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A mathematical model for the management of a Service Center
Mathematical and Computer Modelling 53 (2011) 2005–2014
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Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
A mathematical model for the management of a Service Center A. Marasco ∗ , A. Romano Department of Mathematics and Applications ‘‘R. Caccioppoli’’ University of Naples Federico II, Via Cintia, 80126, Naples, Italy
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Article history: Received 10 December 2009 Received in revised form 20 January 2011 Accepted 23 January 2011 Keywords: Project scheduling Planning Hard deadline Evolutive model
abstract In this paper we propose a mathematical model to manage a Service Center (SC ) which is based on a system of ordinary differential equations. By resorting to this model, the manager of the SC can design planning strategies to satisfy customer orders, under strict deadlines and human resource constraints. After describing the model, we introduce criteria which optimize the processing time and supply a more accurate description of working behavior. Finally, we conclude presenting some numerical simulations which demonstrate the usefulness of the proposed model to reach correct decisions in managing a SC. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction This paper originates from a research project1 funded both to formulate a mathematical model and to implement a software for the management of a Service Center (SC ). The problem we faced can be sketched as follows: the manager of a SC must decide which customer orders with strict deadlines can be chosen in a set of potential orders; then, he must determine a schedule for all the selected orders in such a way that all the orders are dealt with before their due dates. The set of orders (jobs) is a priori partitioned into a set of job families {JF1 , JF2 , . . . , JFm } so that all the jobs of each family has the same deadline, although the arrival time of any job could be different.2 For processing the jobs in the due times, the manager can resort to a set of heterogeneous human resources, to which he could decide to add part time workers or hired extra resource units. This problem belongs to the dynamic framework in which inevitable and unpredictable real-time events may cause a change in the schedule plans. For instance, these events could refer to the resources (i.e. ill operators, unavailability or tool failures, etc.), and to the jobs (i.e. cancellation of an order, early or delayed arrival of jobs, a change in job priority, a change in job processing time, etc.). The model we propose has the purpose of guiding the manager of the SC to choose among the potential orders in such a way as to answer the following questions:
• Let us suppose that a given number of orders have already been accepted by the SC. When the different deadlines and the workload are taken into account, is it possible to determine the date in which they will be completed?
• Is it possible to know if a further order can be accepted when the workload is considered? • If the processing times are not compatible with the deadlines of the different orders, how must the schedule plans be modified?
∗
Corresponding author. E-mail addresses: [email protected] (A. Marasco), [email protected] (A. Romano).
1 This project was funded by the Italian Ministry of Economic Development, and it refers to ‘‘Misura 2.1.a—PON Nazionale Pia Innovazione I bando Anno 2003’’, for the SC Creditalia s.a.s. 2 Since in our case, the time list of arrivals is unknown or given with a probability distribution, we decided to call this list arrival time list instead of release time list. 0895-7177/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2011.01.032
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• If the ordinary conditions, which allow the correct behavior of the already accepted orders in the due times, are modified by unexpected events (for instance the unavailability of resources due to ill operators or shifts of operators to other jobs), what is the resulting delay? How could it be reset? • For a given set of activities to be developed and a workforce formed by different skills, which workers or resources should be assigned to the different activities in order to meet all the deadlines? • If the scarcity of the resources produces bottlenecks and consequent delays in the production cycle, is it possible to adopt suitable strategies to meet the deadlines? The above considerations suggest that the problem can naturally be formulated as a scheduling problem. In fact, we must find for each job the best possible schedule such that all the above constraints are met and a suitable given objective function is optimized (i.e. the total costs are minimized, the maximal lateness of the jobs are minimized, the total net profit is maximized, etc.). Due to the rich variety of different problems within this research field, scheduling is one of the most widely inter-disciplinary research areas (Mathematics, Operations Research, Information Technology and Computer Science, Manufacturing, Management, Business, Engineering, etc.). That is proved by more than 30,000 publications, referring to ‘‘scheduling’’ since the 1990s, that can be found in Web of Science. This situation makes an exhaustive state-of-the-art review on the scheduling problem very difficult. For instance, the recent review papers are addressed to a determined class of scheduling problems (see, for instance, [1] on scheduling problems arising in production industries, [2] on scheduling problems with setup considerations, [3] on dynamic scheduling in manufacturing systems, [4] on multicriteria scheduling, [5] on scheduling with learning effects, and so on). We remark that our problem can be classified as a workforce scheduling problem, for which the numerous approaches existing in literature use many different objective functions and constraints, as well as a wide variety of mathematical models (see [6–13] and reference therein). Moreover, it is well known that almost all the scheduling problems are typically NP-hard, i.e., so far there does not exist any reliable numerical method able to find an optimal solution within a reasonable time. This difficulty pushed researchers to develop heuristic algorithms or approximation procedures (see [1,2,14,15]). Since the manager’s decision of accepting or rejecting an order must be taken in real-time, we were asked to propose a mathematical model to be written in finite arithmetic by using an algorithm compatible with the database and the tool of the Work Flow Management (WFM), already employed by SC.3 In order to solve our problem in real-time, we propose an evolutive mathematical model which evaluates a feasible not optimal scheduling for the job families. In particular, it ensures that all the deadlines are met, provided that the potential customer orders with their characteristics (such as process route, processing times, deadline, etc.), and the job shop configuration (such as resources, their availability, etc.) are assigned. This model is based on a system of ordinary differential equations (ODEs) which, in the first approximation, can analytically be solved. Further, for these equations a numerical procedure is implemented to answer in real-time all the above managerial questions. We conclude by noting that resorting to a mathematical model of a feasible not optimal scheduling does not always represent a real simplification of the problem. For instance, in the case in which to meet the deadline of a particular order requires an increasing of the number of the people involved in the work, it would appear quite spontaneous the choice leading to a positive balance between costs and profits. However, the manager could decide to accept this new order as a business strategy (important customer, prestigious order, etc.), even if there is not a total net profit. In these situations, the choice to adopt is devolved upon the manager since it could be not convenient in pursuing a prefixed objective. The paper is organized into five sections including the introduction. In Section 2 we describe the proposed mathematical model, whereas in Section 3 we present suitable criteria to resect the dead times. Section 4 is devoted to some numerical i ,j simulations relative to the model which is based on the first choice of the involvement coefficients ah defined in Section 3.1. Finally, Section 5 deals with the research perspectives concerning suitable development of the model in a classical framework of combinatorial optimization problems. 2. A mathematical model Before discussing the mathematical model, we describe the fundamental quantities characterizing both the structure of SC and the activities of any job. A set of job families {JF1 , JF2 , . . . , JFm } collect all types of possible orders, and with each job ȷih ∈ JFi , for i = 1, . . . , m and h = 1, . . . , |JFi |, a deadline di and an arrival time thi are associated. In our problem each job belonging to the family JFi consists of a set of ni sequential activities Ai,1 , Ai,2 , . . . , Ai,ni :
Activity Ai,1 Activity Ai,2 Job family JFi ≡ . . . Activity Ai,ni −1 Activity Ai,ni . 3 In other words, the SC wished that the implementation of our model became an upgrade of the software Mentore 5.0 already used by the SC.
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We remark that the same elementary activity can appear in processes which belong to different job families. Moreover, the number ni of the elementary activities depends on the typology JFi . The SC workforce is made up of p workers W1 , . . . , Wp mastering one or several skills, i.e. each of them may be assigned to one or more activities. Then, in searching for a schedule of the activities, we have to determine which resources are the most appropriate to process each activity in each period. Let TiA,j be the absolute time needed to perform the elementary activity Ai,j . This time is evaluated supposing that: (1) there is no obstacle during the development of the activity; (2) the workers related to this activity realize the best performance. Moreover, it is fundamental to assign once for ever the unit time. In our analysis, the time unit is assumed to be 8 h (8 h). If the activity Ai,j has a standard duration of 3 h, then we have that TiA,j = 3/8. Finally, let Ni,j (0) be the initial number of the jobs referring to the activity Ai,j . In other words, the quantities Ni,j (0) represent the workload in any activity Ai,j at the beginning of the simulation. All the above quantities are the input data of the new WFM system implementing the mathematical model we are introducing. Let Ni,j = Ni,j (t ) be the job number at the instant t of the activity Ai,j . Then, we assume that for each Ni,j (t ) the following balance equation holds: dNi,j dt
= σi,j−1 (t , N, 9) − σi,j (t , N, 9),
i = 1, . . . , m, j = 1, . . . , ni ;
(1)
where N ≡ Nh,k , h = 1, . . . , m, k = 1, . . . , nh , 9 denotes a set of suitable parameters we list below, σi,j−1 (t , N, 9) is the number of jobs belonging to the family JFi and that at the instant t come from activity Ai,j−1 and go to Ai,j in the unit time; σi,j (t , N, 9) is the number of jobs in the job family JFi that at the instant t overcome the step Ai,j and arrive at Ai,j+1 in the unit time. Shortly, σi,j (t , N, 9) denotes the flux of jobs sent from Ai,j to Ai,j+1 in the unit time. We remark that:
• for j = 0 and i arbitrary, σi,0 (t ) denotes the number of jobs in the family JFi which are accepted by the SC at the instant t; consequently, σi,0 (t ) is an input datum only depending on t. It is evident that the constraint consisting in meeting all the deadlines could compel the manager not to accept other jobs from a certain instant Tiin on, so that σi,0 (t ) = 0, for t ≥ Tiin , provided that the business strategy does not suggest other choices (see Section 1); • when σi,j−1 (t , N, 9) − σi,j (t , N, 9) > 0, the jobs relative to Ai,j are accumulating and Ni,j increases. System (1) allows us to evaluate Ni,j (t ) when the net flux of jobs σi,j−1 (t , N, 9)−σi,j (t , N, 9) is given. Therefore, to complete the mathematical model, we must assign the constitutive equation of the flux σi,j , i.e., we have to express this quantity as a function of t , N, and the parameters 9 that take into account
• • • • •
the structural parameters of the SC ; the absolute time TiA,j ; the efficiency level of the operators working on the activity Ai,j ; the total or partial involvement of the worker in the activity Ai,j ; some corrective parameters which take into account delay factors modifying the absolute time, which is an ideal time.
