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Enterprise resource planning implementation decision & optimization models*
Journal of Systems Engineering and Electronics Vol. 19, No. 3, 2008, pp.513–521
Enterprise resource planning implementation decision & optimization models∗ Wang Shaojun, Wang Gang, L¨ u Min & Gao Guoan Advanced Manufacturing Technology Center, Harbin Inst. of Technology, Harbin 150001, P. R. China (Received July 13, 2007)
Abstract: To study the uncertain optimization problems on implementation schedule, time-cost trade-off and quality in enterprise resource planning (ERP) implementation, combined with program evaluation and review technique (PERT), some optimization models are proposed, which include the implementation schedule model, the timecost trade-off model, the quality model, and the implementation time-cost-quality synthetic optimization model. A PERT-embedded genetic algorithm (GA) based on stochastic simulation technique is introduced to the optimization models solution. Finally, an example is presented to show that the models and algorithm are reasonable and effective, which can offer a reliable quantitative decision method for ERP implementation.
Keywords: optimization model, ERP, chance-constrained programming, PERT, genetic algorithm, time, cost, quality.
1. Introduction ERP is becoming increasingly the necessary integrated information platform in enterprise business management. Implementing ERP can bring huge benefits to enterprises, at the same time, the enterprises also face huge risks on the process of ERP implementation. These risks include decision risk, implementation risk, and application risk[1] . A successful ERP project depends on the ERP software, which occupies 30% in significance, but owes 70% to correct implementation. The process of project implementation is one of the key steps to ERP success. The project executors will meet some uncertain factors such as techniques, human resource, enterprise environment, and so on, which will lead to the project implementation time, cost, and quality uncertainty. It was investigated by some literatures that during the process of ERP implementation, the implementation schedules of 90% of all enterprises were over the limited time, the implementation cost of 80% of all enterprises was over the project budget, and 60%–70% of all enterprises did not complete system integration[2]. The universal problems were that the implementation qualities were uncertain and short of effective project manage-
ment. Since the ERP project is a system engineering, the ERP project needs explicit objective, scientific programming, and strict project management[3] . Therefore, when the project starts, it is necessary to evaluate and analyze the implementation schedule proposed by ERP supplier or consulting company. To reach the goals of reduction of implementation risks and assurance of quality, it is important to optimize the implementation schedule and cost. PERT was used to deal with the problems that the logic relationships of activities were certain but their durations were uncertain in the project management. Thus, PERT can be used in project evaluation and programming now. To solve the uncertainty problem of time, cost, and quality in the process of implementation, combined with the chance constrained programming based on stochastic simulation (Monte Carlo emulation), several optimization models are established in this article, which give enterprise some methods to make decision in ERP implementation.
2. Decision problem description and hypothesis & preconditions 2.1 Decision problem description The process of ERP project optimization and deci-
* This project was supported by the National High-Tech. R & D Program for CIMS, China (2003AA413210).
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sion is shown in Fig.1.
Fig.1
The logic decision structure on ERP implementation optimization
The implementation time, cost, quality, and risks are the considerably important factors in making an ERP implementation project. The time, cost, and quality are the basic critical success factors, but they always conflict each other, and it is impossible to optimize the three factors simultaneously[4] . Enhancing quality makes investment increase or schedule prolong, whereas compressing schedule will lead projects to increase the direct implementation cost; on the contrary, controlling the cost to the lowest and compressing schedule to the shortest will lead the quality to lower, thereby causing project risks in cost, time, and quality. Therefore, ERP project should be feasible and coincident with the mutual constrained conditions. The executors always hope that the implementation time will be shortest, the cost will be under the budget, and the quality implementation result will achieve the expected level. In short, the problem can be described as ⎧ ⎪ ⎪ min T ime ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ min Cost ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ max Quality ⎪ ⎪ ⎨ s.t. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T ime TP ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Cost CP ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Quality QE where TP , CP denote the project time limit and bud-
get, respectively, and QE denotes the expected project quality. 2.2
Hypothesis & preconditions of models
This article proposes the following basic hypothesis & preconditions based on the classical PERT net model. (1) As the existence of uncertain factors, implementation time, cost, and quality are stochastic variables under some kinds of probability distributions. (2) The resources for every activity are limited. (3) The direct activity cost is inversely proportional to the activity time, namely, the longer the continued time, the higher is the direct cost; the indirect activity cost is directly proportional to the activity time, the longer the continued time, the higher is the indirect cost. (4) Under certain implementation conditions, during the necessary period of activity time, activity quality is directly proportional to activity time; the longer the continued time, the better will be the implementation quality. The activity quality is also directly proportional to the activity cost; the higher the cost, the better will be the quality.
3. Chance constrained programming ERP implementation optimization is to optimize the objective function in uncertain circumstance. Chance constrained programming mainly deals with such situation that constraint conditions contain stochastic variables, and decision must be made before the
Enterprise resource planning implementation decision & optimization models results of the stochastic variables’ realization are observed[5]. To consider the decision may do not coincide with the constrain conditions sometimes when the situation is not good, so allowing the decision making do not coincide with the constraint conditions in certain extent, but the probability that the decision makes constraint conditions tenable must not be less than a believable degree α. The chance constrained programming model can be expressed as follows. ⎧ ⎪ min f ⎪ ⎪ ⎪ ⎪ ⎨ s.t. ⎪ ⎪ Pr f (x, ξ) f β ⎪ ⎪ ⎪ ⎩ Pr gj (x, ξ) f α where x is an n dimensions variable, ζ is a stochastic variable, f (x, ξ) is an objective function, gj (x, ξ) is a stochastic constrained function, j = 1, 2, . . . , p, Pr{·} denotes the probability that makes event {·} come into being, α is the believable degree of constraint conditions decided beforehand, β is the believable degree of objective functions, f is the minimum of objective functions when the believable degree is not less than β, and f = min{f | Pr{f (x, ξ) f } β}.
