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A NEW APPROACH OF FLEXIBILITY PLANNING A REALITY OR A SEARCH OF THE HOLY GRAIL!
IFAC MCPL 2007 The 4th International Federation of Automatic Control Conference on Management and Control of Production and Logistics September 27-30, Sibiu - Romania
A NEW APPROACH OF FLEXIBILITY PLANNING A REALITY OR A SEARCH OF THE HOLY GRAIL! Khaoula El Bedoui-Maktouf1, Zied Bahroun1, Mohamed Moalla1, Jean-Pierre Campagne2 1
LIP2, Faculty of Sciences of Tunis, Tunisia 2 LIESP, INSA-Lyon France
Abstract: In the new context of “contract-order”, companies are engaged with their customers on a global program of delivery orders. Each order concerns the deliver of quantities (varying between a minimum and a maximum value) of products at precise moments. The exact quantities are known only few days before the delivery. This variability of delivery quantities needs the management in an efficient way of some flexibility levers in order to guarantee the service engagement and minimize the costs. In this paper, a new approach of flexibility planning will be presented. This approach tends to satisfy these goals and takes into account internal and external constraints of these companies. The most important internal constraint is the evidence presence of a bottleneck resource. An external constraint must be obviously satisfied which is the imposed regulations. Copyright © 2007 IFAC Keywords: contract-order, flexibility, optimisation, bottleneck resource, regulation.
1. INTRODUCTION
flexibility planning will be exposed. Finally, the evaluation of this approach is provided through some experimental results.
Nowadays, companies are faced to the need of flexibility management as they were asked to for quality ten or twenty years ago. This effective need of flexibility management is argued to be the crucial component for the surviving of the company in a complex and a difficult environment which is characterised by its variability and its uncertainty.
2. PROBLEMATIC As it was mentioned, in this new context, enterprises are dynamically and continuously looking for stability. This search of the equilibrium state is carried out by the management of different flexibility levers. Obviously, these enterprises are asked to determine which flexibility levers are to use.
In fact, to maintain a minimum level of stability, a new relationship is established between companies and their customers called “contract-order”. Enterprises exchange, then, a service guarantee against a work guarantee from costumers. The guarantee of service concerns a delivery schedule with a quantity varying between a minimum and a maximum value for each delivery and requires, so, the management of different flexibility levers.
In general framework, the choice of flexibility levers is not easy but, in the context of contract-order, the choice must provide a flexible production capacity, to deal with the variability of the demand. This production capacity is based on the human and material resources. So, enterprises must, in one hand, manage the human resource as an important mean of flexible capacity and, in the other hand, they must
In the following part, the problematic of these enterprises is presented. The literature review is then presented. After that, the proposed approach of 733
not ignore the evident role of material or machine resources in this case.
The resolution of the second and the third question is based on the distinction of two types of resources in the production system: bottlenecks and nonbottlenecks. The bottleneck resources can be either machine or manpower. In this paper, the case of bottleneck machine is considered. So, the number of basic employee needed on this bottleneck resource is not a parameter to determine but a data to use. The total number of needed employees is calculated: • from the load induced, on the rest of the production system, from the working hours of the bottleneck resource (as in most cases the load induced is calculated in number of manpower working hours), • and from the functioning mode i.e. the number of shifts to use on the bottleneck resource in order to vary its production capacity.
Both human and material resources are submitted to some constraints. In fact, the use of human resources is submitted to the regulations constraints, whereas, material resources are submitted to the presence of a bottleneck resource which determines the whole production capacity. Bottleneck resource is generally suffering from a high production load which exceeds its production capacity and is characterised by the fact that it is expensive and/or rare. Traditionally, enterprises have solved the need of flexible production capacity, essentially, by the use of human resource in overtime and/or interim hours. The use of overtime is submitted to the regulations and is very difficult to manage at long horizon. In addition, the use of interim is not evident and imposes different other constraints such as the need of training of the interim employees.
To illustrate the first point, the following example can be presented: if the amount of machine working hours on the bottleneck is x hours, then the number of induced manpower working hours on the rest of the production system is proportional to this x. The rate of proportionality is ≥1. This is due to the fact that the most bottleneck resource in the whole production system is considered and the rest of production system is supposed as non-bottleneck resource. So, the load in the bottleneck will be less than that induced on the rest of production system.