In order to determine the constitutive relation σi,j (t , N, 9), we start supposing that pi,j employers are working on the activity i ,j
Ai,j . To each employer Wh we associate a bounded function ah (t , N, 9), with h = 1, . . . , pi,j i,j
0 ≤ ah (t , N, 9) ≤ 1,
(2)
which denotes the involvement level of the employer Wh in activity Ai,j . Clearly, the involvement coefficients of an employer Wh working in different activities verify the following condition
−
i ,j
ah (t , N, 9) = 1.
i,j
In Section 3, we give some suggestions to model these coefficients. i,j In view of the meaning of the involvement coefficients ah (t , N, 9), the function: i ,j
ri,j (t , N, 9) = a1 (t , N, 9) + · · · + aip,ij,j (t , N, 9)
(3)
will denote the effective number of workers involved in the performance of the activity Ai,j . For instance, suppose that the involvement coefficients are constants and chosen by the manager on the bases of the structure of the SC. Then, if an operator performs only the activity Ai,j for all his working time, then pi,j = 1 and ri,j = 1. i,j
Differently, if two workers are assigned to the activity Ai,j for 3/4 and 1/2 of their working time, then pi,j = 2, a1 = 3/4, i,j
a2 = 1/2 and ri,j = 5/4. Finally, if a single worker spends 1/2 of his working time for the activity Ai,j and the remaining i,j
h ,k
half for the activity Ah,k , then it results a1 = a1
= 1/2.
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The above considerations lead us to adopt the following constitutive equation of σi,j (t , N, 9):
σi,j (t , N, 9) = ri,j (t , N, 9) σi,0 (t , N, 9) = σi,0 (t ),
1 TiA,j
,
i = 1, . . . , m, j = 1, . . . , ni ,
(4)
i = 1, . . . , m.
(5) i,j
We can conclude that the model is complete when the functions ah (t , N, 9) are given. In fact, if the involvement coefficients are assigned, then the relation (3) provides the quantities ri,j (t , N, 9), and finally σi,j (t , N, 9) are determined by (4)–(5). In conclusion, adding the initial data Ni,j (0) to the system (1), and recalling (3) and (4)–(5), we face with the following Cauchy problem
1 dNi,1 dt = σi,0 (t ) − ri,1 (t , N, 9) T A , i ,1 dNi,j
i = 1, . . . , m,
1
1
= ri,j−1 (t , N, 9) A − ri,j (t , N, 9) A , dt Ti,j−1 Ti,j Ni,j (0) = Ni0,j , i = 1, . . . , m, j = 1, . . . , ni .
(6)
i = 1, . . . , m, j = 2, . . . , ni ,
The integration of (6) supplies the functions Ni,j (t ) whose analysis allows us to verify whether the objectives of SC will be realized or not. When the results of this analysis show that the expectations will be not satisfied, the manager must elaborate i ,j a new strategy modifying the involvement coefficients ah (t , N, 9) until the objectives are reached. i,j
In the next section we show that, in choosing the involvement coefficients ah (t , N, 9), the manager must satisfy a fundamental criterion which improves the work planning resecting the dead times in performing all the jobs. Moreover, we are looking for a schedule which is able to accommodate disruption without exceeding the deadlines set by the customer’s orders. i,j
3. Two criteria to choose the coefficients ah i ,j
3.1. First choice: ah are constant and their assignment is based on resecting the dead times i,j
Let us suppose that the manager, on the basis of its experience, assigns the constant values to the coefficients ah and, consequently, to the quantities ri,j . In this case, the Cauchy problem (6) can be analytically integrated since it becomes
1 dNi,1 = σi,0 (t ) − ri,1 A , dt Ti,1 1
dNi,j
i = 1 , . . . , m,
1
= ri,j−1 A − ri,j A , i = 1, . . . , m, j = 2, . . . , ni , dt Ti,j−1 Ti,j Ni,j (0) = Ni0,j , i = 1, . . . , m, j = 1, . . . , ni .
(7)
Consequently, it is easy to verify if all the deadlines are met. However, even if a lucky choice allows us to perform all the jobs before the due times, it could be not convenient if the work were carried out much in advance with respect to the deadlines. We now introduce a criterion to resect the dead times. This criterion also guides the manager in the assignment i,j of the coefficients ah when his initial choice does not permit him to reach the objective. In other words, we use the solution of (7) to determine the conditions that the involvement coefficients must satisfy so that (1) the deadlines are met; j
(2) the instants Ti to complete the activity Ai,j verify the condition Ti1 ≤ · · · ≤ Tim (absence of dead times). To explain the idea underlying this criterion, we resort to the following hydraulic model. Let us consider a system formed by n reservoirs Aj , placed side by side to each other. Any reservoir has an entrance filler and an exit filler. The exit filler of Aj is also the entrance filler of the reservoir Aj+1 . We suppose that, at the instant t , Aj contains Nj (t ) liters of water and a device which discharges towards the adjacent reservoir a constant number σj of liters in the unit time. Moreover, Aj receives σj−1 liters from Aj−1 , provided that Aj−1 is not empty. We denote by t = 0 the arbitrary instant in which we start our analysis of the working process. We suppose that:
• at the instant t = 0 each reservoir Aj is empty or contains a known amount of water; • during time interval [0, T in ], a certain number of liters of water are introduced in the first reservoir A1 . Starting from the above data, we evaluate
• the constant exit water fluxes in such a way that the reservoirs expel their water content subsequently in time; • the instant T f > T in in which all the reservoirs are empty.
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Let σ0 (t ) be a function such that σ0 (t ) ≥ 0 in the time interval t ∈ [0, T in ], and identically vanishing for t ≥ T in . This function describes the time evolution of water flux entering the first reservoir A1 until the instant T in . Moreover, let N10 , . . . , Nn0 be the liters of water contained in the n reservoirs at the initial time. Finally, T in
∫
N in =
σ0 (t )dt ,
(8)
0
is the total number of liters entering A1 in the time interval [0, T in ], whereas N in
σM =
(9)
T in
is the entering mean flux. The number N1 (t ) of liters of A1 is the solution of the following Cauchy problem (see (7))
dN1
= σ0 (t ) − σ1 , dt N1 (0) = N10 ,
t ∈ [0, T in ]
(10)
so that it is given by the relation N1 (t ) =
t
∫
σ0 (t )dt − σ1 t + N10 ,
t ∈ [0, T in ].
(11)
0
From (11), when (8) and (9) are taken into account, it follows that, at the instant T in , A1 contains N1 (T in ) = (σM − σ1 )T in + N10
(12)
liters. For time values greater than T in , the function N1 (t ) is the solution of the following Cauchy problem:
dN1
= −σ1 , t ∈ [T in , T 1 ] dt N1 (T in ) = (σM − σ1 )T in + N10 ,
(13)
where T 1 is the instant in which N1 (T 1 ) = 0, i.e., A1 is empty. The solution of (13) has the form N1 (t ) = −σ1 t + σM T in + N10 ,
t ∈ [T in , T 1 ]
(14)
1
and consequently T is given by the relation T1 =
σM in 1 0 T + N , σ1 σ1 1
(15)
so that
σ1 =
r1 σM T in + N10 ≡ 1
1 T1
(16)
TA
where TA1 is defined as the absolute emptying time of the reservoir A1 . We note that T in ≤ T 1 ⇒ σ1 ≤ σM +
N10
(17)
T in
and the first reservoir becomes empty at the instant T in , when the input of water in the first reservoir ends provided that (17) is satisfied.4
4 In fact, from (15) we obtain T 1 − T in =
σM 1 − 1 T in + N10 σ1 σ1
and then T 1 − T in ≥ 0 ⇐⇒ σM − σ1 ≥ −
N10 T in
⇐⇒ σ1 ≤ σM +
N10 T in
.
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The behavior of the function N2 (t ), which gives the number of liters contained in the second reservoir A2 , is a solution of the following Cauchy problem
dN2
= σ1 − σ2 ,
t ∈ [0, T 1 ]
dt N2 (0) = N20 .