4. ERP implementation optimization models 4.1 ERP implementation schedule expression The whole ERP implementation process can be decomposed to limited activities (sub projects), which form a PERT net according to the time logic sequence. Let arrow (or directed arc) denote activity, and let node denote the beginning or the end of an activity (or event). It is used to be illustrated by Active-onArrow method (A-O-A)[6] . To construct an A-O-A net, the following rules should be observed. (1) Before an activity begins, the adjacent former activities must be complete. (2) Each activity can only connect two events, and there is only one activity between two events. (3) There is only one original node and one final node in a net; they represent the beginning and end of the project, respectively. (4) Feedback circles and loops are not allowed in the net. We can regard the PERT net as a directed loopless
515
graph; the node in the graph represents an event or decision time, and the directed arc represents activity. Let G = (V, A) denote a directed loopless graph, and V = {1, 2, . . . , n} be the set of nodes; A denotes the set of arcs, and aij ∈ A denotes an arc from node i to node j in G. The order among nodes is denoted by an adjacent matrix M , M = (γij )n×n , and ⎧ ⎨ 1, {i} ≺ {j} . {i} ≺ {j} denotes node γij = ⎩ 0, {i} not ≺ {j} {i} borders on node {j}, and {i} is the adjacent former node in activity (i, j), while {j} is the adjacent subsequent node in activity (i, j). 4.2
Implementation schedule optimization modelbased on critical path
Suppose the net is composed of k activities in all. The critical path is defined as the route with the longest implementation period. Let L = {γij = 1|aij ∈ A, i = 1, 2, . . . , n − 1; j = 2, 3, . . . , n} denote a set of paths from the beginning note 1 to the final note n. Let stochastic variable xij denote the duration of activity (i, j), namely, the length of arc aij . The total time of a random path from beginning γij xij . node {1} to final node {n} is TL (xij ) = aij ∈L
Thus, the total time of the critical path is denoted as TK (xij ) = max TL (xij ) = max γij xij . Let aij ∈L
stochastic variable xi denote the net node time, and x1 denote the earliest beginning time of the project, and xn denote the earliest finish time. In the condition only considering the implementation schedule, the implementation schedule optimization model based on the critical path is established as follows. ⎧ ⎪ min T (x ) = min max γij xij K ij ⎪ ⎪ ⎪ ⎪ aij ∈L ⎪ ⎪ ⎪ ⎪ ⎪ s.t. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ xj − xi xij ⎪ ⎪ ⎪ ⎪ ⎨ x −x T n 1 p
(1) ⎪ ⎪ Pr max γij xij Tp βT ⎪ ⎪ ⎪ ⎪ aij ∈L ⎪ ⎪ ⎪ ⎪ ⎪ Pr {d xij Dij } αT ij ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i = 1, 2, . . . , n − 1; j = 2, 3, . . . , n ⎪ ⎪ ⎪ ⎩ ∀aij ∈ A γij ∈ {0, 1},
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where TP denotes the expected implementation period of the project, Dij denotes the normal implementation period of the activity (i, j), dij denotes the minimum compressed period of the activity (i, j). Pr{·} denotes the probability that the event in{·} is tenable, αT is the believable degree of constraint conditions decided beforehand, and βT is the believable degree of objective functions. The constrain, xj − xi xij , reflects the sequence constraint among activities. 4.3
Implementation time-cost optimization model
The ERP implementation costs mainly include direct cost and indirect cost. The basically direct cost comes from the human resource fee and equipment cost, while the indirect cost mainly comes from the routine management expenditures. Suppose the direct cost has linear progressive decrease relationship with the implementation schedule in the range xij ∈[dij , Dij ], namely, when the activity time is compressed, the direct cost decreases. The indirect cost has linear progressive increase relationship with the implementation schedule, namely, with the activity time increase, the indirect cost also increases. In some cases, the relationship that obtains implementation time decrease by increasing cost is called the time-cost tradeoff[6,7] . The time-cost trade-off model is an optimization model on minimizing the implementation cost under the project implementation period constraint conditions. Let δij denote the time-cost slope of activity (i, j), and the stochastic variable V xij denote the period decrement of activity (i, j) by compression. The time-cost function of direct cost increment can be described as ⎧ ⎪ +∞, V xij Dij − dij ⎪ ⎨ gi (V xij ) = δij V xij , 0 V xij Dij − dij ⎪ ⎪ ⎩ 0, other i = 1, 2, . . . , n − 1; j = 2, 3, . . . , n The decrement of indirect cost is hij V xij when the implementation period of activity (i, j) is compressed, and hij is the indirect cost per unit time (per day). Let CP denote the project budget, and the stochastic variable Cij denote the normal implementation cost of activity (i, j). Suppose the total number of activities is k. In the situation that the schedule compression
is considered, the total project implementation cost function is f (Cij , V xij ), and f (Cij , V xij ) =
k
(Cij + δij V xij − hij V xij ) =
aij ∈A k
[Cij + (δij − hij )V xij ]
aij ∈A
Thus, the project implementation time-cost chance constrained programming model can be established as ⎧ ⎪ ⎪ ⎪ min f (Cij , V xij ) ⎪ ⎪ ⎪ ⎪ ⎪ s.t. ⎪ ⎪ ⎪ ⎪ ⎪ 0 V xij Dij − dij ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k
⎪ ⎪ ⎨ Pr [Cij + (δij − hij )V xij ] CP βC aij ∈A ⎪ ⎪ ⎪
⎪ ⎪ ⎪ Pr max γij (xij − V xij ) Tp αT ⎪ ⎪ ⎪ ⎪ aij ∈L ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i = 1, 2, . . . , n − 1; j = 2, 3, . . . , n ⎪ ⎪ ⎪ ⎪ ⎩ γij ∈ {0, 1}, ∀aij ∈ A (2) 4.4
Implementation quality optimization model
On the premise that the ERP supplier and implementation circumstance are determined, according to the hypothesis mentioned above, the activity quality is directly proportional to the activity time and cost. Since the activity cost is a function of time, the activity quality can be built as a function of time. For xij , the duration of activity (i, j), is a stochastic variable, and the quality of activity (i, j) is also a stochastic variable. The function on the quality of activity (i, j) can be built as follows. ⎧ ⎪ 0, xij < dij ⎪ ⎨ C Qij (xij ) = qij + ηij xij , dij xij Dij ⎪ ⎪ ⎩ N qij , xij Dij i = 1, 2, . . . , n − 1; j = 2, 3, . . . , n N C − qij qij C , where, qij is the minimum qualDij − dij ity when the implementation period of activity (i, j) N is compressed to the limitation, and qij is the normal quality in the normal implementation period of
and ηij =
Enterprise resource planning implementation decision & optimization models
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activity (i, j). The function Qij (xij ) shows that the activity is impossible to be completed when the implementation period is less than the compression limitation, and Qij = 0; on the other hand, when the implementation period is over the normal period, the activity quality maintains stability and without increasing with the implementation period extension. As the whole project quality is determined by the minimum quality of the weakest activity, the quality function of the project is Q(xij ) = min Qij (xij ), i = 1, 2, . . . , n − 1; j = 2, 3, . . . , n. The chance constrained programming model on project implementation quality is established as follows ⎧ ⎪ max Q(xij ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s.t. ⎪ ⎨ (3) Pr{min Qij (xij ) QE } βQ ⎪ ⎪ ⎪ ⎪ ⎪ Pr{dij xij Dij } αT ⎪ ⎪ ⎪ ⎩ i = 1, 2, . . . , n − 1; j = 2, 3, . . . n
established as follows: ⎧ ⎪ min F (xij , V xij , Cij ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s.t. ⎪ ⎪
⎪ ⎪ ⎪ ⎪ γ x T βT Pr max ⎪ ij ij p ⎪ ⎪ ⎪ aij ∈L ⎪ ⎪ ⎪ ⎪ k
⎪ ⎪ ⎪ ⎪ Pr [Cij + (δij − hij )V xij ] CP βC ⎪ ⎪ ⎨ aij ∈A
where QE denotes the expected activity quality.
The project managers also expect to evaluate the success or risk probability in advance before the project is complete. According to the stochastic programming optimum value obtained from the above optimization models, the success or risk probability can be obtained through stochastic simulation (Monte Carlo emulation). Let ET denote the event
max γij xij Tp ; let EC denote the event
4.5
Implementation time-cost-quality combined tradeoff optimization model
In practice project implementation, the executors always consider the implementation schedule, cost, and quality together, and expect to achieve the shortest time, the lowest cost, and the best quality. The problem belongs to multi-objective optimization. However, the three factors constrain each other, and to solve the conflict, the usual method is to adopt the tradeoff model to transform multi-objective functions into a single-objective function by adding weight to each objective function. F (xij , V xij , Cij ) = w1 TK (xij ) + w2 f (Cij , V xij ) − w3 Q(xij ) = w1 max
aij ∈L
γij xij + w2
k aij ∈A
[Cij + (δij − hij )V xij ] − w3 min Qij (xij ) Under the constraints of implementation schedule, budget cost, and expected quality, the chance constrained time-cost-quality combined tradeoff model is
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
Pr{min Qij (xij ) QE } βQ Pr{dij xij Dij } αT
Pr max γij (xij − V xij ) Tp αT aij ∈L
i = 1, 2, . . . , n − 1; γij ∈ {0, 1},
j = 2, 3, . . . , n
∀aij ∈ A
(4) In the model shown above, w1 , w2 , w3 denote the weight of implementation schedule, cost, and quality, respectively, and w1 + w2 + w3 = 1. 4.6
The probability evaluation on success and risk
k
aij ∈L
[Cij + (δij − hij )V xij ] CP ; let EQ denote
aij ∈A
the event {min Qij QE }. The probability of success can be described as Pr{success} = Pr{ET ∩ EC ∩ EQ } = Pr{EQ |ET ∩ EC } · Pr{EC |ET } · Pr{ET }. The equation illuminates that the project can be regarded as successful only when the schedule is within the time limit of project, the cost is under the budget, and the implementation results get the expected quality. Thus, the correspondent implementation risk can be described as Pr{risk} = 1 − Pr{success}.
