), type-checking the inner expression, and then (1) asserting that there is a free variable with the name of the bound variable and (2) storing its type.  go (ESum (Untyped name) expr) = do expr' <- withoutBinding name$ typecheckExpression expr
delFree name >>= \case
ty@(ParamCustom _) -> pure (ESum (Typed name ty) expr')
aTy -> eExpected "param<$custom>" name (Just aTy)  EIndex requires checking an argument list. I’m not allowing quantifying over model variables, so in the expression EIndex name args, then name must refer to a global. All globals are of known types, so we can look up the type of the argument list from the global environment.  go (EIndex name args) = getTy name >>= \case Just (ModelBinary argtys) -> do typecheckArgList args argtys pure (EIndex name args) aTy -> eExpected "model<binary(_)>" name aTy  EVar uses a variable directly, in which case the variable must be an integer. This is handled by looking for a binding and, if there isn’t one, introducing a new free variable.  go (EVar name) = do assertType name ParamInteger pure (EVar name)  EConst, EMul, and EAdd are pretty simple and just involve recursive calls to typecheckExpression.  go (EConst k) = pure (EConst k) go (EMul k expr) = do expr' <- typecheckExpression expr pure (EMul k expr') go (EAdd expr1 expr2) = do expr1' <- typecheckExpression expr1 expr2' <- typecheckExpression expr2 pure (EAdd expr1' expr2')  The input to typecheckExpression is an UntypedExpression and the output is a TypedExpression. We get there by rewriting ESum constructs to contain the inferred type of the quantifier variable. This will be useful when generating code. Typechecking constraints: Typechecking a constraint is pretty much the same as typechecking an expression. CForall is like ESum, CCheck is like EAdd. The only new thing is that a CForall can have a predicate constraint… but that’s typechecked in the same way as an EIndex: get the argument types of the predicate from the environment, and check that against the argument variables. Here it is: typecheckConstraint :: UntypedConstraint -> TcFun TypedConstraint typecheckConstraint c0 = decorate c0 (go c0) where go (CForall (Untyped name) (Just (rname, rargs)) c) = getTy rname >>= \case Just (ParamPredicate argtys) -> do typecheckArgList rargs argtys c' <- withoutBinding name$ typecheckConstraint c
ty <- delFree name
pure (CForall (Typed name ty) (Just (rname, rargs)) c')
aTy -> eExpected "param<predicate(_)>" rname aTy
go (CForall (Untyped name) Nothing c) = do
c' <- withoutBinding name $typecheckConstraint c ty <- delFree name pure (CForall (Typed name ty) Nothing c') go (CCheck op expr1 expr2) = do expr1' <- typecheckExpression expr1 expr2' <- typecheckExpression expr2 pure (CCheck op expr1' expr2')  While typecheckConstraint works, it leaves something to be desired. Here’s a slightly nicer interface: typecheckConstraint_ :: [Binding] -> UntypedConstraint -> Either String TypedConstraint typecheckConstraint_ env0 c0 = check =<< runExcept (runStateT (runReaderT (typecheckConstraint c0) env0) []) where check (c, []) = Right c check (_, free) = Left ("Unbound free variables: " ++ strings (sort (map fst free)) ++ ".")  This: • Takes the global bindings as an argument. • Does away with the TcFun, it returns a plain Either. • Checks that no free variables leak out. Some utility functions used above are: remove :: Eq a => a -> [(a, b)] -> [(a, b)] remove a = filter ((/=a) . fst) decorate :: Show a => a -> TcFun b -> TcFun b decorate e = goR where goR m = ReaderT (goS . runReaderT m) goS m = StateT (goE . runStateT m) goE = withExcept (\err -> err ++ "\n in " ++ show e) where  ## Example: type checking and inference Here’s a little function to print out the inferred type, or type error, for all of our constraints from the running example: demoTypeInference :: IO () demoTypeInference = for_ constraints$ \constraint -> do
case typecheckConstraint_ globals constraint of
Right c' -> print c'
Left err -> putStrLn err
putStrLn ""


Behold!

