CN101241562A - Method for optimizing items scheduling discount cash flow by ant colony algorithm - Google Patents

Abstract

The present invention discloses a method for optimizing discounted cash flow of project scheduling using ant colony algorithm. The method is used for solving resource constrained scheduling problem of discounted cash flow with multi-mode (MRCPSPDCF). Firstly, MRCPSPDCF model is built, the model considers overhead costs and a punishment mechanism besides ordinary cash inflow and outflow. Then a structural map is built for MRCPSPDCF model, accordingly, the scheduling problem is changed into a search problem based on diagram. Ant constructs solution of problems step by step with pheromone and heuristic information in structural map by using serial schedule generation system (SSGS). The algorithm comprehensive uses the pheromone and heuristic information based on time and cost to guide ant search. The algorithm can obtain better result when solving problem by comparison with existing other algorithm.

Description

The utilization ant group algorithm is optimized the method for the discounted cash flow (DCF) in the project scheduling

Technical field:

The present invention relates to the cash flow management and the intelligence computation two big fields of project level, relate generally to a kind of method of using ant group algorithm to optimize the discounted cash flow (DCF) in the project scheduling.

Technical background:

Cash flow is meant the income and the expenditure of enterprise's different phase cash in certain project or a certain transaction, from start to finish in each link of enterprise.If there is not positive cash flow, enterprise just is difficult to provide salary and gives supplier with cash payment, even can cause the bankruptcy of whole enterprise.The economist calls cash flow life of enterprise and his like, therefore, the management and the optimization of cash flow is seemed particularly important.In the aspect of project level, whether feasible cash flow be to estimate a project a major criterion.When run out of cash takes place, go bankrupt even therefore the project of a high profit also very possible.So in the last few years, the project level cash flow management becomes the focus of a research.An important development in this field is that cash flow management is reduced famous resource constraint project scheduling problem (RCPSP).

What RCPSP will solve is under the condition that satisfies sequential and resource constraint, each activity that has cash inflow or outflow in the project to be dispatched, thereby reach the shortest purpose of finished item duration.Need to consider renewable and non-renewable two class resources among the RCPSP.Wherein renewable resource is meant that the resource available quantity keeps constant resource in each time period of the whole project implementation process, as human resources; Non-renewable resources are meant that the resource available quantity is given when project begins, and the resource that reduces gradually along with consumption, as fund.The precedence constraint of operation usually uses arrow (AoA) network and single code name (AoN) network to represent.In the AoA network, arrow is represented activity and the node presentation of events, and node is represented activity and arrow is represented corresponding priority ranking in the AoN network.In industrial project, RCPSP has obtained using for example building construction and software development etc. widely.

But traditional RCPSP has ignored the standard of estimating a project in the cash flow.Russell has at first considered in the project scheduling problem the not cash flow evaluation criterion of belt restraining condition.Comprise a series of cash inflow and outflow in his model, wherein the inflow of cash comprises that the activity of payment finishes funds, and the outflow of cash comprises the expense of resource etc.The time value to money in the cash flow of discounting takes in, and the inflow of the cash of wherein discounting and the difference of outflow are defined as the net present value (NPV) of whole project.The target of model is exactly reasonably to arrange each activity to make the NPV maximum of cash flow from project.If among resource constraint adding problem, just become the resource constraint project scheduling problem (RCPSPDCF) that contains discounted cash flow (DCF).The result that the scheduling result that obtains with the NPV criterion often obtains with the shortest project duration is very different, and the former practicality makes RCPSPDCF still all obtain paying close attention to widely on engineering is used in scientific research.

More comprehensive and practical type of RCPSPDCF is consideration time a---expense, time---resource and resource---balance of resource.Each activity can be finished under various pattern.In the time---in the middle of the balance of expense, a kind of duration, longer execution pattern may only need lower expense, and short execution pattern of duration is wanted more money.In the time---in the middle of the balance of resource, consume less resource than farm labourer's phase, and the casual labourer needs to consume ample resources the phase.At last, in resource---in the middle of the balance of resource, in different execution patterns, a kind of resource can be used another kind of substitution of resources.The problem that exists these balances to accept or reject is called multi-mode RCPSPDCF (MRCPSPDCF).

Famous solve job shop scheduling problems is exactly a kind of special case of RCPSP, so RCPSP is the very complicated difficult problem of NP-.Owing to need calculate NPV in a nonlinear model, the calculated amount that RCPSPDCF needs is bigger.And the pattern that operation that not only will the consideration activity in MRCPSPDCF also will the consideration activity be carried out, so MRCPSPDCF than before two problems complicated more.Kolisch has proved that in the problem that contains a kind of non-renewable resources at least a feasible solution seeking out MRCPSPDCF just has been NP-[1] fully.Though existing a lot of heuritic approaches are used to find the solution RCPSPDCF, for example Zero-one integer programming, branch-and-bound method, genetic algorithm, tabu search and simulated annealing etc., the algorithm that is used to find the solution MRCPSPDCF is but very rare.Aspect this, Icmeli and Erenguc propose a time---balanced expense problem, wherein in the time of the normal weak point of the duration of activity ratio, just need to pay the expense of overworking.They have used a heuristic procedure that has embedded sequence rule to address this problem [2].