(18)
Therefore, we have that N2 (t ) = (σ1 − σ2 )t + N20 ,
t ∈ [0, T 1 ]
(19)
up to the instant T 1 in which the first reservoir is empty. In this instant the value of N2 is N2 (T 1 ) = (σ1 − σ2 )T 1 + N20 .
(20)
From T 1 on, the behavior of N2 (t ) is the solution of the following other Cauchy problem:
dN2
= −σ2 , t ∈ [T 1 , T 2 ] dt N2 (T 1 ) = (σ1 − σ2 )T 1 + N20 ,
(21)
where T 2 is the instant in which N2 (T 2 ) = 0 and the second reservoir is empty. The solution of (21) is N2 (t ) = −σ2 t + σ1 T 1 + N20 ,
t ∈ [T 1 , T 2 ]
(22)
2
and the instant T is expressed by the relation T2 =
σ1 1 1 0 T + N . σ2 σ2 2
(23)
Then, we have
σ2 =
1 T2
r2 σ1 T 1 + N20 ≡ 2 ,
(24)
TA
where TA2 is the absolute emptying time of the reservoir A2 . If we want that the second reservoir becomes empty after the first one has became empty, i.e., if T 1 ≤ T 2,
(25)
then the following condition has to be verified
σ2 ≤ σ1 +
N20 T1
.
(26)
The relations (23) and (24), in view of (15), can also be written as it follows:
σM in 1 T + (N 0 + N20 ), σ2 σ2 1 1 r2 σ2 = 2 σM T in + N10 + N20 ≡ 2 . T2 =
T
TA
(27) (28)
Following the same reasoning for the other reservoirs, we obtain the emptying time of the j-th reservoir: Tj =
j σM in 1 − 0 T + N , σj σj h=1 h
(29)
and the j-th constant flux
σj =
1 Tj
σM T + in
j − h =1
Nh0
≡
rj j
TA
,
(30)
j
where TA is the absolute emptying time of the reservoir Aj . This time is less than the emptying time of the previous reservoir if
σj ≤ σj−1 +
Nj0 T j −1
,
j = 2, . . . , n.
(31)
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Now, to come back to our problem, it will be sufficient to identify the reservoir Aj with the activity Ai,j ∈ JFi , for all i, the j
number Nj (t ) of liters with the number of customer jobs Ni,j (t ), the fluxes σj with σi,j , TA with the absolute time TiA,j , n with f
ni , and T f with the instant Ti in which all the jobs belonging to JFi are completed. Therefore, when we assign all the input data σ
,
in i,M Ti
,
Ni0,j
,
TiA,j
,
i ,j ah ,
in view of (30), the inequalities (31) supplies the conditions which allow us to verify the absence f
of dead times. Moreover, owing to the relation (29) for j = n, it is also determined the instant Ti in which all the orders in the job family JFi are fulfilled, so that if f
Ti ≤ di ,
∀i
(32)
all the deadlines di are met. The model based on this strategy, can lead to the following possibilities: (1) If (31) and (32) are verified, then the model supplies a scheduling which satisfies all the constraints imposed to the jobs. However, if this scheduling is not optimum, it could require an improvement. (2) It could happen that one or more conditions (31) and (32) are not verified. In this case, it becomes necessary to define the input data in a convenient way. In this latter case, further strategies can be adopted to reach the fixed objectives, depending on the company policy and the current situation. For instance, if it is not possible to modify the data σi,M , Tiin since the orders have already been accepted, i,j
then the model should give the new values of the involvement coefficients ah which allow to deal with all the orders in the i,j ah
due times. In other cases, the coefficients could be determined on the basis of the real working capacity, so that it is not possible to improve them. In this case, the model should give information about the parameters σi,M , Tiin in order to meet i,j
the deadlines. In the absence of information about the values of the coefficients ah , or when inevitable and unpredictable real-time events cause a change in the schedule plans (i.e. ill operators, cancellation of an order, a change in job priority, etc.), the model should proceed to a new evaluation of these parameters. In the next section, we propose some simple numeric simulations based on the above choice of the involvement coefficients. We recall that this model has been implemented in the upgrade of the WFM software Mentore 10. i ,j
3.2. Second choice: ah are not constant, but depending on N, and TA i,j
A more flexible model can be obtained by improving the structure of the coefficients ap as follows. Let us suppose that a worker Wp is involved in three different activities Ai,j , Ah,k , and Ar ,s each of them requiring the absolute times TiA,j , ThA,k and TrA,s , respectively. The involvement coefficients of this worker related to the different activities can be defined by the following relations api,j (N, TA ) = aph,k (N, TA ) = apr ,s (N, TA ) =
Ni,j TiA,j Ni,j TiA,j
+ Nh,k ThA,k + Nr ,s TrA,s
,
Nh,k ThA,k Ni,j TiA,j + Nh,k ThA,k + Nr ,s TrA,s Nl,m TrA,s Ni,j TiA,j + Nh,k ThA,k + Nr ,s TrA,s
,
(33)
,
where TA ≡ TAu,v , u = 1, . . . , m, v = 1, . . . , nu . These formulae can be improved by multiplying them by a factor αpu,v , 0 < αpu,v ≤ 1, which takes into account the efficiency level and the reliability of the worker Wp involved in the activity Au,v . Let us suppose that the workers W1 , . . . , Wpi,j are involved in the activity Ai,j . Applying the above formulae to each worker and introducing the resulting expressions into (3), we obtain the new quantities ri,j . By recalling (33), the Cauchy problem (6) becomes
dNi,1 1 = σi,0 (t ) − ri,1 (N, TA ) A , dt Ti,1 dNi,j
1
i = 1, . . . , m, 1
= ri,j−1 (N, TA ) A − ri,j (N, TA ) A , i = 1, . . . , m, j = 2, . . . , ni , dt Ti,j−1 Ti,j 0 Ni,j (0) = Ni,j , i = 1, . . . , m, j = 1, . . . , ni .
(34)
It is important to remark that, since the system (34) is nonlinear and nonautonomous, its solution could not be expressed in a closed form. However, its numerical integration is a standard computational problem so that, also in this case, it is possible to numerically verify if the conditions (31) and (32) are satisfied. We conclude remarking that, although this last criterion supplies a more sophisticated mathematical model, it could be less convenient from a computational point of view. For this reason we have implemented the model presented in Section 3.1 in the software Mentore 10.
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4. Numerical simulation In order to show the several possibilities of management offered by the model, in this section we present some numerical simulations based on the approach presented in Section 3.1. 4.1. Simulation 1 We suppose that SC ’s manager decide to accept Nhin = 100 orders, with the same deadline dh = 90, in the time interval [0, 30]. These orders refer to the job family JFh characterized by the data listed in the following table (see Section 2)
Activity Ah,1 : Nh,1 (0) = 40, Activity Ah,2 : Nh,2 (0) = 30, Activity Ah,3 : Nh,3 (0) = 70, JFh ≡ Activity Ah,4 : Nh,4 (0) = 10, Activity Ah,5 : Nh,5 (0) = 20, Activity Ah,6 : Nh,6 (0) = 45,
ThA,1 ThA,2 ThA,3 ThA,4 ThA,5 ThA,6
= 4; = 6; = 3; = 6; = 6; = 5. f
In this simulation we determine the coefficients rh,1 , . . . , rh,6 and the instant Th in which all the orders in JFh are fulfilled, in such a way to satisfy the constraints (31) and (32), and to perform the activities by available and appropriate resources. First, we remark that, in view of relation (16), all the values rh,1 such that rh,1 ≤ 18.6667 allow us to satisfy (17). Similarly, when the value of the coefficient rh,1 is given, Eq. (24) supplies the upper bound of rh,2 for which the constraint (26) is verified. It is evident how we can go on (see relations (31)). We note that the values of the coefficients rh,i satisfying relation (31) f
allow us to determine uniquely the instant Th in which all the orders in the job family JFh are fulfilled. Therefore, we are f
interested in a set of coefficients rh,i such that Th ≤ 90. Performing some simulations using Mathematica we obtain the f
outputs listed in the following tables. We note that in all simulations the end time Th = 87.5 is the same. (1) In the first simulation we have that Activity Ah,1 : rh,1 = 16, Activity Ah,2 : rh,2 = 15,
Activity Ah,3 : rh,3 = 10,
Activity Ah,4 : rh,4 = 18,
Activity Ah,5 : rh,5 = 19, Activity Ah,6 : rh,6 = 18,
f
Th,1 = 35;
f Th,2 f Th,3 f Th,4 f Th,5 f Th,6
= 68; = 72; = 83.3333; = 85.2632; = 87.5.