5. Models solution The conventional chance programming solution is to transform chance constraints into corresponding cer-
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tain equivalent classes according to the given believable degree in advance, and then solve the certain equivalent class model with classical methods[5] . However, it is impossible to find the corresponding equivalent classes for several complex chance constraint programmings. Fortunately, the appearance of the genetic algorithm makes it possible for us to solve this kind of model easily. The genetic algorithm (GA) is a method that searches for the optimization solution by simulating the nature evolution process. GA has the following characteristics: all fields search ability, good robust, easy to extend, and easy to program; besides, it does not demand that the objective function must be continuous and derivationable[8] . It is these properties that make GA suitable to solve the optimization problem of PERT nets stochastic programming. Aiming at the models established above, a PERT-embedded genetic algorithm based on stochastic simulation is proposed to solve these models. 5.1
The method of chromosome coding
To adapt the solution vectors, the float vector is adopted to represent a chromosome. The length of the chromosome is not fixed and it is the same length as the corresponding solution vector. The vector X = (x1 , x2 , . . . , xn ) denotes the solution of optimization, and n is the dimension number; thus, the chromosome is expressed as V = (v1 , v2 , . . . , vn ). 5.2
Verifying stochastic system constraint
Consider chance constraint Pr{gj (x, ξ) 0, j = 1, 2, . . . , k} α. ξ is the stochastic variable, for which the probability density distribution function is Φ(ξ). For a given decision variable x, the stochastic simulation (Monte Carlo emulation) is applied to verify if the constraint is tenable. The methods producing the stochastic number are shown in Ref. [7]. The test steps are as follows. Step 1 Step 2 probability Step 3 N + +; Step 4 gether; Step 5
Set N = 0; Produce a stochastic variable ξ from the distribution function Φ(ξ); If gj (x, ξ) 0, j = 1, 2, . . . , k, then,
return to “untenable”. 5.3
Objective value calculation
Consider the objective function Pr{f (x, ξ) f } β with stochastic parameters; the following algorithm can find the minimum f that satisfies the conditions above: Step 1 Produce N stochastic variables {ξ1 , ξ2 , . . . , ξn } from Φ(ξ); Step 2 Set fi = f (x, ξi ), i = 1, 2, . . . , N ; Step 3 Set N as the integer part of βN ; Step 4 Return to the N th minimum element located in sequence {f1 , f2 , . . . , fN }. 5.4
The process of initiation
First, to determine every activity’s normal implementation period, the compressed period and the probability distribution of each activity period are determined. To satisfy the project period constraint, the critical path must be determined. Since the activity period and compression decrement are random in PERT net, the critical path is not fixed. As a chromosome, vector V = (v1 , v2 , . . . , vk ) represents a path from node 1 to n. The algorithm that converts a path from node 1 to n into a chromosome can be given by Step 1 Set l = 0, v0 = 1; Step 2 able; Step 3 Step 4 Step 5
Find suffix m that makes γvl ,m = 1 tenl ← l + 1, and vl = m; Repeat Step 2 and Step 3 until vl = n; Get a chromosome V = (v1 , v2 , . . . , vl−1 ).
Then, initiate PERT feasible chromosomes by the heuristic algorithm. Step 1 Set population size: pop size, set N = 0; Step 2 Set l = 0, v0 = 1; Step 3 Random select a suffix m satisfying (vl , m) ∈ L, where L is the set of net paths mentioned above; Step 4 Step 5 Step 6
l ← l + 1, and vl = m; Repeat Step 2 and Step 4 till vl = n; Get a chromosome V = (v1 , v2 , . . . , vl−1 );
Repeat Step 2 and Step 3 N times alto-
Step 7 N = N + 1, repeat Steps 2 to 7 till N = pop size;
If N /N α, return to “tenable”, else
Step 8 Get V1 , V2 , . . . , Vpop size , total N = pop size chromosomes.
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Suppose that the current generation chromosomes are V1 , V2 , . . . , Vpop size ; a sequence relationship for them is given. The more optimum the objective value of the chromosome, the smaller is the sequence number of the chromosome, which makes the chromosomes reorder from the superior to the inferior; the ones ordered in front are more competitive than the ones behind. Define the evaluation function based on sequence as eval(Vi ) = a(1 − a)i−1 , i = 1, 2, . . . , pop size, where, a is called the fitness coefficient, and a ∈ (0, 1). When i=1, the chromosome V1 is the most optimum, and when i = pop size, the chromosome Vpop size is the most inferior.
chromosomes by stochastic simulation; Step 3 Perform crossover and mutation operation to chromosomes, and verify the feasibility of these offspring chromosomes by stochastic simulation; Step 4 Calculate all chromosomes’ objective value; Step 5 According to the objective values, compute every chromosome’s fitness by the evaluation function based on sequence; Step 6 Rotate the roulette wheel, select chromosomes; Step 7 Repeat Step 3 to Step 6 until the given loop times are complete. Step 8 Select the best chromosome as the optimum solution.