Considered a project that has different expenditure ratios, activity is wherein carried out in various modes to keep positive balance.He has proposed a special heuritic approach dispatches project, to reach finished item on time and to maximize the purpose [3] of NPV.Ulusoy has proposed a kind of genetic algorithm and has been used to solve MRCPSPDCF and has considered four kinds of different payment model [4].

The present invention applies to ant group algorithm among the MRCPSPDCF.Ant group algorithm is that the foraging behavior that is subjected to ant inspires and proposes.At occurring in nature, ant carries out each other interchange by a kind of special chemical substance that is called pheromones.By the release and the perception of pheromones on the path, ant can be found a shortest path from the ant cave to food source.The inspiration that ant group algorithm is subjected to this phenomenon just proposes, and successfully is applied in the finding the solution of various combinatorial optimization problems.Ant group algorithm is very potential in finding the solution MRCPSPDCF, this be because: one, ant group algorithm has been proved to be highly effective on based on the search problem of chart, for example traveling salesman problem and routing issue etc., and MRCPSPDCF can be converted into the search problem based on chart at an easy rate; Its two, be different from other heuritic approach, ant group algorithm is a kind of construction algorithm, wherein separates step by step to construct.Because the cash flow in the project is also studied step by step, this makes ant group algorithm can utilize some effective heuristic informations to improve search speed; Its three, people such as Merkle have applied to ant group algorithm in the finding the solution of RCPSP, and have obtained good effect.

List of references:

[1]R.Kolisch：Project scheduling under resource constraints，efficient heuristics forseveral problem classes.Physica，Heidelberg，1995.

[2]O.Icmeli，S.S.Erenguc：The resource constrained time cost tradeoff projectscheduling problem with discounted cash flows.Journal of Operations Management，vol.14，pp.255-275，1996.

[3]L.Ozdamar：On scheduling project activities with variable expenditure rates.IIETransactions，vol.30，no.8，pp.695-704，1998.

[4]G.Ulusoy：Four payment models for the multi-mode resource constrained projectscheduling problem with discounted cash flows.Annals of Operations Research，vol.102，pp.237-261，2001.

Summary of the invention:

This paper has proposed an ant group algorithm that is used to solve MRCPSPDCF.At first, set up the model of MRCPSPDCF, wherein except common cash inflow and flowing out, this model also will consider indirect cost and one and rewards and penalty mechanism.Model conversation with MRCPSPDCF is a structural map of this problem then, thereby scheduling problem is converted into a search problem based on chart.Ant group algorithm comprises based on the search step of this structural map:

(1) each parameter of initialization algorithm, the pheromones initial value of establishing on the structural map is τ ₀

(2) in the ant according to pheromones and heuristic information, by utilization the separating of serial progress generation method (SSGS) construction problem.Suppose the current mode of being in of ant _i, it selects next mode m ode _jFormula be

Wherein, q ∈ [0,1] is a random number, q ₀For selecting parameter, β is the parameter of decision heuristic information proportion.τ _IlAnd η _IlRepresent from mode respectively _iTo mode _lDirected edge edge (i, l) pheromones on and heuristic information.Each step is carried out the renewal of local pheromones after selecting to finish.Suppose the current mode of being in of ant _iAnd next step has selected mode _j, then the plain rule of upgrading of local message is given by the following formula

τ _ij＝(1-ξ)τ _ij+ξτ ₀

Wherein, ξ ∈ [0,1] is a parameter.

(3) after all ants have all constructed and separated, calculate the NPV that these are separated.The NPV of operation S is with following Equation for Calculating

NPV(S)＝f _IN(S)-f _OUT(S)+f _RP(S)

Wherein, the final net present value (NPV) of NPV (S) expression operation cash flow.f _IN(S), f _OUTAnd f (S), _RP(S) represent cash inflow respectively, the cash outflow and the rewards and punishments amount of money.And S must satisfy priority ranking constraint and resource constraint.

(4) after all ants are all finished the structure of separating, carry out the plain renewal of global information to strengthen best operation.Have only the pheromones on the limit of historical optimum solution or contemporary optimum solution just can be strengthened

Wherein ρ ∈ [0,1] is a parameter, and ω is the positive integer of a historical optimum update strategy of decision and contemporary optimum update strategy utilization ratio.Δ _Ij ^BsValue and NPV (S ^Bs) relevant, be expressed as

${\mathrm{\Δ}}_{\mathrm{ij}}^{\mathrm{bs}}=(\frac{\mathrm{NPV}\left(S^{\mathrm{bs}}\right)-\mathrm{first}}{\mathrm{firstBest}-\mathrm{first}}+1)\·\mathrm{\ψ}$

In following formula, first is meant the NPV of first feasible solution that finds in the algorithm implementation.FirstBest is meant the NPV of the historical optimum solution that first is feasible, is noted that firstBest must be greater than first.ψ is a parameter of the plain ratio of control information.Same, Δ _Ij ^IbAlso can use following formulate

${\mathrm{\Δ}}_{\mathrm{ij}}^{\mathrm{ib}}=(\frac{\mathrm{NPV}\left(S^{\mathrm{ib}}\right)-\mathrm{first}}{\mathrm{firstBest}-\mathrm{first}}+1)\·\mathrm{\ψ}$

If algorithm does not find feasible solution (first is undefined), will can not carry out the plain renewal of global information so.If found feasible solution, but not definition of firstBest is then set ${\mathrm{\Δ}}_{\mathrm{ij}}^{\mathrm{bs}}=\mathrm{\ψ}$ Perhaps ${\mathrm{\Δ}}_{\mathrm{ij}}^{\mathrm{ib}}=\mathrm{\ψ}.$

(5) if reach termination condition, then export optimum operation, otherwise get back to step (2).

