Constructing schedules for sports leagues with divisional and round-robin tournaments

Appendix A:

From HAP set to template

To attempt assign numbers to a given HAP set, we can find a feasible point satisfying the following constraints. Let N be the set of numbers to be assigned to the HAP set, let N₁ (resp. N₂) be the first (resp. second) half of elements in N, and let P be the set of periods within the HAP set divided into Parts I, II, and, III (represented by P₁, P₂, and P₃).

– Variables:

1. A variable to decide who hosts whom in a given period.

xijp={1: if team  i  hosts team  j  in period  p,0: otherwise,  ∀i, j, ∈N, p∈P.

– Parameters:

1. A parameter denoting whether the HAP set calls for a home, away, or bye game.

Aip={   1: if team  i  plays at home in period  p,  0:if team  i  has a bye in period  p,−1: if team  i  plays away in period  p  ∀i,∈N, p∈P

– Constraints:

1. Ensure that no team plays during a bye period.

∑jxijp+∑jxjip=0  ∀i∈N, p∈P  s.t.  Anp=0.

2. Ensure that no team is home during an away game or away during a home game.

∑jxijp=0  ∀i∈N, p∈P  s.t.  Anp=−1.∑jxijp=1  ∀i∈N, p∈P  s.t.  Anp=1.

3. Ensure at most one game per round.

∑jxijp+∑jxjip≤1  ∀i∈N, p∈P.

4. Ensure numbers in N₁ meet in Part I (and similarly for numbers in N₂).

∑p∈P1(xijp+xjip)=1  ∀i, j∈N1  s.t.  i≠j.∑p∈P1(xijp+xjip)=1  ∀i, j∈N2  s.t.  i≠j.

5. Ensure that each number hosts each other number in Parts II or III.

∑p∈P2∪P3xijp=1  ∀i, j∈N, i≠j

6. Ensure numbers in N₁ (resp. N₂) meet home-away or away-home in periods P₁ and P₂.

∑p∈P1∪P2xijp=1  ∀i, j∈N1,  i≠j.∑p∈P1∪P2xijp=1  ∀i, j∈N2,  i≠j.

7. Ensure a Part III mirrors Part II.

xij8=xji33, xij9=xji32, …, xij20=xji21  ∀i, j∈N

Appendix B: From template to schedule

To assign teams to numbers in the template, we can form an IP model of small enough size to be easily solved for any size league (at most, a few hundred binary variables). Let T be the set of teams (indexed by t), let P be the set of periods (indexed by p), and let N be the set of numbers in the template (indexed by n). Let T₁ (T₂) be the first (second) division of teams and let N₁ (N₂) be the first (second) group of numbers, each containing 3 pairs of complementary teams. Lastly, introduce sets D_p and M_p for all periods p where entires in D_p are pairs of teams that are desired to meet in period p and M_p contains pairs of numbers that meet in period p.

– Variables:

1. A variable to determine which number each team is assigned to.

xtn={1:if team  t  is assigned to number  n,0:otherwise,    ∀t∈T, n∈N.

2. A variable to determine which group of numbers each division is assigned to.

Δ={1: if division  T1  is assigned to the group of numbers  N1,0: otherwise. 

3. A variable indicating if a desired game occurs during a given period.

δ(t1, t2), p, (n1, n2)={1: if teams  (t1, t2)  assigned to  (n1, n2)  play a desired match in period  p,0: otherwise,    ∀(t1, t2)∈Dp,∀p∈P, (n1, n2)∈Mp.

– Parameters:

1. A parameter denoting hard venue unavailabilities.

Atp={0:if venue  t  is unavailable throughout period  p,1:otherwise,  ∀t∈T, p∈P.

2. A parameter denoting soft venue unavailabilities.

Stp={1: if venue  t  would prefer not to host during period  p,0: otherwise,   ∀t∈T, p∈P.

3. A numerical value for HAP set entries.

Hnp={   1: if number  n  plays at home during period  p,  0:−1: if number  n  has a bye during period  p, if number  n  plays away during period  p,  ∀n∈N, p∈P.

4. A parameter to count the number of soft violations.

Vtn=∑p(HnpStp)+  ∀t∈T, ∀n∈N.

5. Define α∈(0,1) as the trade-off between minimizing soft conflicts and maximizing desired games. In practice, the IP can be solved for a variety of values of α to provide a few schedules. The league can then be made aware of these options to decide the relative merit of such choices. For example, is 10 desired games worth one additional soft conflict.

– Constraints:

1. Ensure each number is assigned a team.

∑txtn=1  ∀n∈N.

2. Ensure each team is assigned a number.

∑nxtn=1  ∀t∈T.

3. Ensure teams in T₁ are in the same subgroup of numbers.

∑t∈T1∑n∈Nxtn=Δ|N1|.

4. Ensure hard venue unavailabilities are not violated.

∑nxtnHnp≤Atp  ∀p∈P, ∀t∈T.

5. Ensure δ(t1, t2), p, (n1, n2) can only be 1 if a desired matchup occurs between teams t₁ and t₂ in period p.

xt1n1+xt2n2+xt1n2+xt2n1≥2δ(t1, t2), p, (n1, n2)∀(t1, t2)∈Dp, p∈P, (n1, n2)∈Mp.

– Objective Function:

minimizex(1−α)∑t∑nxtnVtn−α∑p∑(t1,t2)∈Dp∑(n1,n2)∈Mpδ(t1,t2),p,(n1,n2).