We suppose that the above distribution is compatible with the available resources. In this case, the manager can dispose all the workers employed in the first activity after 35 work days, workers employed in the second activity after 68 work days and so on. All the orders will be fulfilled before the deadline. (2) We suppose that the above distribution cannot be adopted by the manager since the workforce allows the following maximum values rh,2 = 14, rh,3 = 9. Introducing this restriction we obtain the following other table Activity Ah,1 : rh,1 = 16,
Activity Ah,2 : rh,2 = 14,
Activity Ah,3 : rh,3 = 9,
Activity Ah,4 : rh,4 = 18, Activity Ah,5 : rh,5 = 19,
Activity Ah,6 : rh,6 = 18,
f
Th,1 = 35; f
Th,2 = 72.8571;
f Th,3 f Th,4 f Th,5 f Th,6
= 80; = 83.3333; = 85.2632; = 87.5. f
In this case, the work ends at the instant Th = 87.5, but the workers employed in the activities Ah,2 and Ah,3 will be f f again at the disposal of the manager after the instants Th,2 = 72.8571 and Th,3 = 80, respectively. (3) When some workers, which usually are devoted to the activity Ah,1 , are not available at the initial instant, the above table is modified as follows f
Activity Ah,1 : rh,1 = 10, Th,1 = 56; f
Activity Ah,2 : rh,2 = 14, Th,2 = 72.8571; f
Activity Ah,3 : rh,3 = 9, Th,3 = 80; f
Activity Ah,4 : rh,4 = 18, Th,4 = 83.3333; f
Activity Ah,5 : rh,5 = 19, Th,5 = 85.2632; f
Activity Ah,6 : rh,6 = 18, Th,6 = 87.5.
A. Marasco, A. Romano / Mathematical and Computer Modelling 53 (2011) 2005–2014
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f
Once again the work is completed at instant Th = 87.5, but the workers of activity Ah,1 become available only after 56 work days. These examples show that the model is able to propose many possible scheduling which meet manager’s requirements. In the last example, the manager could decide to increase the number of workers devoted to the first activity hiring 4 full-time workers in a period of 35 days. In this way, the internal staff of the SC will become free before 56 days. We remark that, in any proposed scheduling, the cost of any activity is not modified provided that the salary of each operator remains constant. In fact, in any case, we have
f rh,i Th,i
=
ThA,i
Nhin
+
i −
Nh,k (0)
≡ const,
k =1
for any h and i. 4.2. Simulation 2 Suppose that the SC ’s manager decides to accept in a month Nkin = 150 orders with deadline dk = 90. All the orders belong to the family JFk which is composed by nk = 15 sequential activities characterized by the following absolute times TAk = {4, 6, 2, 6, 7, 5, 4, 6, 2, 6, 9, 5, 4, 6, 3}. Moreover, let Nk (0) = {10, 30, 15, 10, 25, 23, 20, 30, 15, 10, 8, 15, 10, 20, 22} be the corresponding workload initial set. f In order to determine the coefficients rk,1 , . . . , rk,15 and the instant Th such that all the constrains (31) and (32) are verified, we remark that the upper bound of rk,1 is 21.3333. About this case, we propose the following simulations. (1) We assume that the maximum value for any coefficient rk,i is 35. Performing a numerical simulation with the following values for the coefficients rk,i rk = {21.3, 35, 12.58, 35, 35, 27, 23.2, 35, 12.2, 35, 35, 20.2, 16.6, 26.2, 13.8}, we obtain the following end times f
Tk = {30.0469, 32.5714, 32.5914, 36.8571, 48, 48.7037, 48.7931, 53.6571, 53.7705, 57.9429, 88.9714, 89.3564, 89.3976, 89.542, 89.7826}, for which the constrains (31) and (32) are satisfied. (2) We suppose that SC has a set of operators for which the maximum value of any coefficient rk,i is 25. However we f
modify the input parameters, taking into account the constraints (31), we always obtain that Tk > 90. For instance, the numerical simulation with the following values for the coefficients rk,i rk = {21.3, 25, 8.9, 25, 25, 19.5, 16.5, 25, 8.7, 25, 25, 14.4, 11.8, 18.6, 9.8},
(35)
leads us to the following end times f
Tk = {30.0469, 45.6, 46.0674, 51.6, 67.2, 67.4359, 68.6061, 75.12, 75.4023, 81.12, 124.56, 125.347, 125.763, 126.129, 126.429}.
(36)
We observe that for the activities Ak,3 , Ak,6 , Ak,7 , Ak,9 , Ak,12 , Ak,13 , Ak,14 , Ak,15 the values of the coefficients rk,i are chosen in such a way that the corresponding constrains (31) are verified (see the underlined values in (35)). Moreover, in (36) f it is shown that from any Ak,j , with j > 11, it results Tk,12 > 90. In order to meet the deadline, the manager could decide to accept a smaller number of orders. All the simulations show that only for Nkin = 50 the deadlines are met. In fact, if we assume that Nkin = 50,
rk = {8, 17.5, 6.8, 22, 25, 20.7, 18.5, 25, 8.9, 25, 25, 14.7, 12.2, 19.65, 10.56},
we obtain f
Tk = {30, 30.8571, 30.8824, 31.3636, 39.2, 39.372, 39.5676, 51.12, 51.236, 57.12, 88.56, 88.7755, 88.8525, 88.855, 88.9205}. We conclude remarking that when Nk (0) = 0 for the following input data Nin k = 150,
rk = {20, 25, 8, 24, 25, 17.8, 14, 20, 6.5, 19, 25, 13.5, 10.5, 15.5, 7.5},
we obtain f
Tk = {30, 36., 37.5, 37.5, 42, 42.1348, 42.8571, 45, 46.1538, 47.3684, 54, 55.5556, 57.1429, 58.0645, 60} and the deadline is met in a period of 60 days.
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5. Conclusions and research perspectives This paper provides a detailed description of a model to manage SC which is based on a system of ordinary differential equations. Numerical simulations show that the proposed model offer several possibilities of management in very short running time. As future work, we plan to extend our model to an even more realistic problem that will reflect real life scenarios including several constraints in addition to those considered in this paper. Moreover, we plan also to formulate the problem as combinatorial optimization problem (see [14]). Special emphasis will be given to a possible mathematical formulation of the problem as scheduling problem and to the design of efficient mathematical programming methods in order to compare the two approaches. In our opinion, many topics of interest remain open for further research in this area. First of all, a hybrid approach could be designed that will include the main ingredients of the two pure methods. Secondly, since most of these problems are computationally intractable, besides the study of exact methods, a further fruitful research line would be towards the design of approximation algorithms and metaheuristics. References [1] C.N. Potts, V.A. Strusevich, Fifty years of scheduling: a survey of milestones, Journal of the Operational Research Society 60 (2009) S41–S68. [2] A. Allahverdi, C.T. Ng, T.C.E. Cheng, M.Y. Kovalyov, A survey of scheduling problems with setup times or costs, European Journal of Operational Research 187 (2008) 985–1032. [3] D. Ouelhadj, S. Petrovic, A survey of dynamic scheduling in manufacturing systems, Journal of Scheduling 12 (2009) 417–431. [4] H. Hoogeveen, Multicriteria scheduling, European Journal of Operational Research 167 (2005) 592–623. [5] D. Biskup, A state-of-the-art review on scheduling with learning effects, European Journal of Operational Research 188 (2008) 315–329. [6] H.K. Alfares, Survey, categorization, and comparison of recent tour scheduling literature, Annals of Operations Research 127 (2004) 145–175. [7] P. Cowling, N. Colledge, K. Dahal, S. Remde, The trade off between diversity and quality for multi-objective workforce scheduling, evolutionary computation in combinatorial optimization, Lecture Notes in Computer Science 3906 (2006) 13–24. [8] O. Lambrechts, E. Demeulemeester, W. Herroelen, Proactive and reactive strategies for resource-constrained project scheduling with uncertain resource availabilities, Journal of Scheduling 11 (2008) 121–136. [9] L.-E. Drezeta, J.-C. Billauta, A project scheduling problem with labour constraints and time-dependent activities requirements, International Journal of Production Economics 112 (2008) 217–225. [10] T.A. Guldemond, J.L. Hurink, J.J. Paulus, J.M.J. Schutten, Time-constrained project scheduling, Journal of Scheduling 11 (2008) 137–148. [11] V. Tiwari, J.H. Patterson, V.A. Mabert, Scheduling projects with heterogeneous resources to meet time and quality objectives, European Journal of Operational Research 193 (2009) 780–790. [12] V. Valls, A. Pérez, S. Quintanilla, Skilled workforce scheduling in service centres, European Journal of Operational Research 193 (2009) 791–804. [13] C.-S. Chen, S. Mestry, P. Damodarana, C. Wang, The capacity planning problem in make-to-order enterprises, Mathematical and Computer Modelling 50 (2009) 1461–1473. [14] P. Festa, R. De Leone, E. Marchitto, A new meta-heuristic for the bus driver scheduling problem: GRASP combined with rollout, in: IEEE on Computational Intelligence in Scheduling, 2007. doi:10.1109/SCIS.2007.367689. [15] R. De Leone, P. Festa, E. Marchitto, A bus driver scheduling problem: a new mathematical model and a GRASP approximate solution, Journal of Heuristics (2011) doi:10.1007/s10732-010-9141-3.
