Cooperative evolutionary heterogeneous simulated annealing algorithm for google machine reassignment problem

Abstract

This paper investigates the Google machine reassignment problem (GMRP). GMRP is a real world optimisation problem which is to maximise the usage of cloud machines. Since GMRP is computationally challenging problem and exact methods are only advisable for small instances, meta-heuristic algorithms have been used to address medium and large instances. This paper proposes a cooperative evolutionary heterogeneous simulated annealing (CHSA) algorithm for GMRP. The proposed algorithm consists of several components devised to generate high quality solutions. Firstly, a population of solutions is used to effectively explore the solution space. Secondly, CHSA uses a pool of heterogeneous simulated annealing algorithms in which each one starts from a different initial solution and has its own configuration. Thirdly, a cooperative mechanism is designed to allow parallel searches to share their best solutions. Finally, a restart strategy based on mutation operators is proposed to improve the search performance and diversification. The evaluation on 30 diverse real-world instances shows that the proposed CHSA performs better compared to cooperative homogeneous SA and heterogeneous SA with no cooperation. In addition, CHSA outperformed the current state-of-the-art algorithms, providing new best solutions for eleven instances. The analysis on algorithm behaviour clearly shows the benefits of the cooperative heterogeneous approach on search performance.

Appendix: Problem formulation

The description of this problem is adopted from [1, 51]. Let M= $\{m_{1},m_{2},\ldots ,m_{k}\}$ be the set of machines and P=$\{p_{1},p_{2},\ldots ,p_{w}\}$ be the set of processes. The solution can represented by a vector Map with length equal to w, where Map(p) corresponds to the machine assigned to process p. The constant and variable symbols of the problem are:

• M The set of machines, $M=\{m_1,m_2,\dots ,m_k\}, |M|=k$.

• P The set of processes, $P=\{p_1,p_2,\dots ,p_w\}, |P|=w$.

• R The set of resources, $R=\{r_1,r_2,\dots ,r_d\}, |R|=d$.

• TR: The subset of resources which need transient usage $TR=\{tr_1,tr_2,\ldots ,tr_h\}, |TR|=h, TR \subseteq R$.

• S The set of services, $S=\{s_1,s_2,\dots ,s_f\}, |S|=f$. Processes are partitioned into services, all services are disjoint, i.e., $\cup _{s\in S}s=P, \forall s_i,s_j \in S, s_i \ne s_j, s_i \cap s_j=\oslash$.

• L The set of locations, $L=\{l_1,l_2,\dots ,l_g\},|L|=g$. Machines are partitioned into locations, all locations are disjoint, i.e., $\cup _{l\in L}l=M,\forall l_i,l_j \in L, l_i \ne l_j, l_i \cap l_j=\oslash$.

• N The set of neighbourhoods, $N=\{n_1,n_2,\dots ,n_v\},|N|=v$. Machines are partitioned into neighbourhoods, all neighbourhoods are disjoint, i.e., $\cup _{n\in N}n=M,\forall n_i,n_j \in N, n_i \ne n_j, n_i \cap n_j=\oslash$.

• SD The set of service dependencies defined in $S^2. SD=\{sd_1,sd_2,\ldots ,sd_z\} |SD|=z$.

• B The set of triples defined in $R^2x N$, N is the set of natural numbers. $B=\{b_1,b_2,\ldots ,b_e\},|B|=e$.

• NE(m) The neighbourhood of machine $m \in M$.

• C(m, r) The capacity of resource $r\in R$ on machine $m \in M$.

• R(p, r) The requirement of resource $r\in R$ for process $p \in P$.

• $S\_min(s)$ The minimum number of distinct locations where at least one process of service $s\in S$ should run.

• SC(m, r) The safety capacity of resource $r\in R$ on machine $m \in M$.

• PMC(p) The cost of moving process $p \in P$ from its original machine $Map_0(p)$.

• $MMC(m,m')$ The cost of moving any process $p \in P$ from machine $m \in M$ to machine $m' \in M$. $\forall m \in M, MMC(m,m)=0$.

• $wt_1(r)$ The weight of load cost for resource $r\in R$.

• $wt_2(b)$ The weight of balance cost for triple $b\in B$.