Many researches have approached the human resource management such as those of (Chaabouni, 2001; Kane et al., 1999; Edieal et al., 2003; Hung R., 1999; Oz and Rune, 2003). Both (Chaabouni, 2001) and (Kane et al., 1999) proposed a mathematical approach to the work-time management and focused on the overtime and interim treatment. (Edieal et al, 2003) represented a variety of contracts, including temporary workers, on-call workers with guaranteed minimum pay, and comp-time arrangement. The model determines the regular and contingent worker pool sizes that minimize. The bottleneck resource is mistreated.
The answer of the fourth question will be given in the last section of this paper. Besides, in this paper it is supposed that the management of human resources is imposed to some regulations, especially the use of overtime hours which is subject to many legal constraints. In our work we consider the French case and precisely we treat the annualised hours which impose to take into account the maximal limits of the global working overtime hours by week, by period of four consecutive weeks and by year.
Moreover, the treatment of the management flexibility under the presence of a bottleneck resource has not been the object of deep investigations. The proposed approach is supposed to answer the following questions: 1. How to well manage the flexibility levers in order to satisfy the costumers demand and so optimise the production planning 2. How many employees are required to meet the induced capacity need in manpower 3. How to vary the production capacity of bottleneck resource and essentially how to increase it 4. In which manner these informations can be helpful to the decision maker
Furthermore, it is very interesting to apply the management of the human resources and the bottleneck resource in the long term, since this management is not only useful for preparing the implementation in the short term, but it also contributes to bring better costs optimization. The most intuitive question is now the achievement of all such goals under all these constraints. Are we looking for the Holy Grail?
To answer to the first question, it is necessary to prepare the use of flexibility levers to really exploit them in an efficient way. So, the use of flexibility levers is integrated to the process of production planning. In others words, the proposed approach will determine the amount of overtime and/or interim hours to use in parallel to the determination of the quantities to produce to satisfy costumers demands.
3. MATHEMATICAL MODEL To formulate the proposed approach, mathematical model is developed in which the production planning is carried out on a horizon discretized in T periods. The elementary period is noted t. Let’s consider:
734
• For products: N : total number of different products i : product identifier Csi: inventory holding cost of product i per period Si0 : initial stock of product i Cri : manufacturing set-up cost of product i ri :set-up time of product i dit :demand of product i at period t pi : processing time of product i in the bottleneck
• For employees: wTotal : total number of needed employee for the whole horizon On the bottleneck resource: wGoulett: total number of employees succeeding on the bottleneck at period t Hsupt: total number of overtime hours at period t for the bottleneck H1t: number of day overtime hours of the first range per employee on the bottleneck resource and at period t H2t: number of day overtime hours of the second range per employee on the bottleneck resource and at period t HJt: total number of day overtime hours at period t on the bottleneck HNt: total number of night overtime hours at period t Hintt: total number of interim hours at period t HintJt: total number of day interim hours at period t HintNt:total number of night interim hours at period t
• For system production: : maximum number of different functioning modes m : functioning mode identifier Capm: bottleneck capacity if the functioning mode is the mode m per period Hregm : number of regular hours of the functioning mode m per period and per employee HregB : number of indebted regular hours per period and per employee Chmodn,k : changing cost from mode n to mode k Rap : the proportion coefficient between bottleneck resource and the rest of production system M
On the rest of production system: wAtelt: number of permanent employee needed on the rest of production system in regular working wAtelSt: number of permanent employee needed on the rest of production system in overtime hours HsupAtelt: total number of overtime hours induced on the rest of production system at period t H1Atelt: number of overtime hours induced on the rest of production system at period t for the first range H2Atelt: number of overtime hours induced on the rest of production system at period t for the second range HintAtelt: number of interim hours induced on the rest of production system at period t