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Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
A mathematical model for the management of a Service Center A. Marasco ∗ , A. Romano Department of Mathematics and Applications ‘‘R. Caccioppoli’’ University of Naples Federico II, Via Cintia, 80126, Naples, Italy
article
info
Article history: Received 10 December 2009 Received in revised form 20 January 2011 Accepted 23 January 2011 Keywords: Project scheduling Planning Hard deadline Evolutive model
abstract In this paper we propose a mathematical model to manage a Service Center (SC ) which is based on a system of ordinary differential equations. By resorting to this model, the manager of the SC can design planning strategies to satisfy customer orders, under strict deadlines and human resource constraints. After describing the model, we introduce criteria which optimize the processing time and supply a more accurate description of working behavior. Finally, we conclude presenting some numerical simulations which demonstrate the usefulness of the proposed model to reach correct decisions in managing a SC. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction This paper originates from a research project1 funded both to formulate a mathematical model and to implement a software for the management of a Service Center (SC ). The problem we faced can be sketched as follows: the manager of a SC must decide which customer orders with strict deadlines can be chosen in a set of potential orders; then, he must determine a schedule for all the selected orders in such a way that all the orders are dealt with before their due dates. The set of orders (jobs) is a priori partitioned into a set of job families {JF1 , JF2 , . . . , JFm } so that all the jobs of each family has the same deadline, although the arrival time of any job could be different.2 For processing the jobs in the due times, the manager can resort to a set of heterogeneous human resources, to which he could decide to add part time workers or hired extra resource units. This problem belongs to the dynamic framework in which inevitable and unpredictable real-time events may cause a change in the schedule plans. For instance, these events could refer to the resources (i.e. ill operators, unavailability or tool failures, etc.), and to the jobs (i.e. cancellation of an order, early or delayed arrival of jobs, a change in job priority, a change in job processing time, etc.). The model we propose has the purpose of guiding the manager of the SC to choose among the potential orders in such a way as to answer the following questions:
• Let us suppose that a given number of orders have already been accepted by the SC. When the different deadlines and the workload are taken into account, is it possible to determine the date in which they will be completed?
• Is it possible to know if a further order can be accepted when the workload is considered? • If the processing times are not compatible with the deadlines of the different orders, how must the schedule plans be modified?
∗
Corresponding author. E-mail addresses: [email protected] (A. Marasco), [email protected] (A. Romano).
1 This project was funded by the Italian Ministry of Economic Development, and it refers to ‘‘Misura 2.1.a—PON Nazionale Pia Innovazione I bando Anno 2003’’, for the SC Creditalia s.a.s. 2 Since in our case, the time list of arrivals is unknown or given with a probability distribution, we decided to call this list arrival time list instead of release time list. 0895-7177/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2011.01.032
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• If the ordinary conditions, which allow the correct behavior of the already accepted orders in the due times, are modified by unexpected events (for instance the unavailability of resources due to ill operators or shifts of operators to other jobs), what is the resulting delay? How could it be reset? • For a given set of activities to be developed and a workforce formed by different skills, which workers or resources should be assigned to the different activities in order to meet all the deadlines? • If the scarcity of the resources produces bottlenecks and consequent delays in the production cycle, is it possible to adopt suitable strategies to meet the deadlines? The above considerations suggest that the problem can naturally be formulated as a scheduling problem. In fact, we must find for each job the best possible schedule such that all the above constraints are met and a suitable given objective function is optimized (i.e. the total costs are minimized, the maximal lateness of the jobs are minimized, the total net profit is maximized, etc.). Due to the rich variety of different problems within this research field, scheduling is one of the most widely inter-disciplinary research areas (Mathematics, Operations Research, Information Technology and Computer Science, Manufacturing, Management, Business, Engineering, etc.). That is proved by more than 30,000 publications, referring to ‘‘scheduling’’ since the 1990s, that can be found in Web of Science. This situation makes an exhaustive state-of-the-art review on the scheduling problem very difficult. For instance, the recent review papers are addressed to a determined class of scheduling problems (see, for instance, [1] on scheduling problems arising in production industries, [2] on scheduling problems with setup considerations, [3] on dynamic scheduling in manufacturing systems, [4] on multicriteria scheduling, [5] on scheduling with learning effects, and so on). We remark that our problem can be classified as a workforce scheduling problem, for which the numerous approaches existing in literature use many different objective functions and constraints, as well as a wide variety of mathematical models (see [6–13] and reference therein). Moreover, it is well known that almost all the scheduling problems are typically NP-hard, i.e., so far there does not exist any reliable numerical method able to find an optimal solution within a reasonable time. This difficulty pushed researchers to develop heuristic algorithms or approximation procedures (see [1,2,14,15]). Since the manager’s decision of accepting or rejecting an order must be taken in real-time, we were asked to propose a mathematical model to be written in finite arithmetic by using an algorithm compatible with the database and the tool of the Work Flow Management (WFM), already employed by SC.3 In order to solve our problem in real-time, we propose an evolutive mathematical model which evaluates a feasible not optimal scheduling for the job families. In particular, it ensures that all the deadlines are met, provided that the potential customer orders with their characteristics (such as process route, processing times, deadline, etc.), and the job shop configuration (such as resources, their availability, etc.) are assigned. This model is based on a system of ordinary differential equations (ODEs) which, in the first approximation, can analytically be solved. Further, for these equations a numerical procedure is implemented to answer in real-time all the above managerial questions. We conclude by noting that resorting to a mathematical model of a feasible not optimal scheduling does not always represent a real simplification of the problem. For instance, in the case in which to meet the deadline of a particular order requires an increasing of the number of the people involved in the work, it would appear quite spontaneous the choice leading to a positive balance between costs and profits. However, the manager could decide to accept this new order as a business strategy (important customer, prestigious order, etc.), even if there is not a total net profit. In these situations, the choice to adopt is devolved upon the manager since it could be not convenient in pursuing a prefixed objective. The paper is organized into five sections including the introduction. In Section 2 we describe the proposed mathematical model, whereas in Section 3 we present suitable criteria to resect the dead times. Section 4 is devoted to some numerical i ,j simulations relative to the model which is based on the first choice of the involvement coefficients ah defined in Section 3.1. Finally, Section 5 deals with the research perspectives concerning suitable development of the model in a classical framework of combinatorial optimization problems. 2. A mathematical model Before discussing the mathematical model, we describe the fundamental quantities characterizing both the structure of SC and the activities of any job. A set of job families {JF1 , JF2 , . . . , JFm } collect all types of possible orders, and with each job ȷih ∈ JFi , for i = 1, . . . , m and h = 1, . . . , |JFi |, a deadline di and an arrival time thi are associated. In our problem each job belonging to the family JFi consists of a set of ni sequential activities Ai,1 , Ai,2 , . . . , Ai,ni :
Activity Ai,1 Activity Ai,2 Job family JFi ≡ . . . Activity Ai,ni −1 Activity Ai,ni . 3 In other words, the SC wished that the implementation of our model became an upgrade of the software Mentore 5.0 already used by the SC.
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We remark that the same elementary activity can appear in processes which belong to different job families. Moreover, the number ni of the elementary activities depends on the typology JFi . The SC workforce is made up of p workers W1 , . . . , Wp mastering one or several skills, i.e. each of them may be assigned to one or more activities. Then, in searching for a schedule of the activities, we have to determine which resources are the most appropriate to process each activity in each period. Let TiA,j be the absolute time needed to perform the elementary activity Ai,j . This time is evaluated supposing that: (1) there is no obstacle during the development of the activity; (2) the workers related to this activity realize the best performance. Moreover, it is fundamental to assign once for ever the unit time. In our analysis, the time unit is assumed to be 8 h (8 h). If the activity Ai,j has a standard duration of 3 h, then we have that TiA,j = 3/8. Finally, let Ni,j (0) be the initial number of the jobs referring to the activity Ai,j . In other words, the quantities Ni,j (0) represent the workload in any activity Ai,j at the beginning of the simulation. All the above quantities are the input data of the new WFM system implementing the mathematical model we are introducing. Let Ni,j = Ni,j (t ) be the job number at the instant t of the activity Ai,j . Then, we assume that for each Ni,j (t ) the following balance equation holds: dNi,j dt
= σi,j−1 (t , N, 9) − σi,j (t , N, 9),
i = 1, . . . , m, j = 1, . . . , ni ;
(1)
where N ≡ Nh,k , h = 1, . . . , m, k = 1, . . . , nh , 9 denotes a set of suitable parameters we list below, σi,j−1 (t , N, 9) is the number of jobs belonging to the family JFi and that at the instant t come from activity Ai,j−1 and go to Ai,j in the unit time; σi,j (t , N, 9) is the number of jobs in the job family JFi that at the instant t overcome the step Ai,j and arrive at Ai,j+1 in the unit time. Shortly, σi,j (t , N, 9) denotes the flux of jobs sent from Ai,j to Ai,j+1 in the unit time. We remark that:
• for j = 0 and i arbitrary, σi,0 (t ) denotes the number of jobs in the family JFi which are accepted by the SC at the instant t; consequently, σi,0 (t ) is an input datum only depending on t. It is evident that the constraint consisting in meeting all the deadlines could compel the manager not to accept other jobs from a certain instant Tiin on, so that σi,0 (t ) = 0, for t ≥ Tiin , provided that the business strategy does not suggest other choices (see Section 1); • when σi,j−1 (t , N, 9) − σi,j (t , N, 9) > 0, the jobs relative to Ai,j are accumulating and Ni,j increases. System (1) allows us to evaluate Ni,j (t ) when the net flux of jobs σi,j−1 (t , N, 9)−σi,j (t , N, 9) is given. Therefore, to complete the mathematical model, we must assign the constitutive equation of the flux σi,j , i.e., we have to express this quantity as a function of t , N, and the parameters 9 that take into account
• • • • •
the structural parameters of the SC ; the absolute time TiA,j ; the efficiency level of the operators working on the activity Ai,j ; the total or partial involvement of the worker in the activity Ai,j ; some corrective parameters which take into account delay factors modifying the absolute time, which is an ideal time.