• $wt_3,wt_4,wt_5$ The weights of process move cost, service move cost and machine move cost.

• U(m, r) The usage of machine $m\in M$ for resource $r\in R$, i.e., $\cup (m,r)=\sum _{p\in P}bool(Map(p)=m).R(p,r)$.

• A(m, r) The available capacity of resource $r\in R$ on machine $m\in M$, i.e., $A(m,r)=C(m,r)-\cup (m,r)$.

• A(r) The total available capacity of resource $r\in R$ over all machines, i.e., $A(r)=\sum _{m\in M}A(m,r)$.

• TU(m, r) The transient usage of machine $m\in M$ for resource $r\in R$, i.e., ${\left\{ \begin{array}{ll}\sum _{p\in P}bool(Map_0(p)=m \wedge Map(p)\ne m).R(p,r), r \in TR \\ 0, \quad r\in R-TR \end{array}\right. }$.

• CO(m) The capacity overload on machine $m\in M$, i.e., $CO(m)=\sum _{r \in R} max(TU(m,r)-A(m,r),0)$.

where bool is the truth indicator function. It takes value 1 if the proposition is true and takes 0 if not. The allocation of processes must not violate the following five hard constraints:

• Capacity constraints $\forall m\in M, r\in R, \cup (m,r)\le C(m,r)$.

• Conflict constraints $\forall s \in S,(p_i,p_j) \in s^2, p_i\ne p_j\Rightarrow Map(p_i)\ne Map(p_j)$.

• Transient usage constraints: $\forall m \in M,tr \in TR, \cup (m,tr)+ TU(m,tr)\le C(m,tr)$.

• Spread constraints $\forall s\in S,\sum \limits _{l \in L}min(1,|\{p \in s|Map(p)\in l\}|)\ge S\_min(s)$.

• Dependency constraints $\forall (s^a,s^b) \in SD, \forall p^a \in s^a, \exists p^b \in s^b \Rightarrow NE(Map(p^a))= NE(Map(p^b))$.

A solution, which meets all the above hard constraints, is considered feasible. In comparison soft constraints do not affect the feasibility but the quality of a solution. Soft constraints violation is measurable and should be as low as possible. The soft constraints are summarised as follows:

• Load cost. SC(m, r) represents the safety capacity of a resource $r \in R$ on a machine $m \in M$. The load cost is defined per resource and corresponds to the used capacity above the safety capacity; more formally:

  $$\begin{aligned} f_1(r)=\sum \limits _{m \in M}max(0,U(m,r)-SC(m,r)) \end{aligned}$$

• Balance cost In order to balance the availability of resources, let the set of triples is $B \subseteq R^2 X N$ [1]. For a given triple $b=(r^b_1,r^b_2, target^b) \in B$, the balance cost is:

  $$\begin{aligned} f_2(b)=\sum \limits _{m \in M}max(0, target^b . A(m,r^b_1)-A(m,r^b_2)) \end{aligned}$$

• Process move cost represents the cost of moving a process from its current machine to a new one which can be defined as follow:

  $$\begin{aligned} f_3=\sum \limits _{p \in P}bool(Map_0(p)\not =Map(p))PMC(p) \end{aligned}$$

• Service move cost represents the maximum number of moved processes over services which can be defined as follow:

  $$\begin{aligned} f_4=\smash {\displaystyle \max _{s \in S}}(|\{p \in s |Map(p)\not =Map_0(p)\}|) \end{aligned}$$

• Machine move cost represents the sum of all moves weighted by relevant machine cost which can be defined as follow:

  $$\begin{aligned} f_5=\sum \limits _{p \in P}MMC(Map_0(p),Map(p)) \end{aligned}$$

The total objective cost is a weighted sum of all previous costs which is calculated as follows:

$$\begin{aligned} f(Map)=\sum \limits _{r \in R}wt_1(r). f_1(r)+ \sum \limits _{b \in B}wt_2(b)+f_2(b)+wt_3.f_3+wt_4.f_4+wt_5.f_5 \end{aligned}$$

(2)

where $wt_1(r)$, $wt_2(b)$, $wt_3$, $wt_4$ and $wt_5$ define the importance of each individual cost.