• For employee: HsupA: maximum number of overtime hours allowed par year and per employee HsupM: maximum number of overtime hours allowed per month and per employee HsupL: limit of overtime hours per month of the first range Tb: bonus rate for the first overtime range ≤ HsupL and doing in day Tm: bonus rate for the day overtime above HsupL and doing in day Tn: bonus rate of the night overtime hours wGouletB : number of basic employees needed on the bottleneck resource pI: productivity coefficient of the interim employees Cb: cost of a regular hour CIJ: cost of interim hour in day CIN: cost of interim hour at night TRSG: the rate of service efficiency of permanent employees on the bottleneck resource TRSA: the rate of service efficiency of permanent employees on the rest of the production system Acc: the rate of employees who accept the overtime
In order to facilitate the model, we define the following variables: ChGoulett: load on the bottleneck resource at period t ChAtelt: load on the rest of production system at period t SChAtelt: overload on the rest of production system at period t The objective function to optimize (minimize) is: Min
The set of decision variables is:
N
T
∑∑
• For products: Nli: number of production orders of product i at [1..T] qit: quantity to produce of product i at period t σ it : 1 if qit >0 0 otherwise Sit: stock of product i at the end of period t
Inventory holding cost
Cs i * S it
t =1 i =1
N
+ ∑ Cri * Nl i
Manufacturing set-up cost
i =1 T
+ ∑ HregB * wTotal * Cb t =1
Permanent employee cost + T
• δmt : 1 0
∑ ((1 + Tb) * H1 + (1 + Tm) * H2
For system production: if the functioning mode at period t is m otherwise
t
t
+ (1 + Tn) * HN t )) * Cb * wGouletB
t =1
Bottleneck overtime cost + 735
T
∑ ((1 + Tb) * H1Atel
t
(13) ∀t H1t ≤ HsupL The overtime hours of the first range on the bottleneck at a period is less than the permitted limit t + 12 (14) ∀t Hsup ≤ Hsup A
+ (1 + Tm) * H2Atel t ) * Cb
t =1
Overtime cost on the rest of production system
∑
+ T
∑ (HintJt * CIJ + HintNt * CIN)* wGouletB
The sum of the overtime hours throughout the year must be less than the amount authorised by the regulation (15) ∀t ChAtelt= Rap *ChGoulett The load induced on the rest of production system is proportional of that of bottleneck ∀t SChAtelt=max(0,ChAteltTRSA*HregB*wAtelt) (16) This constraint calculate the overload on the rest of the manufacturing system ∀t SChAtelt= TRSA*HsupAtelt + pI*HintAtelt (17) The overload on the rest of production system will be covered by the use of overtime and interim hours (18) ∀t HsupAtelt=H1Atelt+H2Atelt The total number of overtime hours on the rest of the production system is the sum of the amount of the first and the second range (19) ∀t WatelSt= Acc * wAtel t
t =1
Bottleneck interim cost T
+ ∑ HintAtel * CIJ t t =1
Interim cost on the rest of production system T −1 M
+∑
M
∑ ∑ Chmod
* δ nt * δ k(t +1) )
nk
t =1 n =1 k =1
Changing-mode cost The following constraints are considered: (1) For t=1, Si1 = Si0+ qi1 – di1 (2) For t>1, Sit = Si(t-1)+ qit – dtit These constraints express the relation between stocks at successive periods of the product i M
∑δ
∀t
=1
mt
(3)
m =1
This constraint expresses the number of employees who accept overtime (20) ∀t HsupAtelt ≤ HsupM*wAtelSt The total number of overtime hours on the rest of production system at a period is less than the permitted amount by month multiplied per the number of employee accepting the overtime (21) ∀t H1Atelt ≤ HsupL*wAtelSt The total number of overtime hours of the first range on the rest of production system at a period is less than the permitted amount multiplied per the number of employee accepting the overtime t + 12 ∀t HsupAtel /wAtelS ≤ Hsup A (22)
Only one mode should be taken at a period T
∑σ
∀i = 1..N
it
(4)
= Nl i
t =1
The total number of production orders for a product i is calculated from its state. N
∀t ChGoulett=
∑( p * q i
it
+ σ it * ri )
(5)
i =1
This constraint calculate the load on the bottleneck resource M ChGoulet ≤ TRSG* ( Hreg + Hsup ) + pI * Hint (6) t
∑
m
t
t
m=1
∑
The load induced on the bottleneck at a period by products must be lower than its capacity (7) ∀t Hsupt = HJt +HNt The total number of overtime hours on the bottleneck is the sum of those doing in day and those doing at night (8) ∀t HJt = H1t +H2t The total number of overtime hours on the bottleneck doing in day is the sum of the amount of the first and the second range (9) ∀t Hintt=HintJt + HintNt The total number of interim hours on the bottleneck is the sum of those doing in day and those doing at night ∀t
M
HJt+HintJt≤max(0,416- ∑ Hreg
m
t
The total number of overtime hours of the first range on the rest of production system at a period is less than the permitted amount multiplied per the number of employee accepting the overtime ∀t
wGoulett=wGouletB*
M
∑
m *δmt
(23)
m =1
This constraint expresses the number of employee on the bottleneck (24) ∀t wTotal= wGoulett+WAtelt This constraint expresses the total number of needed employee As it is noticeable, this model is non-linear. Its nonlinearity comes from the changing mode cost and from constraints. Besides, this model deals with the case of multi-product system, the case of monoproduct is obtained from this model by considering N=i=1. Then, we assume that is not necessary to consider either a set-up time (i.e ri=0) and or a manufacturing set-up cost (i.e Cri=0).