In order to determine the constitutive relation σi,j (t , N, 9), we start supposing that pi,j employers are working on the activity i ,j
Ai,j . To each employer Wh we associate a bounded function ah (t , N, 9), with h = 1, . . . , pi,j i,j
0 ≤ ah (t , N, 9) ≤ 1,
(2)
which denotes the involvement level of the employer Wh in activity Ai,j . Clearly, the involvement coefficients of an employer Wh working in different activities verify the following condition
−
i ,j
ah (t , N, 9) = 1.
i,j
In Section 3, we give some suggestions to model these coefficients. i,j In view of the meaning of the involvement coefficients ah (t , N, 9), the function: i ,j
ri,j (t , N, 9) = a1 (t , N, 9) + · · · + aip,ij,j (t , N, 9)
(3)
will denote the effective number of workers involved in the performance of the activity Ai,j . For instance, suppose that the involvement coefficients are constants and chosen by the manager on the bases of the structure of the SC. Then, if an operator performs only the activity Ai,j for all his working time, then pi,j = 1 and ri,j = 1. i,j
Differently, if two workers are assigned to the activity Ai,j for 3/4 and 1/2 of their working time, then pi,j = 2, a1 = 3/4, i,j
a2 = 1/2 and ri,j = 5/4. Finally, if a single worker spends 1/2 of his working time for the activity Ai,j and the remaining i,j
h ,k
half for the activity Ah,k , then it results a1 = a1
= 1/2.
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The above considerations lead us to adopt the following constitutive equation of σi,j (t , N, 9):
σi,j (t , N, 9) = ri,j (t , N, 9) σi,0 (t , N, 9) = σi,0 (t ),
1 TiA,j
,
i = 1, . . . , m, j = 1, . . . , ni ,
(4)
i = 1, . . . , m.
(5) i,j
We can conclude that the model is complete when the functions ah (t , N, 9) are given. In fact, if the involvement coefficients are assigned, then the relation (3) provides the quantities ri,j (t , N, 9), and finally σi,j (t , N, 9) are determined by (4)–(5). In conclusion, adding the initial data Ni,j (0) to the system (1), and recalling (3) and (4)–(5), we face with the following Cauchy problem
1 dNi,1 dt = σi,0 (t ) − ri,1 (t , N, 9) T A , i ,1 dNi,j
i = 1, . . . , m,
1
1
= ri,j−1 (t , N, 9) A − ri,j (t , N, 9) A , dt Ti,j−1 Ti,j Ni,j (0) = Ni0,j , i = 1, . . . , m, j = 1, . . . , ni .
(6)
i = 1, . . . , m, j = 2, . . . , ni ,
The integration of (6) supplies the functions Ni,j (t ) whose analysis allows us to verify whether the objectives of SC will be realized or not. When the results of this analysis show that the expectations will be not satisfied, the manager must elaborate i ,j a new strategy modifying the involvement coefficients ah (t , N, 9) until the objectives are reached. i,j
In the next section we show that, in choosing the involvement coefficients ah (t , N, 9), the manager must satisfy a fundamental criterion which improves the work planning resecting the dead times in performing all the jobs. Moreover, we are looking for a schedule which is able to accommodate disruption without exceeding the deadlines set by the customer’s orders. i,j
3. Two criteria to choose the coefficients ah i ,j
3.1. First choice: ah are constant and their assignment is based on resecting the dead times i,j
Let us suppose that the manager, on the basis of its experience, assigns the constant values to the coefficients ah and, consequently, to the quantities ri,j . In this case, the Cauchy problem (6) can be analytically integrated since it becomes
1 dNi,1 = σi,0 (t ) − ri,1 A , dt Ti,1 1
dNi,j
i = 1 , . . . , m,
1
= ri,j−1 A − ri,j A , i = 1, . . . , m, j = 2, . . . , ni , dt Ti,j−1 Ti,j Ni,j (0) = Ni0,j , i = 1, . . . , m, j = 1, . . . , ni .
(7)
Consequently, it is easy to verify if all the deadlines are met. However, even if a lucky choice allows us to perform all the jobs before the due times, it could be not convenient if the work were carried out much in advance with respect to the deadlines. We now introduce a criterion to resect the dead times. This criterion also guides the manager in the assignment i,j of the coefficients ah when his initial choice does not permit him to reach the objective. In other words, we use the solution of (7) to determine the conditions that the involvement coefficients must satisfy so that (1) the deadlines are met; j
(2) the instants Ti to complete the activity Ai,j verify the condition Ti1 ≤ · · · ≤ Tim (absence of dead times). To explain the idea underlying this criterion, we resort to the following hydraulic model. Let us consider a system formed by n reservoirs Aj , placed side by side to each other. Any reservoir has an entrance filler and an exit filler. The exit filler of Aj is also the entrance filler of the reservoir Aj+1 . We suppose that, at the instant t , Aj contains Nj (t ) liters of water and a device which discharges towards the adjacent reservoir a constant number σj of liters in the unit time. Moreover, Aj receives σj−1 liters from Aj−1 , provided that Aj−1 is not empty. We denote by t = 0 the arbitrary instant in which we start our analysis of the working process. We suppose that:
• at the instant t = 0 each reservoir Aj is empty or contains a known amount of water; • during time interval [0, T in ], a certain number of liters of water are introduced in the first reservoir A1 . Starting from the above data, we evaluate
• the constant exit water fluxes in such a way that the reservoirs expel their water content subsequently in time; • the instant T f > T in in which all the reservoirs are empty.
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Let σ0 (t ) be a function such that σ0 (t ) ≥ 0 in the time interval t ∈ [0, T in ], and identically vanishing for t ≥ T in . This function describes the time evolution of water flux entering the first reservoir A1 until the instant T in . Moreover, let N10 , . . . , Nn0 be the liters of water contained in the n reservoirs at the initial time. Finally, T in
∫
N in =
σ0 (t )dt ,
(8)
0
is the total number of liters entering A1 in the time interval [0, T in ], whereas N in
σM =
(9)
T in
is the entering mean flux. The number N1 (t ) of liters of A1 is the solution of the following Cauchy problem (see (7))
dN1
= σ0 (t ) − σ1 , dt N1 (0) = N10 ,
t ∈ [0, T in ]
(10)
so that it is given by the relation N1 (t ) =
t
∫
σ0 (t )dt − σ1 t + N10 ,
t ∈ [0, T in ].
(11)
0
From (11), when (8) and (9) are taken into account, it follows that, at the instant T in , A1 contains N1 (T in ) = (σM − σ1 )T in + N10
(12)
liters. For time values greater than T in , the function N1 (t ) is the solution of the following Cauchy problem:
dN1
= −σ1 , t ∈ [T in , T 1 ] dt N1 (T in ) = (σM − σ1 )T in + N10 ,
(13)
where T 1 is the instant in which N1 (T 1 ) = 0, i.e., A1 is empty. The solution of (13) has the form N1 (t ) = −σ1 t + σM T in + N10 ,
t ∈ [T in , T 1 ]
(14)
1
and consequently T is given by the relation T1 =
σM in 1 0 T + N , σ1 σ1 1
(15)
so that
σ1 =
r1 σM T in + N10 ≡ 1
1 T1
(16)
TA
where TA1 is defined as the absolute emptying time of the reservoir A1 . We note that T in ≤ T 1 ⇒ σ1 ≤ σM +
N10
(17)
T in
and the first reservoir becomes empty at the instant T in , when the input of water in the first reservoir ends provided that (17) is satisfied.4
4 In fact, from (15) we obtain T 1 − T in =
σM 1 − 1 T in + N10 σ1 σ1
and then T 1 − T in ≥ 0 ⇐⇒ σM − σ1 ≥ −
N10 T in
⇐⇒ σ1 ≤ σM +
N10 T in
.
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The behavior of the function N2 (t ), which gives the number of liters contained in the second reservoir A2 , is a solution of the following Cauchy problem
dN2
= σ1 − σ2 ,
t ∈ [0, T 1 ]
dt N2 (0) = N20 .
