An Algebraic Approach to Scheduling Problems in Project Management Nikolai Krivulin Faculty of Mathematics and Mechanics Saint Petersburg State University E-mail: nkk < at > math.spbu.ru URL: http://www.math.spbu.ru/user/krivulin/ Annual International Workshop on Advances in Methods of Information and Communication Technology Petrozavodsk, 2010
Outline Motivating Example Activity Network Model Schedule Development Problem Idempotent Algebra Notation and References Solution to Motivating Example Linear Equations of the First Kind Example 2: Start-to-Start Constraints Activity Network Model Schedule Development Problem Linear Equations of the Second Kind Example 3: Mixed Constraints Schedule Development Problem Conclusions Acknowledgments N. Krivulin (SPbSU) An Algebraic Approach AMICT’2010 2 / 22
Motivating Example Activity Network Model Motivating Example: Activity Network Model Start-to-Finish Precedence Relationship ◮ Consider a project consisting of n activities ◮ Every activity finishes as soon as some work is performed within some other activities ◮ For each activity i = 1 , . . . , n we introduce the notation x i , the initiation time ; y i , the completion time ; a ij , the time activity j takes to do the work that has to be completed before the completion of activity i ◮ The completion time of activity i can be represented as y i = max( x 1 + a i 1 , . . . , x n + a in ) N. Krivulin (SPbSU) An Algebraic Approach AMICT’2010 3 / 22
Motivating Example Activity Network Model Model Transformation ◮ Consider the precedence relationship equations y i = max( x 1 + a i 1 , . . . , x n + a in ) , i = 1 , . . . , n ◮ Substitution of the symbol ⊕ for max , and ⊗ for + gives y i = a i 1 ⊗ x 1 ⊕ · · · ⊕ a in ⊗ x n , i = 1 , . . . , n ◮ With the symbol ⊗ omitted, the equations takes the form y i = a i 1 x 1 ⊕ · · · ⊕ a in x n , i = 1 , . . . , n (note a formal similarity to equations in the conventional algebra y i = a i 1 x 1 + · · · + a in x n , i = 1 , . . . , n ) N. Krivulin (SPbSU) An Algebraic Approach AMICT’2010 4 / 22
Motivating Example Activity Network Model Vector Representation ◮ The matrix-vector notation a 11 · · · a 1 n x 1 y 1 . . . . ... . . . . A = x = y = , , . . . . a n 1 · · · a nn x n y n ◮ The precedence relationship equation in the vector form y = A x (matrix-vector multiplication is performed in the usual way with the standard addition and multiplication replaced with ⊕ and ⊗ ) N. Krivulin (SPbSU) An Algebraic Approach AMICT’2010 5 / 22
Motivating Example Activity Network Model A Network and Its Matrix ◮ An activity network ✓✏ ✓✏ ✓✏ ✓✏ x 1 x 2 x 3 x 4 ✒✑ ❍❍❍❍❍❍❍❍❍❍❍ 10 ✒✑ 12 4 ✒✑ 8 ✟ ✒✑ ✟ � ❅ � � ✟ 6 ✟ � ❅ � � 8 5 11 ✟ 7 12 ✟ � � ❅ � ✟ ✟ � � ❅ � ✟ ❄ ❄ ❘ ❄ ❄ ✓✏ ✓✏ ✓✏ ✓✏ ✟ ✠ � ✠ � ❅ � ✠ ✟ ✙ ❥ y 1 y 2 y 3 y 4 ✒✑ ✒✑ ✒✑ ✒✑ ◮ The network precedence relationship matrix ( ✵ = −∞ ) 8 10 ✵ ✵ 5 4 8 ✵ A = 6 12 11 7 12 ✵ ✵ ✵ N. Krivulin (SPbSU) An Algebraic Approach AMICT’2010 6 / 22
Motivating Example Schedule Development Problem Schedule Development Problem Schedule Development Under Late Finish Date Constraints ◮ Suppose each activity i = 1 , . . . , n is subject to the time constraint b i , the late finish date ◮ The vector notation: b = ( b 1 , . . . , b n ) T Problem ◮ Find the vector x of start dates to meet the condition y = b ◮ The solution satisfies the linear equation of the first kind A x = b in a semiring with the operations ⊕ = max and ⊗ = + N. Krivulin (SPbSU) An Algebraic Approach AMICT’2010 7 / 22
Idempotent Algebra Notation and References Idempotent Algebra: Notation and References Idempotent Semiring ❘ max , + ◮ Idempotent semiring (semifield) ❘ max , + = � ❳ , ✵ , ✶ , ⊕ , ⊗� ◮ The set: ❳ = ❘ ∪ {−∞} ◮ The operations: ⊕ = max and ⊗ = + ◮ Null and identity elements: ✵ = −∞ and ✶ = 0 ◮ The inverse: for each x ∈ ❘ there exists x − 1 ( − x in conventional algebra) ◮ The power: for each x, y ∈ ❘ one can define x y ( xy in conventional algebra) N. Krivulin (SPbSU) An Algebraic Approach AMICT’2010 8 / 22