The method of invention has the following advantages: 1, considered conditions such as indirect expense, bonus and fine and cash flow constraint in the model of MRCPSPDCF, improved the actual value of model.2, utilize the characteristics of ant group algorithm, designed based on the time with based on the heuristic information of expense and guided the search of ant, thereby improved the optimization Algorithm performance.

Description of drawings:

Figure 1A oA network diagram

Fig. 2 priority ranking constraint synoptic diagram

The synoptic diagram of the MoN figure of Fig. 3 AoA network and correspondence

The process flow diagram of Fig. 4 ant group algorithm

Embodiment:

The invention will be further described below in conjunction with accompanying drawing.

The MRCPSPDCF that the present invention considered is defined as follows.Given one has n activity, the project of some renewable resources and some non-renewable resources, wherein each movable a _iCorresponding limited execution pattern M _iIn the scheduling of activity, provide priority ranking constraint and resource constraint.The inflow of cash and outflow take place during the project implementation.Target is to find out optimum operation, the NPV maximum that makes the project discounted cash flow (DCF).

Following table has provided the concrete definition of the symbol that relates in the problem

Symbol	Definition
		A＝{a 0，a 1，...，a n，a n+1}	Movable set; a 1，...，a nExpression is real movable; a 0And a n+1Represent the dummy activity of project head and the tail respectively; N is illustrated in the number of authentic activity in the project;

S i.st S i.et TE i A [T LOW，T UP] I [ti，ti+1]λ and θ γ and δ

S iStart time, i=0 ..., n+1,0=S 0.st＝S 1.st≤S 2.st≤...≤S n+1.st＝S.ET； S iConcluding time, i=0 ..., n+1, S 0.st＝S 0.et＝0， S n+1.st＝S n+1.et=S.ET; The time point that incident i takes place, i=1 ..., | E|; Discount rate; If the deadline of expection is S.ET＜T LOW, will obtain extra award, if instead S.ET＞T UP, as punishment, the remuneration that obtains will reduce; At unit interval [t i，t i+ 1] mean terms purpose indirect expense; Expense advance payment ratio lambda and milestone payment ratio θ; Award ratio γ and punishment ratio δ;

MRCPSPDCF has following characteristics:

(1) priority ranking constraint---this paper uses that arrow (AoA) network G=(A E) represents priority ranking constraint among the MRCPSPDCF.The AoA network is a kind of directed acyclic graph, the corresponding movable set A of the set of arrow wherein, the set E of the corresponding incident of the set of node.Fig. 1 is the synoptic diagram of AoA network.Be noted that and in the AoA network, increased by two virtual movable a ₀And a ₁₇, the beginning and the ending of project represented in these two activities respectively.Based on the AoA network, the priority ranking constraint can provide by rule shown in Figure 2.Just, incident e _iAfter activity was finished before all were tight, incident e _iTake place at once.Only at incident e _iAfter the generation, the activity after it is tight could begin to carry out.

(2) multi-mode---each activity can be from finite aggregate $M_i=\{{\mathrm{<figure-callout\; id="1"\; label="m\; i"\; filenames="S2008100264786E00011.png,S2008100264786E00021.png"\; state="\{\{state\}\}">m</figure-callout>}}_i,\&CenterDot;\&CenterDot;\&CenterDot;,m_{\mathrm{ij}},\&CenterDot;\&CenterDot;\&CenterDot;m_{i\left|M_i\right|}\}$ In select a kind of execution pattern.The time that each execution pattern is required and the resource of consumption all provide in advance.Movable a _iIn mode m _IjUnder execution time be d _Ij, consume rr ^k _IjThe renewable resource k of unit, nr ^l _IjThe non-renewable resources l of unit, and its fixing expense is CE _IjTherefore, the same activity under the different execution patterns may consume different money, resource and time.---resource and resource---balance of resource that wherein needs consideration time---expense, the time has only optimum pattern just can be used for finishing corresponding movable a in the operation scheduling _i

(3) power of not trying to be the first---as movable a _iIn mode m _Ij, (m _Ij∈ M _i) under begin to carry out the back and just must not interrupt.

(4) resource constraint---there are renewable resource and non-renewable resources in the project.Renewable resource use amount of unit interval in the project implementation process is limited, and the total use amount of non-renewable resources in whole project is limited.

(5) cash flow analysis---the net present value (NPV) of analysis project discounted cash flow (DCF), the objective function of problem are maximization NPV.

In addition, also additionally considered following three characteristics in the present invention:

(1) indirect expense (comprising overhead cost, intermediary fee etc.)---this paper also includes these indirect expenses in model.