* δ mt ) (10)
This constraint determines the amount of hours which can be done in day ∀t M HNt+HintNt≤min(304,720Hreg * δ ) (11) m
t
t
m =1
∑
t
t
mt
m =1
This constraint determines the amount of hours which can be done at night (12) ∀t Hsupt ≤ HsupM The overtime hours on the bottleneck at a period is less than the permitted amount by month 736
4. EXPERIMENTAL RESULTS
For each demand profile, the influence of every parameter of the model is analysed by varying the considered parameter and keeping constant the other parameters. The detailed results of these experimentations are furnished in ElBedoui-Maktouf 2007. A part of these results are shown in Fig.2. Notice that D1, D2, D3 and D4 correspond respectively to the fourth demand profiles presented in the beginning of this section.
The proposed approach was tested on four demand profiles. Three of them deal with the demand fluctuation. These profiles are: • Seasonal (Fig. 1-a), as f(t) = a sin (w t+ b), • Seasonal with amplitude and average perturbation (Fig. 1-b), as f(t) = (a+ d* random()) sin (w t+ b) + random() *c, • Seasonal with amplitude and average perturbation and with tendency (Fig. 1-c), as f(t) = (a+ d* random()) sin (w t+ b) + random() *c +e * t, • Arbitrary (Fig. 1-d), as f(t) = b*random(). N.B. random () is a random function. 3 30
800
3 20
700
The validation of this model is proved through its right behaviour and its coherence. The results show that: • The increase of inventory cost leads to the decrease of the stock level and to increase the production capacity of bottleneck resource by the increase of functioning mode, overtime and interim hours. Also the number of the permanent employee is increased. • the increase of the interim efficiency leads to reduce the rate of interim hours. • The increase of basic cost leads to the diminishing of the functioning mode (i. e. the regular hours) and the overtime and to the increase of the interim hours. The number of permanent employee is decreased. • The increase of the proportionality coefficient leads to the increase of the total number of permanent employees and to the increase of the induced hours on the rest of the production system and so the average of overtime and interim hours on.
600
310
500
3 00
400
2 90
300
2 80
200
2 70
100 0
2 60 1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
(b)
(a) 700 600 500 400 300 200 100 0
700 600 500 400 300 200 100 0 1
2
3
4
5
6
7
8
9
10 11 12
1
2
3
4
5
6
7
8
9
10 11 12
(d) (c) Fig.1. Types of demand profiles The software used to solve the mathematical model is LINGO, which has four different solvers. These solvers are: a direct solver, a linear solver, a nonlinear solver and a branch-and-bound manager. The demand profiles are generated by EXCEL, and are sent to LINGO to optimize the proposed mathematical model. The results are obtained then as an EXCEL table.