λ> demoTypeInference
forall param<TimeSlot> t; forall param<Role> r; sum{param<Person> p} A[t, p, r] = 1

forall param<TimeSlot> t; forall param<Person> p; sum{param<Role> r} A[t, p, r] <= 1

forall param<Person> p; forall param<TimeSlot> t if is_leave(t, p); forall param<Role> r; A[t, p, r] = 0

forall param<Person> p; sum{param<TimeSlot> t} sum{param<Role> r} A[t, p, r] <= M

forall param<TimeSlot> t; forall param<Person> p; forall param<Role> r; X[p] >= A[t, p, r]

forall param<Person> p; X[p] <= sum{param<TimeSlot> t} sum{param<Role> r} A[t, p, r]

Looks pretty good, all types are inferred as they should be.

Here’s a broken example, which arose when I mistyped one of the constraints:

λ> either putStrLn print $typecheckConstraint_ globals (CForall (Untyped "t") Nothing (CForall (Untyped "p") Nothing (CForall (Untyped "r") Nothing (CCheck OLEq (EIndex "X" ["p"]) (ESum (Untyped "t") (ESum (Untyped "r") (EIndex "A" ["t", "p", "r"]))))))) Could not infer a type for 'r'. in forall r; X[p] <= sum{t} sum{r} A[t, p, r] in forall p; forall r; X[p] <= sum{t} sum{r} A[t, p, r] in forall t; forall p; forall r; X[p] <= sum{t} sum{r} A[t, p, r] I’d added extra forall t and forall r quantifiers, which are wrong because those variables are bound by sums. So the types of the forall-bound variables can’t be inferred. ## Code generation I don’t want to write (or learn) bindings to ILP solvers, I already know PuLP so that sounds like a pain. So what I do want to do is generate the PuLP-using Python code the abstract syntax corresponds to. Most of codegenExpression, which produces a Python expression, is straightforward: codegenExpression :: TypedExpression -> String codegenExpression (EIndex name args) = name ++ "[" ++ strings args ++ "]" codegenExpression (EVar name) = name codegenExpression (EConst i) = show i codegenExpression (EMul i expr) = show i ++ " * " ++ codegenExpression expr codegenExpression (EAdd expr1 expr2) = "(" ++ codegenExpression expr1 ++ " + " ++ codegenExpression expr2 ++ ")"  The complex bit is handling ESum, which introduces a generator expression, and multiple ESums are collapsed: codegenExpression (ESum tyname0 expr0) = go [tyname0] expr0 where go vars (ESum tyname expr) = go (tyname:vars) expr go vars e = "pulp.lpSum(" ++ codegenExpression e ++ " " ++ go' (reverse vars) ++ ")" go' [] = "" go' (Typed name (ParamCustom ty):vs) = let code = "for " ++ name ++ " in " ++ ty in if null vs then code else code ++ " " ++ go' vs  I’m making some assumptions about how variables and types are represented in Python: 1. I assume all names are the valid in Python, eg: abstract: EMul 3 (EIndex "A", ["i"]) code: 3 * A[i] 2. I assume user-defined types correspond to Python iterators, eg: abstract: ESum (Typed "x" (ParamCustom "X")) expr code: pulp.lpSum(expr for x in X). These aren’t checked. Assumption (1) could be handled by restricting the characters in names (eg, to alphanumeric only). Assumption (2) would be handled if I were to implement the full abstract syntax, as generated code would be put in a function which takes all the parameter variables as arguments, and which creates the model variables. But this memo only implements expressions and constraints. Generating code for constraints is nothing surprising, the only slight complication is needing to make sure the indentation works out when there are nested CForalls: codegenConstraint :: TypedConstraint -> String codegenConstraint = unlines . go where go (CForall (Typed name (ParamCustom ty)) (Just (rname, rargs)) c) = [ "for " ++ name ++ " in " ++ ty ++ ":" , " if not " ++ rname ++ "(" ++ strings rargs ++ "):" , " continue" ] ++ indent (go c) go (CForall (Typed name (ParamCustom ty)) _ c) = [ "for " ++ name ++ " in " ++ ty ++ ":" ] ++ indent (go c) go (CCheck op expr1 expr2) = let e1 = codegenExpression expr1 e2 = codegenExpression expr2 in ["problem += " ++ e1 ++ " " ++ cgOp op ++ " " ++ e2] cgOp OEq = "==" cgOp op = show op indent = map (" "++)  ## Example: code generation Here’s a little function to print out the generated code, or type error, for all of our constraints from the running example: demoCodeGen :: IO () demoCodeGen = for_ constraints$ \constraint -> do
case typecheckConstraint_  globals constraint of
Right c' -> do
putStrLn (" # " ++ show c')
putStrLn (codegenConstraint c')
Left err -> do
putStrLn err
putStrLn ""