(2) bonus and fine (R-P)---the expectation deadline of project is by [t _LOW, t _UP] provide.If project can be at t _LOWFinish before, just can obtain extra bonus.Otherwise, if project can not be at t _UPFinish, enterprise will pay for and lose money before.

(3) cash flow constraint---RCPSPDCF model is traditionally only considered the NPV that project is last, and has ignored cash flow midway.Because the shortage of fund can cause the interruption of project, control cash flow midway seems particularly important.Provide the constraint of cash flow in model by the free cash amount of flow in restriction a period of time, the cash flow in separating will enough be stablized to avoid financial risks like this.

Based on characteristics tectonic model described above, wherein use " incident is paid (PEO) " method, just defrayment after milestone event takes place.

Below set forth the method for building up of model:

The target of MRCPSPDCF is to seek optimum operation S ^*=S ^* ₀..., S ^* _N+1) to reach the purpose of maximization NPV.The NPV of an operation S can be with following Equation for Calculating

NPV(S)＝f _IN(S)-f _OUT(S)+f _RP(S)

Wherein, the final net present value (NPV) of NPV (S) expression operation cash flow.f _IN(S), f _OUTAnd f (S), _RP(S) represent cash inflow respectively, the cash outflow and the rewards and punishments amount of money.And S must satisfy priority ranking constraint and resource constraint.

1. cash inflow (f _IN)

f _INComprise the expense that all are paid in milestone event.At milestone event e _iThe expense of middle payment is with pay _iExpression.Pay _iValue can discount according to the start time of project and be dpay _if _INCan be divided into three parts: the expense (dpay of payment when project begins ₁), the expense (dpay that pays during end _{| E|}) and the defrayment of milestone event

$f_{\mathrm{IN}}\left(S\right)=\underset{i=1}{\overset{\left|E\right|}{\mathrm{\&Sigma;}}}{\mathrm{dpay}}_i\left(S\right)={\mathrm{dpay}}_1\left(S\right)+\underset{i=2}{\overset{\left|E\right|-1}{\mathrm{\&Sigma;}}}{\mathrm{dpay}}_i\left(S\right)+{\mathrm{dpay}}_{\left|E\right|}\left(S\right)$

The certain proportion (λ) of meeting advance payment total expenses when project begins

dpay ₁(S)＝pay ₁(S)＝λU

Suppose at e _iA milestone event before occurs in TLATEST _i, and e _iOccur in TE _i, payment so midway is

Wherein,

θ is the payment ratio of milestone event, pay _iDiscount according to following formula and to be dpay _i

dpay _i(S)＝pay _i(S)×exp(-α·TE _i)(1＜i＜|E|)

Payment when project finishes is

${\mathrm{dpay}}_{\left|E\right|}\left(S\right)=(U-{\mathrm{pay}}_1\left(S\right)-\underset{i=2}{\overset{\left|E\right|-1}{\mathrm{\&Sigma;}}}{\mathrm{pay}}_i\left(S\right))\exp (-\mathrm{\&alpha;}\&CenterDot;S.\mathrm{ET})$

2. cash outflow (f _OUT)

f _OUTComprise all spendings in project, comprise direct cost (dcost) and indirect expense (icost)

f _OUT(S)＝dcost(S)+icost(S)

Dcost comprises expending of movable expense of fixing and resource.For the movable a that under l kind pattern, carries out _j(m _Jl), the resource cost of time per unit is

${\mathrm{RC}}_{\mathrm{jl}}=\underset{k=1}{\overset{\left|\mathrm{RR}\right|}{\mathrm{\&Sigma;}}}{\mathrm{rr}}_{\mathrm{jl}}^k\&CenterDot;{\mathrm{PRR}}_k+\underset{k=1}{\overset{\left|\mathrm{NR}\right|}{\mathrm{\&Sigma;}}}{\mathrm{nr}}_{\mathrm{jl}}^k\&CenterDot;{\mathrm{PNR}}_k$

With m _IlThe direct cost of discounting accordingly can be calculated with following formula

${\mathrm{DC}}_{\mathrm{jl}}={\mathrm{CE}}_{\mathrm{jl}}+\underset{t^{\&prime;}=0}{\overset{d_{\mathrm{jl}}-1}{\mathrm{\&Sigma;}}}{\mathrm{RC}}_{\mathrm{jl}}\&CenterDot;\exp (-\mathrm{\&alpha;}\&CenterDot;t^{\&prime;})$

Based on the expense of discounting under every kind of pattern, the overall direct cost of operation S is

$d\cos t\left(S\right)=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}D{C}_{(S_i.\mathrm{act})(S_i.\mathrm{mode})}\&CenterDot;\exp (-\mathrm{\&alpha;}\&CenterDot;S_i.\mathrm{st})$

The formula that calculates icost is given as follows

$i\cos t\left(S\right)=\underset{t^{\&prime;}=0}{\overset{S.\mathrm{ET}-1}{\mathrm{\&Sigma;}}}{I}_{[t^{\&prime;},t^{\&prime;}+1]}\exp [-\mathrm{\&alpha;}(t^{\&prime;}+1)]$

In particular cases a kind of, if the indirect expense any time in the unit is constant I, following formula can be reduced to so

$i\cos t\left(S\right)=I\frac{1-\exp (-\mathrm{\&alpha;}\&CenterDot;S.\mathrm{ET})}{\exp \left(\mathrm{\&alpha;}\right)-1}$

3. reward and punishment (f _RP)

f _RPThe deadline (S.ET) of depending on project.Given expectation finish the time limit [T _LOW, T _UP], if S.ET is less than T _LOW, enterprise will obtain extra income so.Otherwise, if S.ET is greater than T _UP, enterprise will pay for.The amount of money of rewarding and punishing also needs to discount.Calculate f _RPFormula as follows

Wherein, γ and δ represent respectively to reward and the punishment ratio.