Other interesting points are deduced: • The demands which have seasonality and which have a high standard deviation are less sensitive to the variation of the parameters than those of low standard deviation • So the variation of the equilibrium state is low for the demand of high standard deviation than for those of low standard deviation • The arbitrary demand with high or low standard deviation has the similar comportment. • The number of permanent employee is situated in neighbourhood of the maximum estimated number • The standard deviation between the minimal estimated number and the maximum estimated number of permanent employee high for the demand with a high standard deviation and low for those with low standard deviation. • Despite of the inactivity of permanent employee in some periods, their work in the other periods is very opportune
In order to avoid essentially the problem of local optimum and to reduce the execution time of experiments, the model was linearised. The result of each experiment determines: • the optimal total number of permanent employees and for each period: • the number of permanent employees on the bottleneck • the number of permanent employees on the rest of production system • the functioning mode i.e. the number of shifts on bottleneck • the rate of overtime and interim hours on the bottleneck • the rate of overtime and interim hours on the rest of production system • the quantities to produce and so the stock of each product
Finally, it is important to note that the knowledge of the optimal number of permanent employee is very interesting for the enterprise in order to estimate its future planning. The anticipation is also very useful and helpful to the decision maker.
737
5. CONCLUSION
Stock level 8000 7000 6000 5000 4000 3000 2000 1000 0
In the context of contract-order, companies always face demand fluctuations. So, they need to assume a well management of flexibility in order to guarantee their engagements with the lowest costs. The integration of flexibility levers (as overtime, interim and functioning modes) in the planning process becomes a really requirement to anticipate decisions, to offer the possibility of costs and needs evaluation and to advance flexibility exploitation, so as to better prepare their implementation.
D4 D3 D2 D1
0
10
1
50
0 10
0 15
0 20
0 30
0 40
0 00 00 10 10
inventory cost
permanent employees number 20 D4 D3 D2 D1
15 10 5
In this paper, the possibility of the management of some flexibility levers under legal constraints and under the constraint of bottleneck resource was studied. This management was starting from the long term by using an estimate of a rate model and this, for two fundamental reasons: - On the one hand, it is important for the enterprise to analyze in advance its capacity to honour its engagements for contract-orders over important horizons. The enterprise has to check for arrangements to execute these contract orders and to evaluate its risks. - On the other hand, such analysis at long term can be useful to prepare and optimize medium and short term decisions such as the recruiting policy and the operators training.
40 0 10 00 10 00 0
30 0
20 0
15 0
50
10 0
1
10
0
0
inventory cost
Regular hours 7000 6000 5000 4000 3000 2000 1000 0
D1 D2 D3 D4
0
10
1
50
60 100 150 200 300 400 600 660 800 000 1 basic cost
Interim hours 4000 3500 3000 2500 2000 1500 1000 500 0
REFERENCES
D1 D2 D3 D4
0
10
1
50
Chaabouni H., “Etude d’une politique optimale de gestion des ressources humaines dans une entreprise travaillant à capacité infinie”. Mémoire de DEA en Informatique, Faculté des Sciences de Tunis, Octobre 2001. Edieal J. Pinker, Richard C. Larson “Discrete optimisation optimising the use of contingent labor when demand is uncertain”. European Journal of Operational Research. Vol. 144, pp39-55, 2003. Hung R. “ Scheduling a workforce under annualised hours”. The International Journal of Production Research. Vol. 37, No. 11. pp 2419-2427, July 1999. Kane H., Baptiste P. and Barakato O., “Approche mathématique pour l’analyse et l’aménagement du temps de travail : cas des heures supplémentaires et d’intérim”. In : CIP’99, Tanger, November 1999, 542-548. ElBdoui-Maktouf K., “Vers un modèle d pilotage global intégrant différents leviers de flexibilité stock et capacité ”thesis in preparation, FST Oz Shy and Rune Stenbacka, “Strategic outsourcing”. Journal of Economic Behaviour & Organisation , Vol. 50.Issue 2, pp 203-224, February 2003.