5.6
6. Example and analysis
5.5
Evaluation function
Genetic operators
The process of chromosome selection adopts the roulette wheel strategy. Single point crossover operator and single point mutation operator are used for genetic operation. The single point crossover is to generate a cut-point randomly; the parts before the point and the parts after the point of the two parents are exchanged to generate an offspring chromosome. The single point mutation is to select a bit randomly in parent chromosome, then change its genic value from 1 to n on the premise that it must satisfy the PERT net route range, namely, (vl , m) ∈ L. If the mutation chromosome is not satisfied with the condition, then it is given up, and the new offspring is generated again by changing random bit until the condition is satisfied. 5.7
GA program based on stochastic simulation
Step 1 Input parameters: the population size: pop size, crossover probability Pc , and mutation probability Pm ; Step 2 Generate pop size chromosomes by heuristic algorithm, and verify the feasibility of these
Fig.2
A network of ERP project implementation in an enterprise is shown in Fig.2. The dashed direct arc denotes dummy activity, which only denotes the sequence constraint and does not spend time and cost, and does not occupy resources, namely, xij = 0, Cij = 0. Table 1 offers the parameters of each activity’s period, cost, and quality and their probability distributions. Suppose the weights in the time-cost-qualitycombined optimization model are w1 = 0.35, w2 = 0.3, w3 = 0.35. The GA based on stochastic simulation is realized by programming with VC++6.0. The concrete parameters are pop size = 40, Pc = 0.40, Pm = 0.15, the fitness coefficient a = 0.05, and the repetition time is 1 000. The concerned parameters and data are then input into the program. The given believable degree levels are αT = 0.90, αT = 0.95, βT = 0.90, βC = 0.90, βQ = 0.90. By program operation, the critical path in the network is determined as 1→2→5→3→6→7→8→9→ 10→13→14→15. The optimized calculation results on time, cost, and quality corresponding to different optimization models, which are under the different
ERP implementation schedule network
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Wang Shaojun, Wang Gang, L¨ u Min & Gao Guoan Table 1
Activity
Activity name
code
ERP schedule network activity parameters
Former
Activity time(day)
activity
Dij
dij
Probability distribution
Activity cost( 10 000) Cij
δij
hij
Probability distribution
Activity quality N qij
C qij
Probability distribution
1
System analysis
0
60
40
u (40, 80)
15
2
0.2
u (20, 1)
1
0.55
u (0.4, 1)
2
System installment
1
14
10
N (15, 1)
50
1
0.1
N (50, 2)
1
0.9
N (1, 0.2)
3
High level cultivation
1
10
7
N (8, 1)
1.5
0.2
0.1
N (1.5, 0.2)
1
0.85
T (0.4,1,0.8)
4
BPR
1
30
20
u (15, 40)
10
2
0.05
u (8, 12)
1
0.6
u (0.5, 1)
5
——
4
0
0
——
0
0
0
——
0
0
——
6
Software installment
2, 5
5
3
T (3, 4, 5)
0.6
0.5
0.08
N (0.5,0.01)
1
0.96
N (0.9,0.05)
7
Information coding
2, 5
40
30
u (20, 50)
10
1.5
0.2
u (8, 12)
1
0.45
u (0.2, 1)
8
Data preparation
7
30
20
N (30, 3)
8
1.2
0.1
N (8,0.1)
1
0.62
T (0.1,1,0.7)
9
Prototype test
6, 8
20
10
N (15, 2)
5
1.2
0.2
N (5, 0.5)
1
0.5
T (0.3,1,0.6)
10
Implementation training
3
14
10
N (13, 1)
1.5
0.5
0.08
N (1.6, 0.2)
1
0.75
N (0.8,0.1)
11
Function simulation
9, 10
7
5
N (5, 1)
1.2
1.3
0.15
u (1, 1.5)
1
0.65
T (0.4,1,0.7)
12
Sys. manager training
9, 10
10
7
T (5,15,10)
1
0.5
0.1
N (1, 0.1)
1
0.8
T (0.4,1,0.8)
13
Customization setting
11
15
10
T (10,15,12)
2
1
0.1
u (1, 2.5)
1
0.7
u (0.5, 1)
14
Making process standard
11
30
20
N (25, 3)
2.5
1
0.08
N (3, 0.2)
1
0.8
N (0.8,0.05)
15
Data transition & test
13
30
25
u (20, 40)
10
2
0.3
u (8, 15)
1
0.4
u (0.1, 1)
16
——
14
0
0
——
0
0
0
——
0
0
——
17
Final user training
12
15
10
T (10,18,15)
1
0.3
0.1
N (1, 0.1)
1
0.8
N (0.8,0.1)
18
System transformation
15,16,17
6
3
u (5, 10)
8
2.5
1.5
T (5, 10, 9)
1
0.68
u (0.2, 1)
19
Project check & accept
18
6
5
T (5,10,8)
2
2
0.3
T (1, 4, 2)
1
0.85
N (1, 0.1)
time and cost constraints, are given in Table 2. The Table 2
success and risk probabilities are also given.
The calculation results of the optimization models
Optimization
Time limit
Budget
Time
Implementation
Quality
Success
Risk
models
constraint (Day)
constraint( 10 000)
(Day)
cost( 10 000)
probability
probability
Schedule
240
——
171.6
——
——
0.863
0.137
optimization model
200
——
159.1
——
——
0.552
0.448
Time-cost trade
240
130
189.3
110.63
——
0.891
0.109
off model
200
100
160.7
98.79
——
0.457
0.525
T-C-Q combined
240
130
199.2
119.57
0.81
0.910
0.09
tradeoff model
200
100
187.4
105.02
0.67
0.668
0.332
It can be seen from the optimization results in Table 2 that when only the time constraint is considered and the implementation schedule optimization model based on the critical route is adopted, the optimized period result is the shortest, and the op timized cost is the lowest when the time-cost tradeoff model is adopted, and the implementation risk is the lest when the time-cost-quality-combined tradeoff optimization model is adopted. In addition, when the project schedule and budget cost are compressed, the project
implementation risks increase obviously. It can be seen from the quality optimum values that the phase of data transition & test and system analysis are the critical steps to the whole system quality, and therefore, these phases should be given considerable importance. Thus, when the project schedule and budget cost need to be adjusted, the adjustment to be made are the activities’ implementation time, cost, and timecost slopes according to the risk probability and the decision variables’ optimum. By adjusting the enter-
Enterprise resource planning implementation decision & optimization models prise resource constraints and implementation schedule, enhancing the implementation ability of ERP supplier and improving the implementation circumstance, the objectives of minimum time, cost, and maximum quality can be achieved.
7. Conclusion
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terprise resource planning: Implementation procedures and critical success factors. European Journal of Operational Research, 2003, 146(2): 241–257. [2] Xue Yajiong, Liang Huaigang, William R Boulton, et al. ERP implementation failures in China: Case studies with implications for ERP vendor.
International Journal of
Production Economics, 2005, 97: 279–295.
Since there are several kinds of uncertainties in the ERP implementation process, the ERP implementation time, cost, and quality are stochastic variables, which coincide with some probability distributions. It is important for ERP project executors to consider sufficiently the stochastic properties of time, cost, quality, and implementation risks in ERP implementation programming. This article proposes the implementation schedule optimization model, time-cost trade-off model, quality optimization model, and time-cost-quality combined tradeoff optimization model, which provide effective methods to solve uncertain programming problems on the process of project implementation, and the solutions by timecost-quality combined tradeoff optimization model have the smaller implementation risk probability. The PERT-embedded genetic algorithm based on stochastic simulation can effectively solve the problems of ERP project stochastic programming models solution. The schedule optimization model based on critical path proposed in this article is more coincident with the characters of project management, and its optimization results accord with the practice considerably well. Finally, the example’s solution illuminates that the models and algorithms are reasonable and effective, and the models and algorithms offer support in theory and practice to ERP project management.
[3] Jaideep Motwani, Ram Subramanian, Pradeep Gopalakrishna. Critical factors for successful ERP implementation: Exploratory findings from four case studies. Computers in Industry, 2005, 6: 529–544. [4] Babu A J G, Suresh N. Project management with time, cost, and quality considerations. European Journal of Operational Research, 1996, 88(2): 320–327. [5] Liu Baoding, Zhao Ruiqing, Wang Gang. Uncertain programming with applications.
Beijing: Tsinghua Univ.
Press, 2003 (in Chinese). [6] Feng Yuncheng. Activity network analysis. Beijing: Beijing University of Aeronautics & Astronautics Press, 1991 (in Chinese). [7] Feng Chunwei, Liu Liang, Burns S A. Stochastic construction time-cost trade-off analysis. Journal of Computing in Civil Engineering, 2000, 14(2): 117–126. [8] Michalewicz J Z. Genetic Algorithms + Data Structures = Evolution Programs. 3rd ed. Berlin: Springer-Verlag, 1996.
Wang Shaojun was born in 1971. He now is the Ph. D. candidate in Harbin Institute of Technology. His major research interests are industry engineering, enterprise modeling, CIMS and decision support systems.
References
Wang Gang was born in 1964. He is a professor in Harbin Institute of Technology. His main research
[1] Umble Elisabeth J, Haft Ronald R, Michael Umble M. En-
interests are enterprise modeling, workflow, CIMS etc.