4. constraint condition

Separating of MRCTCTPSP must be satisfied two constraint conditions, priority ranking constraint just and resource constraint.The priority ranking constraint is represented with following two formula

${\mathrm{ET}}_i={\mathrm{<figure-callout\; id="1"\; label="max"\; filenames="S2008100264786E00011.png,S2008100264786E00021.png"\; state="\{\{state\}\}">max</figure-callout>}}_{1\&le;k\&le;\mathrm{n\; and}{S}_k.\mathrm{act}\&Element;\mathrm{pred}\left(e_i\right)}(S_k.\mathrm{et})(1\&le;i\&le;|E\left|\right)$

This formulate is after activity is finished before all of an incident are tight, and this incident takes place at once.

S _k.st≥ET _i，if S _k.act∈succ(e _i)(1≤i≤|E|，1≤k≤n)

This formulate activity can only could begin to carry out after the incident generation before it is tight.

For the per unit time [t ', t '+1] (0≤t '＜S.ET) and all renewable resource k (1≤k≤| RR|), resource constraint is given by the following formula

$\underset{i=0}{\overset{n}{\mathrm{\&Sigma;}}}\mathrm{iswork}\left(S_i\right)\&CenterDot;{\mathrm{rr}}_{(S_i.\mathrm{act})(S_i.\mathrm{mode})}^k\&le;{\mathrm{RR}}_k$

Wherein, must not be in the resources consumption amount of each section in the time greater than amount usable.

To all non-renewable resources k (1≤k≤| NR|), resource constraint can be expressed as

$\underset{i=0}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{nr}}_{(S_i.\mathrm{act})(S_i.\mathrm{mode})}^k\&le;{\mathrm{NR}}_k$

Sometimes, need also to consider that the constraint of cash flow is to avoid financial risks.Give the lower limit C of the free cash that fixes on time t place _Risk(t), should be in cash inflow and the difference of outflow of t before constantly greater than C _Risk(t).

Below describe the ant group algorithm that is used to find the solution MRCPSPDCF in detail:

Ant group algorithm find the solution MRCPSPDCF focus on set up a structural map for problem, and definition pheromones and heuristic information.In the algorithm of invention, at first the AoA network of problem to be converted into node mode (MoN) figure.Such MoN figure just becomes the structural map of ant group algorithm.Based on structural map, artificial ant just can be transported logical serial progress method of formation (SSGS) and come having separated of search problem.In the process that algorithm is carried out, every ant all can be kept a progress maker, and separating according to pheromones and heuristic information construction problem.

1. structural map

(EDGE MODE) is a kind of complete digraph to MoN figure G=.Set of node is made up of all patterns in all activities, just $\mathrm{MODE}=\underset{i=0}{\overset{n+1}{\&cup;}}{M}_i.$ The set EDGE of directed edge couples together any two nodes among the figure.

Below introducing the AoA network switch is the method for MoN figure.At first, the institute with each activity might execution pattern collect set $\mathrm{MODE}=\underset{i=0}{\overset{n+1}{\&cup;}}{M}_i=\{{\mathrm{<figure-callout\; id="1"\; label="mode"\; filenames="S2008100264786E00011.png,S2008100264786E00021.png"\; state="\{\{state\}\}">mode</figure-callout>}}_1,\&CenterDot;\&CenterDot;\&CenterDot;,{\mathrm{mode}}_{\left|\mathrm{MODE}\right|}\}$ In, and rearrange their call number.Just, each mode m _Ij(0≤i≤n+1,1≤j≤| M _i|, m _Ij∈ M _i) an element mode among the corresponding MODE _kWith | MODE| represents the sum of pattern, mode ₁The pattern that the expression project begins, mode _{| MODE|}The pattern that expression finishes.All other pattern M _i(i=1,2 ..., n) all in order from mode ₂To mode _{| MODE|-1}Carry out index.Get in touch two kinds of different expression modes for convenience, work as mode _k=m _IjThe time, establish Act (mode _k)=i, Mode (mode _k)=j.

After having generated set MODE, (EDGE MODE) can obtain MoN figure G=as follows.Each node among the MoN figure is corresponding with an element among the set MODE.The set EDGE of directed edge couples together any two nodes among the figure, wherein from mode _iTo mode _jThe limit by edge (i, j) expression.Fig. 3 is the synoptic diagram of the MoN figure of a simple AoA network and correspondence.The AoA network that provides among the left figure contains three authentic activities and two dummy activities.Suppose that each authentic activity all has two kinds of patterns, and each dummy activity all has only a kind of pattern, then always have eight nodes among the corresponding right figure of eight kinds of patterns.What note is all limits of not drawing in right figure, and all nodes among the MoN figure are all linked to each other by directed edge.

