Potential Winner Determination and Payment Calculation

When STOi receives service information{f_jγ} and asking price{aij}from one or more MUDs, based on {f_jγ}, {aij}, fiπ, andbi, STO i forms a set of available MUDs as

Γc(πi) ={γj|(Tij ∩T_iu)6=∅, Rγ_j ⊆Rπ_i, and aij ≤bi},

in which Tu

i denotes the un-scheduled time duration for task πi at the current round of

auction. This computation is implemented in lines 2-7 of Algorithm 2.

Potential Winner Determination Initially, the set of potential winners isW(πi) =

∅. To schedule working time, STO i first sorts all the available MUDs in Γc_(π

i) in a non-

decreasing order in terms of their asking prices and gets a sorted set Γc0(πi) (see line 8 in

Algorithm 2). Next, STO i scans the available MUDs in Γc0(πi) and allocates un-scheduled

time slots in a greedy manor. Specifically speaking, if MUD γj’s current available working

Tu

i )∩(

S

γ_j0∈W(πi) Ts

ij0)6= (Tij∩Tiu), MUD γi can be chosen as a potential winner and assigned

a set of time slots that has not been allocated to current potential winners in W(πi), i.e., T_ijs = (Tij ∩T_iu)\(Tij ∩T_iu∩( S

γj0∈W(_πi)

T_ijs0)) (see lines 9-14 in Algorithm 2).

Payment Calculation After completing task scheduling, STO i computes the pay-

ment for each potential winning MUD γj via identifying γj’s critical neighbor, which is defined to be the MUD γk in Γc(πi) where if aij is higher than aik, γj can not be scheduled.

Different from the previous works [9, 17–19, 21, 41, 42] in which each winner has only one critical neighbor, each winning MUDγj in the auction CPAS has one or more critical neigh- bors because the time slots of Ts

ij could be scheduled to one or more other available MUDs

if MUD γj does not join the auction (see lines 17 - 32 of Algorithm 2). Thus, the payment

is calculated according to every critical neighbor of winner γj. In order to find the critical neighbors, STOisorts all the MUDs in Γc

−γj(πi) = Γ c_(π

i)\γj in the non-decreasing order in

terms of their asking prices, selects winners again in the sorted set Γc0_−γj(πi), and schedules

working time to them. Any MUD γk is a critical neighbor of MUDγj if their allocated time durations are overlapping, i.e., T_iks0 ∩ T_ijs0 6= ∅, where T_iks0 is the time duration assigned to MUD γk and Tijs0 records the remaining time duration in Tijs that is not allocated to others.

So the corresponding critical payment is aik|T_iks0∩T_ijs0|. But, if no critical neighbor is found for MUD γj, its critical payment is STO i’s budget bi|Tijs0|.

Then, STO iannounces the auction results {Ts

ij} and {pij}to the MUDs.

Remark: Via Algorithm 2, each potential winning MUD can receive a working time

duration Ts

ij that contains one or more sub-time durations. For example, the time du-

ration of task πi is from 1:00pm to 5:00pm, winner γj’s working time duration is Tijs =

{[2:00pm, 3:00pm], [4:30pm, 5:00pm]} containing two sub-time durations, and the number of working time slots is |Ts

Algorithm 2: Cost-Preferred Task Scheduling & Pricing for Task πi input : fπ

i , bi, Tiu,Γ,{f γ

j},{aij} output: W(πi),{T_ijs},{pij}

1 Set Γc(π_i) = ∅, W(π_i) = ∅,{T_ijs}={∅}, and {p_ij}={0}; 2 for each γ_j ∈Γ with submitted f_jγ and a_ij do

3 Calculate T_ij;

4 if (T_ij ∩T_iu)6=∅, R_jγ ⊆Rπ_i, and a_ij ≤b_i then 5 Γc(π_i) = Γc(π_i)∪γ_j;

6 Sort all MUDs in Γc(π_i) in non-decreasing order based on {a_ij} and obtain the

sorted set Γc0(πi); 7 for j = 1 to |Γc0(πi)|do 8 if (T_ij ∩T_iu)∩( S γj0∈W(πi) Ts ij0)6= (T_ij ∩T_iu)then 9 W(π_i) =W(π_i)∪γ_j; 10 T_ijs = (Tij ∩Tiu)\(Tij ∩Tiu∩( S γj0∈W(_πi) Ts ij0)); 11 for each γj ∈W_iπ do 12 Set {T_iks0}={∅} and T_ijs0 =T_ijs;

13 Sort all the MUDs in Γc(πi)\γj in a non-decreasing order based on {aik} and

obtain the sorted set Γc0_−γ

j(πi); 14 Set k = 1 andW−γ_j(π_i) =∅; 15 while k ≤ |Γc0_−γ j(πi)| and T s0 ij 6=∅ do 16 if (T_ik∩T_iu)∩( S γ_j0∈W−_γj(πi) T_ijs00)6= (T_ik∩T_iu) then 17 W−γj(πi) = W−γj(πi)∪γk, 18 T_iks0 = (T_ik∩T_iu)\(T_ik∩T_iu∩( S γ_j0∈W −_γj(_πi) T_ijs00)); 19 if T_iks0∩T_ijs0 6=∅ then 20 pij =pij +aik|Tiks0∩T s0 ij|, 21 T_ijs0 =T_ijs0\(T_iks0∩T_ijs0); 22 k=k+ 1; 23 if T_ijs0 6=∅then 24 pij =pij +bi|T_ijs0|.