Enterprise resource planning implementation decision & optimization models∗ Wang Shaojun, Wang Gang, L¨ u Min & Gao Guoan Advanced Manufacturing Technology Center, Harbin Inst. of Technology, Harbin 150001, P. R. China (Received July 13, 2007)
Abstract: To study the uncertain optimization problems on implementation schedule, time-cost trade-off and quality in enterprise resource planning (ERP) implementation, combined with program evaluation and review technique (PERT), some optimization models are proposed, which include the implementation schedule model, the timecost trade-off model, the quality model, and the implementation time-cost-quality synthetic optimization model. A PERT-embedded genetic algorithm (GA) based on stochastic simulation technique is introduced to the optimization models solution. Finally, an example is presented to show that the models and algorithm are reasonable and effective, which can offer a reliable quantitative decision method for ERP implementation.
Keywords: optimization model, ERP, chance-constrained programming, PERT, genetic algorithm, time, cost, quality.
1. Introduction ERP is becoming increasingly the necessary integrated information platform in enterprise business management. Implementing ERP can bring huge benefits to enterprises, at the same time, the enterprises also face huge risks on the process of ERP implementation. These risks include decision risk, implementation risk, and application risk[1] . A successful ERP project depends on the ERP software, which occupies 30% in significance, but owes 70% to correct implementation. The process of project implementation is one of the key steps to ERP success. The project executors will meet some uncertain factors such as techniques, human resource, enterprise environment, and so on, which will lead to the project implementation time, cost, and quality uncertainty. It was investigated by some literatures that during the process of ERP implementation, the implementation schedules of 90% of all enterprises were over the limited time, the implementation cost of 80% of all enterprises was over the project budget, and 60%–70% of all enterprises did not complete system integration[2]. The universal problems were that the implementation qualities were uncertain and short of effective project manage-
ment. Since the ERP project is a system engineering, the ERP project needs explicit objective, scientific programming, and strict project management[3] . Therefore, when the project starts, it is necessary to evaluate and analyze the implementation schedule proposed by ERP supplier or consulting company. To reach the goals of reduction of implementation risks and assurance of quality, it is important to optimize the implementation schedule and cost. PERT was used to deal with the problems that the logic relationships of activities were certain but their durations were uncertain in the project management. Thus, PERT can be used in project evaluation and programming now. To solve the uncertainty problem of time, cost, and quality in the process of implementation, combined with the chance constrained programming based on stochastic simulation (Monte Carlo emulation), several optimization models are established in this article, which give enterprise some methods to make decision in ERP implementation.
2. Decision problem description and hypothesis & preconditions 2.1 Decision problem description The process of ERP project optimization and deci-
* This project was supported by the National High-Tech. R & D Program for CIMS, China (2003AA413210).
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sion is shown in Fig.1.
Fig.1
The logic decision structure on ERP implementation optimization
The implementation time, cost, quality, and risks are the considerably important factors in making an ERP implementation project. The time, cost, and quality are the basic critical success factors, but they always conflict each other, and it is impossible to optimize the three factors simultaneously[4] . Enhancing quality makes investment increase or schedule prolong, whereas compressing schedule will lead projects to increase the direct implementation cost; on the contrary, controlling the cost to the lowest and compressing schedule to the shortest will lead the quality to lower, thereby causing project risks in cost, time, and quality. Therefore, ERP project should be feasible and coincident with the mutual constrained conditions. The executors always hope that the implementation time will be shortest, the cost will be under the budget, and the quality implementation result will achieve the expected level. In short, the problem can be described as ⎧ ⎪ ⎪ min T ime ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ min Cost ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ max Quality ⎪ ⎪ ⎨ s.t. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T ime TP ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Cost CP ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Quality QE where TP , CP denote the project time limit and bud-
get, respectively, and QE denotes the expected project quality. 2.2
Hypothesis & preconditions of models
This article proposes the following basic hypothesis & preconditions based on the classical PERT net model. (1) As the existence of uncertain factors, implementation time, cost, and quality are stochastic variables under some kinds of probability distributions. (2) The resources for every activity are limited. (3) The direct activity cost is inversely proportional to the activity time, namely, the longer the continued time, the higher is the direct cost; the indirect activity cost is directly proportional to the activity time, the longer the continued time, the higher is the indirect cost. (4) Under certain implementation conditions, during the necessary period of activity time, activity quality is directly proportional to activity time; the longer the continued time, the better will be the implementation quality. The activity quality is also directly proportional to the activity cost; the higher the cost, the better will be the quality.
3. Chance constrained programming ERP implementation optimization is to optimize the objective function in uncertain circumstance. Chance constrained programming mainly deals with such situation that constraint conditions contain stochastic variables, and decision must be made before the
Enterprise resource planning implementation decision & optimization models results of the stochastic variables’ realization are observed[5]. To consider the decision may do not coincide with the constrain conditions sometimes when the situation is not good, so allowing the decision making do not coincide with the constraint conditions in certain extent, but the probability that the decision makes constraint conditions tenable must not be less than a believable degree α. The chance constrained programming model can be expressed as follows. ⎧ ⎪ min f ⎪ ⎪ ⎪ ⎪ ⎨ s.t. ⎪ ⎪ Pr f (x, ξ) f β ⎪ ⎪ ⎪ ⎩ Pr gj (x, ξ) f α where x is an n dimensions variable, ζ is a stochastic variable, f (x, ξ) is an objective function, gj (x, ξ) is a stochastic constrained function, j = 1, 2, . . . , p, Pr{·} denotes the probability that makes event {·} come into being, α is the believable degree of constraint conditions decided beforehand, β is the believable degree of objective functions, f is the minimum of objective functions when the believable degree is not less than β, and f = min{f | Pr{f (x, ξ) f } β}.