MoN figure is the structural map of separating that ant is set up problem.Pheromones and heuristic information are placed on the limit of connection mode, and wherein (i, j) pheromones on and heuristic information are expressed as τ respectively at edge _IjAnd η _IjEach ant all is the guide according to pheromones and heuristic information, thereby selects a global solution of the problem that constructs by opposite side length by length.Like this, scheduling problem just is converted into a search problem based on chart, and purpose is will find out from start node (mode ₁) to endpoint node (mode _{| MODE|}) a path that has maximum NPV, and satisfy priority ranking constraint and resource constraint.Have a lot of redundant limits of violating the priority ranking constraint in MoN figure, these redundant limits are can not appear to separate.When the solution path of construction problem, ant can be kept the set on a legal limit in each step.Have only those feasible Bian Caihui to be included in the set, those infeasible limits then are left in the basket.

2. progress maker

Based on structural map, each ant is all transported logical serial progress method of formation (SSGS) and sets up separating of problem.Separate S (S for one by the SSGS generation ₀..., S _N+1) be exactly from start node (S ₀=mode ₁) to endpoint node (S _N+1=mode _{| MODE|}) all movable paths of a process.The false code of SSGS is as follows

Procedure SSGS

1 S ₀＝mode ₁

2 S ₀.st＝S ₀.et＝0

3 upgrade eligibleSet

4 for i＝1 to n+1

5 S _i=select () // from the legal pattern of selecting of concentrating

6 with S _iBe dispatched to the earliest time that meets precedence constraint and resource constraint

7 new resources more

8 upgrade eligibleSet

9 end for

10 assessment NPV (S)

end procedure

In the starting stage of SSGS (false code 1-3 is capable), at first with start node (mode ₁) elect first pattern of separating S as, and set mode ₁Zero-time (S ₀.st) and concluding time (S ₀.et) be 0.In addition, but ant also can be kept the set (eligibleSet) of a row mode, and wherein each element can be elected the next pattern of separating as.Just for  a _i∈ succ (e ₁), a _iAll patterns all can be selected into eligibleSet.

In the 4-9 of false code was capable, ant was added the pattern among the eligibleSet among the S step by step by circulation, has covered all activities up to S.Ant is at first selected a feasible Mode S from eligibleSet in k goes on foot _k=mode _l, schedule it to the execution time the earliest of satisfying priority ranking and resource constraint then.What wherein, use in function select () is that a kind of pseudorandom compares case selection method.Upgrade execution pattern mode afterwards _lAfter available resources quantity.EligibleSet is upgraded, remove wherein all and mode m ode _lBelong to the pattern of an activity together.In addition, if mode _lAffiliated activity is incident e _jTight before one of movable, Act (mode just _l) ∈ pred (e _j), and at pred (e _j) in all other activities all executed finish, all belong to succ (e so _j) the pattern of activity all can be added among the eligibleSet.A kind of special circumstances are that if last non-renewable resources are not enough to carry out any unenforced pattern, eligibleSet will become empty set so, so just can not form complete separating.In this case, the progress maker will stop, and the NPV of this infeasible operation also can be set as a very little value.

When one complete separate structure and finish after, its net present value (NPV) NPV (S) just can calculate according to the formula that provides.

3. the program of ant group algorithm and rule

The process flow diagram of the ant group algorithm of invention as shown in Figure 4.In algorithm, each ant each time the circulation in all keep one independently SSGS in order to make up separating of it.Same, ant also can be kept pheromones and heuristic information as the selection information among the SSGS.

(1) system of selection

Next pattern is selected than case selection method according to pseudorandom by function select () in SSGS.Suppose the current mode of being in of ant _i, the next mode m ode that from function select (), returns _jCan be with following two formulates

Generate a random number q ∈ [0,1] and with select parameter q ₀Compare.If q≤q ₀, just be chosen in τ among the eligibleSet _Il(η _Il) ^βMaximum execution pattern mode _l, τ wherein _IlAnd η _IlRepresent from mode respectively _iTo mode _lDirected edge edge (β is the parameter of decision heuristic information proportion for i, l) pheromones on and heuristic information.On the other hand, if q＞q ₀, so just carry out the system of selection of roulette, that is to say and select mode _jProbability and τ _Ij(η _Ij) ^βValue proportional.

(2) processing of pheromones

At the initial phase of ant group algorithm, the pheromones on all directed edges all is set to 1

τ _ij＝τ ₀＝1，(i，j)∈EDGE

Finish after one among SSGS step selected a pattern when ant, just carry out the renewal of local pheromones at once.Suppose the current mode of being in of ant _iAnd next step has selected mode _j, then the plain rule of upgrading of local message is given by the following formula

τ _ij＝(1-ξ)τ _ij+ξτ ₀

Wherein, ξ ∈ [0,1] is a parameter.Because τ ₀Be the lower limit of pheromones value, the plain effect of upgrading of local message is to reduce the pheromones of having selected on the limit, and the ant after making has bigger probability to select other limit.In other words, the locality pheromones is upgraded the diversity that has strengthened algorithm.