(18)
Therefore, we have that N2 (t ) = (σ1 − σ2 )t + N20 ,
t ∈ [0, T 1 ]
(19)
up to the instant T 1 in which the first reservoir is empty. In this instant the value of N2 is N2 (T 1 ) = (σ1 − σ2 )T 1 + N20 .
(20)
From T 1 on, the behavior of N2 (t ) is the solution of the following other Cauchy problem:
dN2
= −σ2 , t ∈ [T 1 , T 2 ] dt N2 (T 1 ) = (σ1 − σ2 )T 1 + N20 ,
(21)
where T 2 is the instant in which N2 (T 2 ) = 0 and the second reservoir is empty. The solution of (21) is N2 (t ) = −σ2 t + σ1 T 1 + N20 ,
t ∈ [T 1 , T 2 ]
(22)
2
and the instant T is expressed by the relation T2 =
σ1 1 1 0 T + N . σ2 σ2 2
(23)
Then, we have
σ2 =
1 T2
r2 σ1 T 1 + N20 ≡ 2 ,
(24)
TA
where TA2 is the absolute emptying time of the reservoir A2 . If we want that the second reservoir becomes empty after the first one has became empty, i.e., if T 1 ≤ T 2,
(25)
then the following condition has to be verified
σ2 ≤ σ1 +
N20 T1
.
(26)
The relations (23) and (24), in view of (15), can also be written as it follows:
σM in 1 T + (N 0 + N20 ), σ2 σ2 1 1 r2 σ2 = 2 σM T in + N10 + N20 ≡ 2 . T2 =
T
TA
(27) (28)
Following the same reasoning for the other reservoirs, we obtain the emptying time of the j-th reservoir: Tj =
j σM in 1 − 0 T + N , σj σj h=1 h
(29)
and the j-th constant flux
σj =
1 Tj
σM T + in
j − h =1
Nh0
≡
rj j
TA
,
(30)
j
where TA is the absolute emptying time of the reservoir Aj . This time is less than the emptying time of the previous reservoir if
σj ≤ σj−1 +
Nj0 T j −1
,
j = 2, . . . , n.
(31)
A. Marasco, A. Romano / Mathematical and Computer Modelling 53 (2011) 2005–2014
2011
Now, to come back to our problem, it will be sufficient to identify the reservoir Aj with the activity Ai,j ∈ JFi , for all i, the j
number Nj (t ) of liters with the number of customer jobs Ni,j (t ), the fluxes σj with σi,j , TA with the absolute time TiA,j , n with f
ni , and T f with the instant Ti in which all the jobs belonging to JFi are completed. Therefore, when we assign all the input data σ
,
in i,M Ti
,
Ni0,j
,
TiA,j
,
i ,j ah ,
in view of (30), the inequalities (31) supplies the conditions which allow us to verify the absence f
of dead times. Moreover, owing to the relation (29) for j = n, it is also determined the instant Ti in which all the orders in the job family JFi are fulfilled, so that if f
Ti ≤ di ,
∀i
(32)
all the deadlines di are met. The model based on this strategy, can lead to the following possibilities: (1) If (31) and (32) are verified, then the model supplies a scheduling which satisfies all the constraints imposed to the jobs. However, if this scheduling is not optimum, it could require an improvement. (2) It could happen that one or more conditions (31) and (32) are not verified. In this case, it becomes necessary to define the input data in a convenient way. In this latter case, further strategies can be adopted to reach the fixed objectives, depending on the company policy and the current situation. For instance, if it is not possible to modify the data σi,M , Tiin since the orders have already been accepted, i,j
then the model should give the new values of the involvement coefficients ah which allow to deal with all the orders in the i,j ah
due times. In other cases, the coefficients could be determined on the basis of the real working capacity, so that it is not possible to improve them. In this case, the model should give information about the parameters σi,M , Tiin in order to meet i,j
the deadlines. In the absence of information about the values of the coefficients ah , or when inevitable and unpredictable real-time events cause a change in the schedule plans (i.e. ill operators, cancellation of an order, a change in job priority, etc.), the model should proceed to a new evaluation of these parameters. In the next section, we propose some simple numeric simulations based on the above choice of the involvement coefficients. We recall that this model has been implemented in the upgrade of the WFM software Mentore 10. i ,j
3.2. Second choice: ah are not constant, but depending on N, and TA i,j
A more flexible model can be obtained by improving the structure of the coefficients ap as follows. Let us suppose that a worker Wp is involved in three different activities Ai,j , Ah,k , and Ar ,s each of them requiring the absolute times TiA,j , ThA,k and TrA,s , respectively. The involvement coefficients of this worker related to the different activities can be defined by the following relations api,j (N, TA ) = aph,k (N, TA ) = apr ,s (N, TA ) =
Ni,j TiA,j Ni,j TiA,j
+ Nh,k ThA,k + Nr ,s TrA,s
,
Nh,k ThA,k Ni,j TiA,j + Nh,k ThA,k + Nr ,s TrA,s Nl,m TrA,s Ni,j TiA,j + Nh,k ThA,k + Nr ,s TrA,s
,
(33)
,
where TA ≡ TAu,v , u = 1, . . . , m, v = 1, . . . , nu . These formulae can be improved by multiplying them by a factor αpu,v , 0 < αpu,v ≤ 1, which takes into account the efficiency level and the reliability of the worker Wp involved in the activity Au,v . Let us suppose that the workers W1 , . . . , Wpi,j are involved in the activity Ai,j . Applying the above formulae to each worker and introducing the resulting expressions into (3), we obtain the new quantities ri,j . By recalling (33), the Cauchy problem (6) becomes
dNi,1 1 = σi,0 (t ) − ri,1 (N, TA ) A , dt Ti,1 dNi,j
1
i = 1, . . . , m, 1
= ri,j−1 (N, TA ) A − ri,j (N, TA ) A , i = 1, . . . , m, j = 2, . . . , ni , dt Ti,j−1 Ti,j 0 Ni,j (0) = Ni,j , i = 1, . . . , m, j = 1, . . . , ni .
(34)
It is important to remark that, since the system (34) is nonlinear and nonautonomous, its solution could not be expressed in a closed form. However, its numerical integration is a standard computational problem so that, also in this case, it is possible to numerically verify if the conditions (31) and (32) are satisfied. We conclude remarking that, although this last criterion supplies a more sophisticated mathematical model, it could be less convenient from a computational point of view. For this reason we have implemented the model presented in Section 3.1 in the software Mentore 10.
2012
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4. Numerical simulation In order to show the several possibilities of management offered by the model, in this section we present some numerical simulations based on the approach presented in Section 3.1. 4.1. Simulation 1 We suppose that SC ’s manager decide to accept Nhin = 100 orders, with the same deadline dh = 90, in the time interval [0, 30]. These orders refer to the job family JFh characterized by the data listed in the following table (see Section 2)
Activity Ah,1 : Nh,1 (0) = 40, Activity Ah,2 : Nh,2 (0) = 30, Activity Ah,3 : Nh,3 (0) = 70, JFh ≡ Activity Ah,4 : Nh,4 (0) = 10, Activity Ah,5 : Nh,5 (0) = 20, Activity Ah,6 : Nh,6 (0) = 45,
ThA,1 ThA,2 ThA,3 ThA,4 ThA,5 ThA,6
= 4; = 6; = 3; = 6; = 6; = 5. f
In this simulation we determine the coefficients rh,1 , . . . , rh,6 and the instant Th in which all the orders in JFh are fulfilled, in such a way to satisfy the constraints (31) and (32), and to perform the activities by available and appropriate resources. First, we remark that, in view of relation (16), all the values rh,1 such that rh,1 ≤ 18.6667 allow us to satisfy (17). Similarly, when the value of the coefficient rh,1 is given, Eq. (24) supplies the upper bound of rh,2 for which the constraint (26) is verified. It is evident how we can go on (see relations (31)). We note that the values of the coefficients rh,i satisfying relation (31) f
allow us to determine uniquely the instant Th in which all the orders in the job family JFh are fulfilled. Therefore, we are f
interested in a set of coefficients rh,i such that Th ≤ 90. Performing some simulations using Mathematica we obtain the f
outputs listed in the following tables. We note that in all simulations the end time Th = 87.5 is the same. (1) In the first simulation we have that Activity Ah,1 : rh,1 = 16, Activity Ah,2 : rh,2 = 15,
Activity Ah,3 : rh,3 = 10,
Activity Ah,4 : rh,4 = 18,
Activity Ah,5 : rh,5 = 19, Activity Ah,6 : rh,6 = 18,
f
Th,1 = 35;
f Th,2 f Th,3 f Th,4 f Th,5 f Th,6
= 68; = 72; = 83.3333; = 85.2632; = 87.5.