4. ERP implementation optimization models 4.1 ERP implementation schedule expression The whole ERP implementation process can be decomposed to limited activities (sub projects), which form a PERT net according to the time logic sequence. Let arrow (or directed arc) denote activity, and let node denote the beginning or the end of an activity (or event). It is used to be illustrated by Active-onArrow method (A-O-A)[6] . To construct an A-O-A net, the following rules should be observed. (1) Before an activity begins, the adjacent former activities must be complete. (2) Each activity can only connect two events, and there is only one activity between two events. (3) There is only one original node and one final node in a net; they represent the beginning and end of the project, respectively. (4) Feedback circles and loops are not allowed in the net. We can regard the PERT net as a directed loopless
515
graph; the node in the graph represents an event or decision time, and the directed arc represents activity. Let G = (V, A) denote a directed loopless graph, and V = {1, 2, . . . , n} be the set of nodes; A denotes the set of arcs, and aij ∈ A denotes an arc from node i to node j in G. The order among nodes is denoted by an adjacent matrix M , M = (γij )n×n , and ⎧ ⎨ 1, {i} ≺ {j} . {i} ≺ {j} denotes node γij = ⎩ 0, {i} not ≺ {j} {i} borders on node {j}, and {i} is the adjacent former node in activity (i, j), while {j} is the adjacent subsequent node in activity (i, j). 4.2
Implementation schedule optimization modelbased on critical path
Suppose the net is composed of k activities in all. The critical path is defined as the route with the longest implementation period. Let L = {γij = 1|aij ∈ A, i = 1, 2, . . . , n − 1; j = 2, 3, . . . , n} denote a set of paths from the beginning note 1 to the final note n. Let stochastic variable xij denote the duration of activity (i, j), namely, the length of arc aij . The total time of a random path from beginning γij xij . node {1} to final node {n} is TL (xij ) = aij ∈L
Thus, the total time of the critical path is denoted as TK (xij ) = max TL (xij ) = max γij xij . Let aij ∈L
stochastic variable xi denote the net node time, and x1 denote the earliest beginning time of the project, and xn denote the earliest finish time. In the condition only considering the implementation schedule, the implementation schedule optimization model based on the critical path is established as follows. ⎧ ⎪ min T (x ) = min max γij xij K ij ⎪ ⎪ ⎪ ⎪ aij ∈L ⎪ ⎪ ⎪ ⎪ ⎪ s.t. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ xj − xi xij ⎪ ⎪ ⎪ ⎪ ⎨ x −x T n 1 p
(1) ⎪ ⎪ Pr max γij xij Tp βT ⎪ ⎪ ⎪ ⎪ aij ∈L ⎪ ⎪ ⎪ ⎪ ⎪ Pr {d xij Dij } αT ij ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i = 1, 2, . . . , n − 1; j = 2, 3, . . . , n ⎪ ⎪ ⎪ ⎩ ∀aij ∈ A γij ∈ {0, 1},
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where TP denotes the expected implementation period of the project, Dij denotes the normal implementation period of the activity (i, j), dij denotes the minimum compressed period of the activity (i, j). Pr{·} denotes the probability that the event in{·} is tenable, αT is the believable degree of constraint conditions decided beforehand, and βT is the believable degree of objective functions. The constrain, xj − xi xij , reflects the sequence constraint among activities. 4.3
Implementation time-cost optimization model
The ERP implementation costs mainly include direct cost and indirect cost. The basically direct cost comes from the human resource fee and equipment cost, while the indirect cost mainly comes from the routine management expenditures. Suppose the direct cost has linear progressive decrease relationship with the implementation schedule in the range xij ∈[dij , Dij ], namely, when the activity time is compressed, the direct cost decreases. The indirect cost has linear progressive increase relationship with the implementation schedule, namely, with the activity time increase, the indirect cost also increases. In some cases, the relationship that obtains implementation time decrease by increasing cost is called the time-cost tradeoff[6,7] . The time-cost trade-off model is an optimization model on minimizing the implementation cost under the project implementation period constraint conditions. Let δij denote the time-cost slope of activity (i, j), and the stochastic variable V xij denote the period decrement of activity (i, j) by compression. The time-cost function of direct cost increment can be described as ⎧ ⎪ +∞, V xij Dij − dij ⎪ ⎨ gi (V xij ) = δij V xij , 0 V xij Dij − dij ⎪ ⎪ ⎩ 0, other i = 1, 2, . . . , n − 1; j = 2, 3, . . . , n The decrement of indirect cost is hij V xij when the implementation period of activity (i, j) is compressed, and hij is the indirect cost per unit time (per day). Let CP denote the project budget, and the stochastic variable Cij denote the normal implementation cost of activity (i, j). Suppose the total number of activities is k. In the situation that the schedule compression
is considered, the total project implementation cost function is f (Cij , V xij ), and f (Cij , V xij ) =
k
(Cij + δij V xij − hij V xij ) =
aij ∈A k
[Cij + (δij − hij )V xij ]
aij ∈A
Thus, the project implementation time-cost chance constrained programming model can be established as ⎧ ⎪ ⎪ ⎪ min f (Cij , V xij ) ⎪ ⎪ ⎪ ⎪ ⎪ s.t. ⎪ ⎪ ⎪ ⎪ ⎪ 0 V xij Dij − dij ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k
⎪ ⎪ ⎨ Pr [Cij + (δij − hij )V xij ] CP βC aij ∈A ⎪ ⎪ ⎪
⎪ ⎪ ⎪ Pr max γij (xij − V xij ) Tp αT ⎪ ⎪ ⎪ ⎪ aij ∈L ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i = 1, 2, . . . , n − 1; j = 2, 3, . . . , n ⎪ ⎪ ⎪ ⎪ ⎩ γij ∈ {0, 1}, ∀aij ∈ A (2) 4.4
Implementation quality optimization model
On the premise that the ERP supplier and implementation circumstance are determined, according to the hypothesis mentioned above, the activity quality is directly proportional to the activity time and cost. Since the activity cost is a function of time, the activity quality can be built as a function of time. For xij , the duration of activity (i, j), is a stochastic variable, and the quality of activity (i, j) is also a stochastic variable. The function on the quality of activity (i, j) can be built as follows. ⎧ ⎪ 0, xij < dij ⎪ ⎨ C Qij (xij ) = qij + ηij xij , dij xij Dij ⎪ ⎪ ⎩ N qij , xij Dij i = 1, 2, . . . , n − 1; j = 2, 3, . . . , n N C − qij qij C , where, qij is the minimum qualDij − dij ity when the implementation period of activity (i, j) N is compressed to the limitation, and qij is the normal quality in the normal implementation period of
and ηij =
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activity (i, j). The function Qij (xij ) shows that the activity is impossible to be completed when the implementation period is less than the compression limitation, and Qij = 0; on the other hand, when the implementation period is over the normal period, the activity quality maintains stability and without increasing with the implementation period extension. As the whole project quality is determined by the minimum quality of the weakest activity, the quality function of the project is Q(xij ) = min Qij (xij ), i = 1, 2, . . . , n − 1; j = 2, 3, . . . , n. The chance constrained programming model on project implementation quality is established as follows ⎧ ⎪ max Q(xij ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s.t. ⎪ ⎨ (3) Pr{min Qij (xij ) QE } βQ ⎪ ⎪ ⎪ ⎪ ⎪ Pr{dij xij Dij } αT ⎪ ⎪ ⎪ ⎩ i = 1, 2, . . . , n − 1; j = 2, 3, . . . n
established as follows: ⎧ ⎪ min F (xij , V xij , Cij ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s.t. ⎪ ⎪
⎪ ⎪ ⎪ ⎪ γ x T βT Pr max ⎪ ij ij p ⎪ ⎪ ⎪ aij ∈L ⎪ ⎪ ⎪ ⎪ k
⎪ ⎪ ⎪ ⎪ Pr [Cij + (δij − hij )V xij ] CP βC ⎪ ⎪ ⎨ aij ∈A
where QE denotes the expected activity quality.
The project managers also expect to evaluate the success or risk probability in advance before the project is complete. According to the stochastic programming optimum value obtained from the above optimization models, the success or risk probability can be obtained through stochastic simulation (Monte Carlo emulation). Let ET denote the event
max γij xij Tp ; let EC denote the event
4.5
Implementation time-cost-quality combined tradeoff optimization model
In practice project implementation, the executors always consider the implementation schedule, cost, and quality together, and expect to achieve the shortest time, the lowest cost, and the best quality. The problem belongs to multi-objective optimization. However, the three factors constrain each other, and to solve the conflict, the usual method is to adopt the tradeoff model to transform multi-objective functions into a single-objective function by adding weight to each objective function. F (xij , V xij , Cij ) = w1 TK (xij ) + w2 f (Cij , V xij ) − w3 Q(xij ) = w1 max
aij ∈L
γij xij + w2
k aij ∈A
[Cij + (δij − hij )V xij ] − w3 min Qij (xij ) Under the constraints of implementation schedule, budget cost, and expected quality, the chance constrained time-cost-quality combined tradeoff model is
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
Pr{min Qij (xij ) QE } βQ Pr{dij xij Dij } αT
Pr max γij (xij − V xij ) Tp αT aij ∈L
i = 1, 2, . . . , n − 1; γij ∈ {0, 1},
j = 2, 3, . . . , n
∀aij ∈ A
(4) In the model shown above, w1 , w2 , w3 denote the weight of implementation schedule, cost, and quality, respectively, and w1 + w2 + w3 = 1. 4.6
The probability evaluation on success and risk
k
aij ∈L
[Cij + (δij − hij )V xij ] CP ; let EQ denote
aij ∈A
the event {min Qij QE }. The probability of success can be described as Pr{success} = Pr{ET ∩ EC ∩ EQ } = Pr{EQ |ET ∩ EC } · Pr{EC |ET } · Pr{ET }. The equation illuminates that the project can be regarded as successful only when the schedule is within the time limit of project, the cost is under the budget, and the implementation results get the expected quality. Thus, the correspondent implementation risk can be described as Pr{risk} = 1 − Pr{success}.