Idempotent Algebra Notation and References Matrix Algebra Over ❘ max , + ◮ Addition and multiplication � { A ⊕ B } ij = { A } ij ⊕ { B } ij , { BC } ij = { B } ik { C } kj k ◮ Identity and null matrices: I = diag( ✶ , . . . , ✶ ) and ✵ = ( ✵ ) ◮ The power: A 0 = I , A k + l = A k A l for all integer k, l ≥ 0 ◮ The norm and trace: for any matrix A = ( a ij ) � � � A � = tr A = a ij , a ii i,j i ◮ The pseudoinvers: for any matrix A = ( a ij ) there exists A − = ( a − ij = a − 1 ij ) with a − ji , if a ji � = ✵ , and a − ij = ✵ , otherwise N. Krivulin (SPbSU) An Algebraic Approach AMICT’2010 9 / 22
Idempotent Algebra Notation and References Early Publications ◮ N.N. Vorob’ev (1963), A.A. Korbut (1965), I.V. Romanovskii (1967) Books ◮ R.A. Cuninghame-Green (1979), B. Carr´ e (1979) ◮ U. Zimmermann (1981), F . Baccelli et al (1992) ◮ V.P . Maslov, V.N. Kolokol’tsov (1994), J.S. Golan (1999) ◮ B. Heidergott et al (2006), N.K. Krivulin (2009) Hundreds of Contributing Papers ◮ V.P . Maslov, G.L. Litvinov, G.B. Shpiz, A.N. Sobolevskii, V.D. Matveenko, S.L. Blyumin ◮ G.J. Olsder, B. Heidergott, S. Gaubert, B. De Schutter, G. Cohen . . . ◮ N. Krivulin (SPbSU) An Algebraic Approach AMICT’2010 10 / 22
Solution to Motivating Example Linear Equations of the First Kind Solution to Example: First Kind Linear Equations Problem ◮ Given a ( m × n ) -matrix A and a vector b ∈ ❘ m , find the solution x ∈ ❘ n of the first kind equation A x = b Theorem (Existence and Uniqueness) 1. The equation has a solution if and only if ( A ( b − A ) − ) − b = ✶ 2. The maximum solution, if any, takes the form x = ( b − A ) − 3. If the columns of A form a minimal set generating b , then the solution is unique N. Krivulin (SPbSU) An Algebraic Approach AMICT’2010 11 / 22
Solution to Motivating Example Linear Equations of the First Kind General Solution ◮ For the matrix A , consider a minimal subset of its columns generating b , and denote the set of the column indices by J ◮ Let J be the set of all the subsets J ◮ Let G J be the diagonal matrix that has its diagonal entry in row i set to ✵ , if i ∈ J , and to ✶ , otherwise Theorem The general solution of the first kind equation is the family x J = ( b − A ⊕ v T G J ) − , v ∈ ❘ n , J ∈ J Corollary The solution of the inequality A x ≤ b is given by x ≤ ( b − A ) − N. Krivulin (SPbSU) An Algebraic Approach AMICT’2010 12 / 22
Example 2: Start-to-Start Constraints Activity Network Model Example 2: Activity Network Model Start-to-Start Precedence Relationship ◮ A project involves n activities ◮ Every activity starts not earlier than some work is performed within some other activities ◮ For each activity i = 1 , . . . , n we introduce the notation x i , the initiation time ; y i , the completion time ; a ij , the time activity j takes to do the work that has to be completed before the start of activity i ◮ The initiation time of activity i satisfies the condition x i ≥ max( x 1 + a i 1 , . . . , x n + a in ) N. Krivulin (SPbSU) An Algebraic Approach AMICT’2010 13 / 22
Example 2: Start-to-Start Constraints Activity Network Model Model Representation ◮ In terms of ❘ max , + , the precedence relationships take the form x i ≥ a i 1 x 1 ⊕ · · · ⊕ a in x n , i = 1 , . . . , n ◮ With the matrix-vector notation, we arrive at the inequality A x ≤ x Problem ◮ Find the vector x that satisfies the precedence constraints ◮ Of particular interest is the solution of the homogeneous linear equation of the second kind A x = x N. Krivulin (SPbSU) An Algebraic Approach AMICT’2010 14 / 22
Example 2: Start-to-Start Constraints Activity Network Model A Network and Its Matrix ◮ An activity network ✓✏ ✓✏ − 2 ✲ x 1 x 2 ✐ P ✒✑ P ✒✑ P P ❅ ■ � ❅ 3 P P − 1 ❅ P � ❅ P P 2 ❅ P � ❅ P − 1 ✓✏ P ✓✏ ❅ ✠ � P ❅ ❘ P ✲ x 3 x 4 ✒✑ ✒✑ − 4 ◮ The network precedence relationship matrix ( ✵ = −∞ ) 0 − 2 ✵ ✵ ✵ 0 3 − 1 A = − 1 0 − 4 ✵ 2 ✵ ✵ 0 N. Krivulin (SPbSU) An Algebraic Approach AMICT’2010 15 / 22
Example 2: Start-to-Start Constraints Schedule Development Problem Schedule Development Problem Schedule Development Under Early Start Date Constraints ◮ Suppose each activity i = 1 , . . . , n is subject to the time constraint b i , the early start date ◮ The vector notation: b = ( b 1 , . . . , b n ) T Problem ◮ Find a vector x so as to meet the conditions A x = x , x ≥ b ◮ The solution satisfies the nonhomogeneous linear equation of the second kind A x ⊕ b = x N. Krivulin (SPbSU) An Algebraic Approach AMICT’2010 16 / 22
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