We suppose that the above distribution is compatible with the available resources. In this case, the manager can dispose all the workers employed in the first activity after 35 work days, workers employed in the second activity after 68 work days and so on. All the orders will be fulfilled before the deadline. (2) We suppose that the above distribution cannot be adopted by the manager since the workforce allows the following maximum values rh,2 = 14, rh,3 = 9. Introducing this restriction we obtain the following other table Activity Ah,1 : rh,1 = 16,
Activity Ah,2 : rh,2 = 14,
Activity Ah,3 : rh,3 = 9,
Activity Ah,4 : rh,4 = 18, Activity Ah,5 : rh,5 = 19,
Activity Ah,6 : rh,6 = 18,
f
Th,1 = 35; f
Th,2 = 72.8571;
f Th,3 f Th,4 f Th,5 f Th,6
= 80; = 83.3333; = 85.2632; = 87.5. f
In this case, the work ends at the instant Th = 87.5, but the workers employed in the activities Ah,2 and Ah,3 will be f f again at the disposal of the manager after the instants Th,2 = 72.8571 and Th,3 = 80, respectively. (3) When some workers, which usually are devoted to the activity Ah,1 , are not available at the initial instant, the above table is modified as follows f
Activity Ah,1 : rh,1 = 10, Th,1 = 56; f
Activity Ah,2 : rh,2 = 14, Th,2 = 72.8571; f
Activity Ah,3 : rh,3 = 9, Th,3 = 80; f
Activity Ah,4 : rh,4 = 18, Th,4 = 83.3333; f
Activity Ah,5 : rh,5 = 19, Th,5 = 85.2632; f
Activity Ah,6 : rh,6 = 18, Th,6 = 87.5.
A. Marasco, A. Romano / Mathematical and Computer Modelling 53 (2011) 2005–2014
2013
f
Once again the work is completed at instant Th = 87.5, but the workers of activity Ah,1 become available only after 56 work days. These examples show that the model is able to propose many possible scheduling which meet manager’s requirements. In the last example, the manager could decide to increase the number of workers devoted to the first activity hiring 4 full-time workers in a period of 35 days. In this way, the internal staff of the SC will become free before 56 days. We remark that, in any proposed scheduling, the cost of any activity is not modified provided that the salary of each operator remains constant. In fact, in any case, we have
f rh,i Th,i
=
ThA,i
Nhin
+
i −
Nh,k (0)
≡ const,
k =1
for any h and i. 4.2. Simulation 2 Suppose that the SC ’s manager decides to accept in a month Nkin = 150 orders with deadline dk = 90. All the orders belong to the family JFk which is composed by nk = 15 sequential activities characterized by the following absolute times TAk = {4, 6, 2, 6, 7, 5, 4, 6, 2, 6, 9, 5, 4, 6, 3}. Moreover, let Nk (0) = {10, 30, 15, 10, 25, 23, 20, 30, 15, 10, 8, 15, 10, 20, 22} be the corresponding workload initial set. f In order to determine the coefficients rk,1 , . . . , rk,15 and the instant Th such that all the constrains (31) and (32) are verified, we remark that the upper bound of rk,1 is 21.3333. About this case, we propose the following simulations. (1) We assume that the maximum value for any coefficient rk,i is 35. Performing a numerical simulation with the following values for the coefficients rk,i rk = {21.3, 35, 12.58, 35, 35, 27, 23.2, 35, 12.2, 35, 35, 20.2, 16.6, 26.2, 13.8}, we obtain the following end times f
Tk = {30.0469, 32.5714, 32.5914, 36.8571, 48, 48.7037, 48.7931, 53.6571, 53.7705, 57.9429, 88.9714, 89.3564, 89.3976, 89.542, 89.7826}, for which the constrains (31) and (32) are satisfied. (2) We suppose that SC has a set of operators for which the maximum value of any coefficient rk,i is 25. However we f
modify the input parameters, taking into account the constraints (31), we always obtain that Tk > 90. For instance, the numerical simulation with the following values for the coefficients rk,i rk = {21.3, 25, 8.9, 25, 25, 19.5, 16.5, 25, 8.7, 25, 25, 14.4, 11.8, 18.6, 9.8},
(35)
leads us to the following end times f
Tk = {30.0469, 45.6, 46.0674, 51.6, 67.2, 67.4359, 68.6061, 75.12, 75.4023, 81.12, 124.56, 125.347, 125.763, 126.129, 126.429}.
(36)
We observe that for the activities Ak,3 , Ak,6 , Ak,7 , Ak,9 , Ak,12 , Ak,13 , Ak,14 , Ak,15 the values of the coefficients rk,i are chosen in such a way that the corresponding constrains (31) are verified (see the underlined values in (35)). Moreover, in (36) f it is shown that from any Ak,j , with j > 11, it results Tk,12 > 90. In order to meet the deadline, the manager could decide to accept a smaller number of orders. All the simulations show that only for Nkin = 50 the deadlines are met. In fact, if we assume that Nkin = 50,
rk = {8, 17.5, 6.8, 22, 25, 20.7, 18.5, 25, 8.9, 25, 25, 14.7, 12.2, 19.65, 10.56},
we obtain f
Tk = {30, 30.8571, 30.8824, 31.3636, 39.2, 39.372, 39.5676, 51.12, 51.236, 57.12, 88.56, 88.7755, 88.8525, 88.855, 88.9205}. We conclude remarking that when Nk (0) = 0 for the following input data Nin k = 150,
rk = {20, 25, 8, 24, 25, 17.8, 14, 20, 6.5, 19, 25, 13.5, 10.5, 15.5, 7.5},
we obtain f
Tk = {30, 36., 37.5, 37.5, 42, 42.1348, 42.8571, 45, 46.1538, 47.3684, 54, 55.5556, 57.1429, 58.0645, 60} and the deadline is met in a period of 60 days.
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5. Conclusions and research perspectives This paper provides a detailed description of a model to manage SC which is based on a system of ordinary differential equations. Numerical simulations show that the proposed model offer several possibilities of management in very short running time. As future work, we plan to extend our model to an even more realistic problem that will reflect real life scenarios including several constraints in addition to those considered in this paper. Moreover, we plan also to formulate the problem as combinatorial optimization problem (see [14]). Special emphasis will be given to a possible mathematical formulation of the problem as scheduling problem and to the design of efficient mathematical programming methods in order to compare the two approaches. In our opinion, many topics of interest remain open for further research in this area. First of all, a hybrid approach could be designed that will include the main ingredients of the two pure methods. Secondly, since most of these problems are computationally intractable, besides the study of exact methods, a further fruitful research line would be towards the design of approximation algorithms and metaheuristics. References [1] C.N. Potts, V.A. Strusevich, Fifty years of scheduling: a survey of milestones, Journal of the Operational Research Society 60 (2009) S41–S68. [2] A. Allahverdi, C.T. Ng, T.C.E. Cheng, M.Y. Kovalyov, A survey of scheduling problems with setup times or costs, European Journal of Operational Research 187 (2008) 985–1032. [3] D. Ouelhadj, S. Petrovic, A survey of dynamic scheduling in manufacturing systems, Journal of Scheduling 12 (2009) 417–431. [4] H. Hoogeveen, Multicriteria scheduling, European Journal of Operational Research 167 (2005) 592–623. [5] D. Biskup, A state-of-the-art review on scheduling with learning effects, European Journal of Operational Research 188 (2008) 315–329. [6] H.K. Alfares, Survey, categorization, and comparison of recent tour scheduling literature, Annals of Operations Research 127 (2004) 145–175. [7] P. Cowling, N. Colledge, K. Dahal, S. Remde, The trade off between diversity and quality for multi-objective workforce scheduling, evolutionary computation in combinatorial optimization, Lecture Notes in Computer Science 3906 (2006) 13–24. [8] O. Lambrechts, E. Demeulemeester, W. Herroelen, Proactive and reactive strategies for resource-constrained project scheduling with uncertain resource availabilities, Journal of Scheduling 11 (2008) 121–136. [9] L.-E. Drezeta, J.-C. Billauta, A project scheduling problem with labour constraints and time-dependent activities requirements, International Journal of Production Economics 112 (2008) 217–225. [10] T.A. Guldemond, J.L. Hurink, J.J. Paulus, J.M.J. Schutten, Time-constrained project scheduling, Journal of Scheduling 11 (2008) 137–148. [11] V. Tiwari, J.H. Patterson, V.A. Mabert, Scheduling projects with heterogeneous resources to meet time and quality objectives, European Journal of Operational Research 193 (2009) 780–790. [12] V. Valls, A. Pérez, S. Quintanilla, Skilled workforce scheduling in service centres, European Journal of Operational Research 193 (2009) 791–804. [13] C.-S. Chen, S. Mestry, P. Damodarana, C. Wang, The capacity planning problem in make-to-order enterprises, Mathematical and Computer Modelling 50 (2009) 1461–1473. [14] P. Festa, R. De Leone, E. Marchitto, A new meta-heuristic for the bus driver scheduling problem: GRASP combined with rollout, in: IEEE on Computational Intelligence in Scheduling, 2007. doi:10.1109/SCIS.2007.367689. [15] R. De Leone, P. Festa, E. Marchitto, A bus driver scheduling problem: a new mathematical model and a GRASP approximate solution, Journal of Heuristics (2011) doi:10.1007/s10732-010-9141-3.