5. Models solution The conventional chance programming solution is to transform chance constraints into corresponding cer-
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tain equivalent classes according to the given believable degree in advance, and then solve the certain equivalent class model with classical methods[5] . However, it is impossible to find the corresponding equivalent classes for several complex chance constraint programmings. Fortunately, the appearance of the genetic algorithm makes it possible for us to solve this kind of model easily. The genetic algorithm (GA) is a method that searches for the optimization solution by simulating the nature evolution process. GA has the following characteristics: all fields search ability, good robust, easy to extend, and easy to program; besides, it does not demand that the objective function must be continuous and derivationable[8] . It is these properties that make GA suitable to solve the optimization problem of PERT nets stochastic programming. Aiming at the models established above, a PERT-embedded genetic algorithm based on stochastic simulation is proposed to solve these models. 5.1
The method of chromosome coding
To adapt the solution vectors, the float vector is adopted to represent a chromosome. The length of the chromosome is not fixed and it is the same length as the corresponding solution vector. The vector X = (x1 , x2 , . . . , xn ) denotes the solution of optimization, and n is the dimension number; thus, the chromosome is expressed as V = (v1 , v2 , . . . , vn ). 5.2
Verifying stochastic system constraint
Consider chance constraint Pr{gj (x, ξ) 0, j = 1, 2, . . . , k} α. ξ is the stochastic variable, for which the probability density distribution function is Φ(ξ). For a given decision variable x, the stochastic simulation (Monte Carlo emulation) is applied to verify if the constraint is tenable. The methods producing the stochastic number are shown in Ref. [7]. The test steps are as follows. Step 1 Step 2 probability Step 3 N + +; Step 4 gether; Step 5
Set N = 0; Produce a stochastic variable ξ from the distribution function Φ(ξ); If gj (x, ξ) 0, j = 1, 2, . . . , k, then,
return to “untenable”. 5.3
Objective value calculation
Consider the objective function Pr{f (x, ξ) f } β with stochastic parameters; the following algorithm can find the minimum f that satisfies the conditions above: Step 1 Produce N stochastic variables {ξ1 , ξ2 , . . . , ξn } from Φ(ξ); Step 2 Set fi = f (x, ξi ), i = 1, 2, . . . , N ; Step 3 Set N as the integer part of βN ; Step 4 Return to the N th minimum element located in sequence {f1 , f2 , . . . , fN }. 5.4
The process of initiation
First, to determine every activity’s normal implementation period, the compressed period and the probability distribution of each activity period are determined. To satisfy the project period constraint, the critical path must be determined. Since the activity period and compression decrement are random in PERT net, the critical path is not fixed. As a chromosome, vector V = (v1 , v2 , . . . , vk ) represents a path from node 1 to n. The algorithm that converts a path from node 1 to n into a chromosome can be given by Step 1 Set l = 0, v0 = 1; Step 2 able; Step 3 Step 4 Step 5
Find suffix m that makes γvl ,m = 1 tenl ← l + 1, and vl = m; Repeat Step 2 and Step 3 until vl = n; Get a chromosome V = (v1 , v2 , . . . , vl−1 ).
Then, initiate PERT feasible chromosomes by the heuristic algorithm. Step 1 Set population size: pop size, set N = 0; Step 2 Set l = 0, v0 = 1; Step 3 Random select a suffix m satisfying (vl , m) ∈ L, where L is the set of net paths mentioned above; Step 4 Step 5 Step 6
l ← l + 1, and vl = m; Repeat Step 2 and Step 4 till vl = n; Get a chromosome V = (v1 , v2 , . . . , vl−1 );
Repeat Step 2 and Step 3 N times alto-
Step 7 N = N + 1, repeat Steps 2 to 7 till N = pop size;
If N /N α, return to “tenable”, else
Step 8 Get V1 , V2 , . . . , Vpop size , total N = pop size chromosomes.
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Suppose that the current generation chromosomes are V1 , V2 , . . . , Vpop size ; a sequence relationship for them is given. The more optimum the objective value of the chromosome, the smaller is the sequence number of the chromosome, which makes the chromosomes reorder from the superior to the inferior; the ones ordered in front are more competitive than the ones behind. Define the evaluation function based on sequence as eval(Vi ) = a(1 − a)i−1 , i = 1, 2, . . . , pop size, where, a is called the fitness coefficient, and a ∈ (0, 1). When i=1, the chromosome V1 is the most optimum, and when i = pop size, the chromosome Vpop size is the most inferior.
chromosomes by stochastic simulation; Step 3 Perform crossover and mutation operation to chromosomes, and verify the feasibility of these offspring chromosomes by stochastic simulation; Step 4 Calculate all chromosomes’ objective value; Step 5 According to the objective values, compute every chromosome’s fitness by the evaluation function based on sequence; Step 6 Rotate the roulette wheel, select chromosomes; Step 7 Repeat Step 3 to Step 6 until the given loop times are complete. Step 8 Select the best chromosome as the optimum solution.