In circulation once, all finish the structure of separating when all ants after, carry out that global information is plain to be upgraded to strengthen best operation.Have only the pheromones on the limit of historical optimum solution or contemporary optimum solution just can be strengthened

Wherein ρ ∈ [0,1] is a parameter, and ω is the positive integer of a historical optimum update strategy of decision and contemporary optimum update strategy utilization ratio.Δ _Ij ^BsValue and NPV (S ^Bs) relevant, be expressed as

${\mathrm{\&Delta;}}_{\mathrm{ij}}^{\mathrm{bs}}=(\frac{\mathrm{NPV}\left(S^{\mathrm{bs}}\right)-\mathrm{first}}{\mathrm{firstBest}-\mathrm{first}}+1)\&CenterDot;\mathrm{\&psi;}$

In following formula, first is meant the NPV of first feasible solution that finds in the algorithm implementation.FirstBest is meant the NPV of the historical optimum solution that first is feasible, is noted that firstBest must be greater than first.ψ is a parameter of the plain ratio of control information.Same, Δ _Ij ^IbAlso can use following formulate

${\mathrm{\&Delta;}}_{\mathrm{ij}}^{\mathrm{ib}}=(\frac{\mathrm{NPV}\left(S^{\mathrm{ib}}\right)-\mathrm{first}}{\mathrm{firstBest}-\mathrm{first}}+1)\&CenterDot;\mathrm{\&psi;}$

If algorithm does not find feasible solution (first is undefined), will can not carry out the plain renewal of global information so.If found feasible solution, but not definition of firstBest is then set ${\mathrm{\&Delta;}}_{\mathrm{ij}}^{\mathrm{bs}}=\mathrm{\&psi;}$ Perhaps ${\mathrm{\&Delta;}}_{\mathrm{ij}}^{\mathrm{ib}}=\mathrm{\&psi;}.$

(3) heuristic information

Heuristic information is used to guide the search of ant.Fully utilized heuristic information in the present invention based on time and expense.

Time-based heuristic information can accelerating project process.Time-based herein heuristic information is called (EFT) heuristic information: η on earliest finish time _Ij=1/EFT _jJust, be positioned at mode _iIn ant select mode _jTendentiousness and mode _jConcluding time (the EFT that meets constraint condition the earliest _j) be inversely proportional to.If mode _jBe movable a _kL kind pattern (mode _j=m _Kl), EFT so _j=EST _j+ d _Kl

Can guide the ant sorting charge with less pattern based on the heuristic information of expense.Minimum charge (MCDA) heuristic information: η when being called the activity of discounting and beginning based on the heuristic information of expense herein _Ij=1/DACOST _jJust, select mode _jTendentiousness and the expense of the pattern activity of discounting when beginning be inversely proportional to.If mode _jBe movable a _kL kind pattern (mode _j=m _Kl), DACOST _jCan be calculated as follows

$\mathrm{DACOS}{T}_j={\mathrm{CE}}_{\mathrm{kl}}+\underset{t^{\&prime;}=0}{\overset{d_{\mathrm{kl}}-1}{\mathrm{\&Sigma;}}}({\mathrm{RC}}_{\mathrm{kl}}+I)\&CenterDot;\exp (-\mathrm{\&alpha;}\&CenterDot;t^{\&prime;})$

Carrying out function select () before, artificial ant is selected one type heuristic information at first at random.{ 0,1,2} is if ran=0 then uses the EFT heuristic information to generate a random integers ran ∈; If ran=1 then uses the MCDA heuristic information; Otherwise, utilization EFT+MCDA heuristic information: ${\mathrm{\&eta;}}_{\mathrm{ij}}=\frac{1}{({\mathrm{EFT}}_j\&CenterDot;I)+{\mathrm{DACOST}}_j},$ Wherein I is a constant, and purpose is two kinds of heuristic informations of balance.

The method that below provides invention is used to find the solution the emulation testing result of MRCPSPDCF:

The algorithm that is used to optimize MRCPSPDCF is very rare, and in existing document, the genetic algorithm of having only Ulusoy to propose can be used for finding the solution of similar problem.Therefore, ant group algorithm and this genetic algorithm with invention compares.Wherein, the parameter of ant group algorithm is set to: n=10, ζ=0.1, ρ=0.1, β=1, q ₀=0.9, ω=1 and ψ=20.Generate 55 projects that have 13 to 98 activities at random, wherein each activity all has 1 to 5 kind of execution pattern, and generates resource and cash flow at random.In these 55 emulation testing examples, ant group algorithm can obtain higher NPV in 47 examples.This method that has proved invention is highly effective.

Claims (3)

1, a kind of method of using ant group algorithm to optimize the discounted cash flow (DCF) in the project scheduling; it is characterized in that; this method has at first been set up the model of MRCPSPDCF, and wherein except common cash inflow and flowing out, this model also will consider indirect cost and one and rewards and penalty mechanism.Model conversation with MRCPSPDCF is a structural map of this problem then, thereby scheduling problem just is converted into a search problem based on chart.Ant group algorithm comprises based on the search step of this structural map:

(1) each parameter of initialization algorithm, the pheromones initial value of establishing on the structural map is τ ₀

(2) ant is according to pheromones and heuristic information, by separating of utilization serial progress generation method (SSGS) construction problem.Suppose the current mode of being in of ant _i, it selects next mode m ode _jFormula be

Wherein, q ∈ [0,1] is a random number, q ₀For selecting parameter, β is the parameter of decision heuristic information proportion.τ _IlAnd η _IlRepresent from mode respectively _iTo mode _lDirected edge edge (i, l) pheromones on and heuristic information.Each step is carried out the renewal of local pheromones after selecting to finish.Suppose the current mode of being in of ant _iAnd next step has selected mode _j, then the plain rule of upgrading of local message is given by the following formula

τ _ij＝(1-ξ)τ _ij+ξτ ₀

Wherein, ξ ∈ [0,1] is a parameter.