Behold, again!

λ> demoCodeGen
# forall param<TimeSlot> t; forall param<Role> r; sum{param<Person> p} A[t, p, r] = 1
for t in TimeSlot:
for r in Role:
problem += pulp.lpSum(A[t, p, r] for p in Person) == 1

# forall param<TimeSlot> t; forall param<Person> p; sum{param<Role> r} A[t, p, r] <= 1
for t in TimeSlot:
for p in Person:
problem += pulp.lpSum(A[t, p, r] for r in Role) <= 1

# forall param<Person> p; forall param<TimeSlot> t if is_leave(t, p); forall param<Role> r; A[t, p, r] = 0
for p in Person:
for t in TimeSlot:
if not is_leave(t, p):
continue
for r in Role:
problem += A[t, p, r] == 0

# forall param<Person> p; sum{param<TimeSlot> t} sum{param<Role> r} A[t, p, r] <= M
for p in Person:
problem += pulp.lpSum(A[t, p, r] for t in TimeSlot for r in Role) <= M

# forall param<TimeSlot> t; forall param<Person> p; forall param<Role> r; X[p] >= A[t, p, r]
for t in TimeSlot:
for p in Person:
for r in Role:
problem += X[p] >= A[t, p, r]

# forall param<Person> p; X[p] <= sum{param<TimeSlot> t} sum{param<Role> r} A[t, p, r]
for p in Person:
problem += X[p] <= pulp.lpSum(A[t, p, r] for t in TimeSlot for r in Role)

## What’s missing?

We’ve come to the end of my little language for defining ILP problems, but there is still more to be done if this were to become a fully-fledged language people could use. Here are some missing bits:

• A parser for the concrete syntax.
• More integer operations:
• Integer ranges, in addition to user-defined set types, for forall and sum.
• Arithmetic on integer indices.
• Comparisons, in addition to predicate functions, in forall guards.
• The rest of the abstract syntax (along with typechecking and code generation): integer matrices, objective functions, and type and variable declarations.

For example, in the GOV.UK support rota, one of the constraints is that someone can’t be on support in two adjacent weeks. With integer ranges and arithmetic on integer indices, that could be expressed like so:

forall t in [1, N), p; (sum{r} A[t, p, r]) + (sum{r} A[t - 1, p, r]) <= 1

There’s also a small problem with the current abstract syntax: it’s a bit too flexible. This is not a valid ILP expression:

sum{foo} (sum{bar} A[foo, bar] + sum{baz} B[foo, baz])

Only direct sum nesting is permitted. There are two ways to solve this. One is to change the abstract syntax to preclude it, maybe something like this:

data Void

data TaggedExpression tag a
= TESum !tag (SumExpression a)
| TEIndex Name [Name]
| TEVar Name
| TEConst Integer
| TEMul Integer (TaggedExpression tag a)
| TEAdd (TaggedExpression tag a) (TaggedExpression tag a)

data SumExpression a
= SENest (TypedName a) (SumExpression a)
| SEBreak (TaggedExpression Void a)


A TaggedExpression Void can’t contain any more TESum constructors, because the Void type is uninhabited. Another option is to add a check, between parsing and typechecking, that there are no invalidly nested ESums.