5.6
6. Example and analysis
5.5
Evaluation function
Genetic operators
The process of chromosome selection adopts the roulette wheel strategy. Single point crossover operator and single point mutation operator are used for genetic operation. The single point crossover is to generate a cut-point randomly; the parts before the point and the parts after the point of the two parents are exchanged to generate an offspring chromosome. The single point mutation is to select a bit randomly in parent chromosome, then change its genic value from 1 to n on the premise that it must satisfy the PERT net route range, namely, (vl , m) ∈ L. If the mutation chromosome is not satisfied with the condition, then it is given up, and the new offspring is generated again by changing random bit until the condition is satisfied. 5.7
GA program based on stochastic simulation
Step 1 Input parameters: the population size: pop size, crossover probability Pc , and mutation probability Pm ; Step 2 Generate pop size chromosomes by heuristic algorithm, and verify the feasibility of these
Fig.2
A network of ERP project implementation in an enterprise is shown in Fig.2. The dashed direct arc denotes dummy activity, which only denotes the sequence constraint and does not spend time and cost, and does not occupy resources, namely, xij = 0, Cij = 0. Table 1 offers the parameters of each activity’s period, cost, and quality and their probability distributions. Suppose the weights in the time-cost-qualitycombined optimization model are w1 = 0.35, w2 = 0.3, w3 = 0.35. The GA based on stochastic simulation is realized by programming with VC++6.0. The concrete parameters are pop size = 40, Pc = 0.40, Pm = 0.15, the fitness coefficient a = 0.05, and the repetition time is 1 000. The concerned parameters and data are then input into the program. The given believable degree levels are αT = 0.90, αT = 0.95, βT = 0.90, βC = 0.90, βQ = 0.90. By program operation, the critical path in the network is determined as 1→2→5→3→6→7→8→9→ 10→13→14→15. The optimized calculation results on time, cost, and quality corresponding to different optimization models, which are under the different
ERP implementation schedule network
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Wang Shaojun, Wang Gang, L¨ u Min & Gao Guoan Table 1
Activity
Activity name
code
ERP schedule network activity parameters
Former
Activity time(day)
activity
Dij
dij
Probability distribution
Activity cost( 10 000) Cij
δij
hij
Probability distribution
Activity quality N qij
C qij
Probability distribution
1
System analysis
0
60
40
u (40, 80)
15
2
0.2
u (20, 1)
1
0.55
u (0.4, 1)
2
System installment
1
14
10
N (15, 1)
50
1
0.1
N (50, 2)
1
0.9
N (1, 0.2)
3
High level cultivation
1
10
7
N (8, 1)
1.5
0.2
0.1
N (1.5, 0.2)
1
0.85
T (0.4,1,0.8)
4
BPR
1
30
20
u (15, 40)
10
2
0.05
u (8, 12)
1
0.6
u (0.5, 1)
5
——
4
0
0
——
0
0
0
——
0
0
——
6
Software installment
2, 5
5
3
T (3, 4, 5)
0.6
0.5
0.08
N (0.5,0.01)
1
0.96
N (0.9,0.05)
7
Information coding
2, 5
40
30
u (20, 50)
10
1.5
0.2
u (8, 12)
1
0.45
u (0.2, 1)
8
Data preparation
7
30
20
N (30, 3)
8
1.2
0.1
N (8,0.1)
1
0.62
T (0.1,1,0.7)
9
Prototype test
6, 8
20
10
N (15, 2)
5
1.2
0.2
N (5, 0.5)
1
0.5
T (0.3,1,0.6)
10
Implementation training
3
14
10
N (13, 1)
1.5
0.5
0.08
N (1.6, 0.2)
1
0.75
N (0.8,0.1)
11
Function simulation
9, 10
7
5
N (5, 1)
1.2
1.3
0.15
u (1, 1.5)
1
0.65
T (0.4,1,0.7)
12
Sys. manager training
9, 10
10
7
T (5,15,10)
1
0.5
0.1
N (1, 0.1)
1
0.8
T (0.4,1,0.8)
13
Customization setting
11
15
10
T (10,15,12)
2
1
0.1
u (1, 2.5)
1
0.7
u (0.5, 1)
14
Making process standard
11
30
20
N (25, 3)
2.5
1
0.08
N (3, 0.2)
1
0.8
N (0.8,0.05)
15
Data transition & test
13
30
25
u (20, 40)
10
2
0.3
u (8, 15)
1
0.4
u (0.1, 1)
16
——
14
0
0
——
0
0
0
——
0
0
——
17
Final user training
12
15
10
T (10,18,15)
1
0.3
0.1
N (1, 0.1)
1
0.8
N (0.8,0.1)
18
System transformation
15,16,17
6
3
u (5, 10)
8
2.5
1.5
T (5, 10, 9)
1
0.68
u (0.2, 1)
19
Project check & accept
18
6
5
T (5,10,8)
2
2
0.3
T (1, 4, 2)
1
0.85
N (1, 0.1)
time and cost constraints, are given in Table 2. The Table 2
success and risk probabilities are also given.
The calculation results of the optimization models
Optimization
Time limit
Budget
Time
Implementation
Quality
Success
Risk
models
constraint (Day)
constraint( 10 000)
(Day)
cost( 10 000)
probability
probability
Schedule
240
——
171.6
——
——
0.863
0.137
optimization model
200
——
159.1
——
——
0.552
0.448
Time-cost trade
240
130
189.3
110.63
——
0.891
0.109
off model
200
100
160.7
98.79
——
0.457
0.525
T-C-Q combined
240
130
199.2
119.57
0.81
0.910
0.09
tradeoff model
200
100
187.4
105.02
0.67
0.668
0.332
It can be seen from the optimization results in Table 2 that when only the time constraint is considered and the implementation schedule optimization model based on the critical route is adopted, the optimized period result is the shortest, and the op timized cost is the lowest when the time-cost tradeoff model is adopted, and the implementation risk is the lest when the time-cost-quality-combined tradeoff optimization model is adopted. In addition, when the project schedule and budget cost are compressed, the project
implementation risks increase obviously. It can be seen from the quality optimum values that the phase of data transition & test and system analysis are the critical steps to the whole system quality, and therefore, these phases should be given considerable importance. Thus, when the project schedule and budget cost need to be adjusted, the adjustment to be made are the activities’ implementation time, cost, and timecost slopes according to the risk probability and the decision variables’ optimum. By adjusting the enter-
Enterprise resource planning implementation decision & optimization models prise resource constraints and implementation schedule, enhancing the implementation ability of ERP supplier and improving the implementation circumstance, the objectives of minimum time, cost, and maximum quality can be achieved.
7. Conclusion
521
terprise resource planning: Implementation procedures and critical success factors. European Journal of Operational Research, 2003, 146(2): 241–257. [2] Xue Yajiong, Liang Huaigang, William R Boulton, et al. ERP implementation failures in China: Case studies with implications for ERP vendor.
International Journal of
Production Economics, 2005, 97: 279–295.
Since there are several kinds of uncertainties in the ERP implementation process, the ERP implementation time, cost, and quality are stochastic variables, which coincide with some probability distributions. It is important for ERP project executors to consider sufficiently the stochastic properties of time, cost, quality, and implementation risks in ERP implementation programming. This article proposes the implementation schedule optimization model, time-cost trade-off model, quality optimization model, and time-cost-quality combined tradeoff optimization model, which provide effective methods to solve uncertain programming problems on the process of project implementation, and the solutions by timecost-quality combined tradeoff optimization model have the smaller implementation risk probability. The PERT-embedded genetic algorithm based on stochastic simulation can effectively solve the problems of ERP project stochastic programming models solution. The schedule optimization model based on critical path proposed in this article is more coincident with the characters of project management, and its optimization results accord with the practice considerably well. Finally, the example’s solution illuminates that the models and algorithms are reasonable and effective, and the models and algorithms offer support in theory and practice to ERP project management.
[3] Jaideep Motwani, Ram Subramanian, Pradeep Gopalakrishna. Critical factors for successful ERP implementation: Exploratory findings from four case studies. Computers in Industry, 2005, 6: 529–544. [4] Babu A J G, Suresh N. Project management with time, cost, and quality considerations. European Journal of Operational Research, 1996, 88(2): 320–327. [5] Liu Baoding, Zhao Ruiqing, Wang Gang. Uncertain programming with applications.
Beijing: Tsinghua Univ.
Press, 2003 (in Chinese). [6] Feng Yuncheng. Activity network analysis. Beijing: Beijing University of Aeronautics & Astronautics Press, 1991 (in Chinese). [7] Feng Chunwei, Liu Liang, Burns S A. Stochastic construction time-cost trade-off analysis. Journal of Computing in Civil Engineering, 2000, 14(2): 117–126. [8] Michalewicz J Z. Genetic Algorithms + Data Structures = Evolution Programs. 3rd ed. Berlin: Springer-Verlag, 1996.
Wang Shaojun was born in 1971. He now is the Ph. D. candidate in Harbin Institute of Technology. His major research interests are industry engineering, enterprise modeling, CIMS and decision support systems.
References
Wang Gang was born in 1964. He is a professor in Harbin Institute of Technology. His main research
[1] Umble Elisabeth J, Haft Ronald R, Michael Umble M. En-
interests are enterprise modeling, workflow, CIMS etc.