(3) after all ants have all constructed and separated, calculate the NPV that these are separated.The NPV of operation S is with following Equation for Calculating

NPV(S)＝f _IN(S)-f _OUT(S)+f _RP(S)

Wherein, the final net present value (NPV) of NPV (S) expression operation cash flow.f _IN(S), f _OUTAnd f (S), _RP(S) represent cash inflow respectively, the cash outflow and the rewards and punishments amount of money.And S must satisfy priority ranking constraint and resource constraint.

(4) after all ants are all finished the structure of separating, carry out the plain renewal of global information to strengthen best operation.Have only the pheromones on the limit of historical optimum solution or contemporary optimum solution just can be strengthened

Wherein ρ ∈ [0,1] is a parameter, and ω is the positive integer of a historical optimum update strategy of decision and contemporary optimum update strategy utilization ratio.Δ _Ij ^BsValue and NPV (S ^Bs) relevant, be expressed as

${\mathrm{\&Delta;}}_{\mathrm{ij}}^{\mathrm{bs}}=(\frac{\mathrm{NPV}\left(S^{\mathrm{bs}}\right)-\mathrm{first}}{\mathrm{firstBest}-\mathrm{first}}+1)\&CenterDot;\mathrm{\&psi;}$

In following formula, first is meant the NPV of first feasible solution that finds in the algorithm implementation.FirstBest is meant the NPV of the historical optimum solution that first is feasible, and firstBest must be greater than first.ψ is a parameter of the plain ratio of control information.Same, Δ _Ij ^IbAlso can use following formulate

${\mathrm{\&Delta;}}_{\mathrm{ij}}^{\mathrm{ib}}=(\frac{\mathrm{NPV}\left(S^{\mathrm{ib}}\right)-\mathrm{first}}{\mathrm{firstBest}-\mathrm{first}}+1)\&CenterDot;\mathrm{\&psi;}$

If algorithm does not find feasible solution (first is undefined), will can not carry out the plain renewal of global information so.If found feasible solution, but not definition of firstBest is then set ${\mathrm{\&Delta;}}_{\mathrm{ij}}^{\mathrm{bs}}=\mathrm{\&psi;}$ Perhaps ${\mathrm{\&Delta;}}_{\mathrm{ij}}^{\mathrm{ib}}=\mathrm{\&psi;}.$

(5) if reach termination condition, then export optimum operation, otherwise get back to step (2).

2, based on the described a kind of method of using ant group algorithm to optimize the discounted cash flow (DCF) in the project scheduling of claim 1, it is characterized in that, in the model of MRCPSPDCF, considered conditions such as indirect expense, bonus and fine and cash flow constraint.

3, based on the described a kind of method of using ant group algorithm to optimize the discounted cash flow (DCF) in the project scheduling of claim 1, it is characterized in that algorithm synthesis has utilized the search that guides ant based on the heuristic information of time and expense.Time-based heuristic information can accelerating project process.The computing method of this heuristic information are: η _Ij=1/EFT _jJust, be positioned at mode _iIn ant select mode _jTendentiousness and mode _jConcluding time (the EFT that meets constraint condition the earliest _j) be inversely proportional to.If mode _jBe movable a _kL kind pattern (mode _j=m _Kl), EFT so _j=EST _j+ d _KlCan guide the ant sorting charge with less pattern based on the heuristic information of expense.The computing method of this heuristic information are: η _Ij=1/DACOST _jJust, select mode _jTendentiousness and the expense of the pattern activity of discounting when beginning be inversely proportional to.If mode _jBe movable a _kL kind pattern (mode _j=m _Kl), DACOST _jCan be calculated as follows

$\mathrm{DACOS}{T}_j={\mathrm{CE}}_{\mathrm{kl}}+\underset{t^{\&prime;}=0}{\overset{d_{\mathrm{kl}}-1}{\mathrm{\&Sigma;}}}({\mathrm{RC}}_{\mathrm{kl}}+I)\&CenterDot;\exp (-\mathrm{\&alpha;}\&CenterDot;t^{\&prime;})$

Before selecting next execution pattern, artificial ant is selected one type heuristic information at first at random.Method is that { 0,1,2} is if ran=0 then uses time-based heuristic information to random integers ran ∈ of generation; If ran=1 then uses the heuristic information based on expense; If ran=2, the then above two kinds of heuristic informations of integrated use

${\mathrm{\&eta;}}_{\mathrm{ij}}=\frac{1}{({\mathrm{EFT}}_j\&CenterDot;I)+\mathrm{DACOS}{T}_j}$

Wherein I is a constant.

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