60 100 150 200 300 400 600 660 800 000 1 basic cost
Overtime 140 120 100 80 60 40 20 0
80 0 10 00
66 0
60 0
40 0
30 0
20 0
15 0
10 0
60
50
10
1
0
D1 D2 D3 D4
Basic cost
Permanent employees number 20 15
D1 D2 D3 D4
10 5 0
0
1
10
50
60 100 150 200 300 400 600 660 800 000 1 Basic cost
Fig.2. A part of results
738
A NEW APPROACH OF FLEXIBILITY PLANNING A REALITY OR A SEARCH OF THE HOLY GRAIL! Khaoula El Bedoui-Maktouf1, Zied Bahroun1, Mohamed Moalla1, Jean-Pierre Campagne2 1
LIP2, Faculty of Sciences of Tunis, Tunisia 2 LIESP, INSA-Lyon France
Abstract: In the new context of “contract-order”, companies are engaged with their customers on a global program of delivery orders. Each order concerns the deliver of quantities (varying between a minimum and a maximum value) of products at precise moments. The exact quantities are known only few days before the delivery. This variability of delivery quantities needs the management in an efficient way of some flexibility levers in order to guarantee the service engagement and minimize the costs. In this paper, a new approach of flexibility planning will be presented. This approach tends to satisfy these goals and takes into account internal and external constraints of these companies. The most important internal constraint is the evidence presence of a bottleneck resource. An external constraint must be obviously satisfied which is the imposed regulations. Copyright © 2007 IFAC Keywords: contract-order, flexibility, optimisation, bottleneck resource, regulation.
1. INTRODUCTION
flexibility planning will be exposed. Finally, the evaluation of this approach is provided through some experimental results.
Nowadays, companies are faced to the need of flexibility management as they were asked to for quality ten or twenty years ago. This effective need of flexibility management is argued to be the crucial component for the surviving of the company in a complex and a difficult environment which is characterised by its variability and its uncertainty.
2. PROBLEMATIC As it was mentioned, in this new context, enterprises are dynamically and continuously looking for stability. This search of the equilibrium state is carried out by the management of different flexibility levers. Obviously, these enterprises are asked to determine which flexibility levers are to use.
In fact, to maintain a minimum level of stability, a new relationship is established between companies and their customers called “contract-order”. Enterprises exchange, then, a service guarantee against a work guarantee from costumers. The guarantee of service concerns a delivery schedule with a quantity varying between a minimum and a maximum value for each delivery and requires, so, the management of different flexibility levers.
In general framework, the choice of flexibility levers is not easy but, in the context of contract-order, the choice must provide a flexible production capacity, to deal with the variability of the demand. This production capacity is based on the human and material resources. So, enterprises must, in one hand, manage the human resource as an important mean of flexible capacity and, in the other hand, they must
In the following part, the problematic of these enterprises is presented. The literature review is then presented. After that, the proposed approach of 733
not ignore the evident role of material or machine resources in this case.
The resolution of the second and the third question is based on the distinction of two types of resources in the production system: bottlenecks and nonbottlenecks. The bottleneck resources can be either machine or manpower. In this paper, the case of bottleneck machine is considered. So, the number of basic employee needed on this bottleneck resource is not a parameter to determine but a data to use. The total number of needed employees is calculated: • from the load induced, on the rest of the production system, from the working hours of the bottleneck resource (as in most cases the load induced is calculated in number of manpower working hours), • and from the functioning mode i.e. the number of shifts to use on the bottleneck resource in order to vary its production capacity.
Both human and material resources are submitted to some constraints. In fact, the use of human resources is submitted to the regulations constraints, whereas, material resources are submitted to the presence of a bottleneck resource which determines the whole production capacity. Bottleneck resource is generally suffering from a high production load which exceeds its production capacity and is characterised by the fact that it is expensive and/or rare. Traditionally, enterprises have solved the need of flexible production capacity, essentially, by the use of human resource in overtime and/or interim hours. The use of overtime is submitted to the regulations and is very difficult to manage at long horizon. In addition, the use of interim is not evident and imposes different other constraints such as the need of training of the interim employees.
To illustrate the first point, the following example can be presented: if the amount of machine working hours on the bottleneck is x hours, then the number of induced manpower working hours on the rest of the production system is proportional to this x. The rate of proportionality is ≥1. This is due to the fact that the most bottleneck resource in the whole production system is considered and the rest of production system is supposed as non-bottleneck resource. So, the load in the bottleneck will be less than that induced on the rest of production system.
Many researches have approached the human resource management such as those of (Chaabouni, 2001; Kane et al., 1999; Edieal et al., 2003; Hung R., 1999; Oz and Rune, 2003). Both (Chaabouni, 2001) and (Kane et al., 1999) proposed a mathematical approach to the work-time management and focused on the overtime and interim treatment. (Edieal et al, 2003) represented a variety of contracts, including temporary workers, on-call workers with guaranteed minimum pay, and comp-time arrangement. The model determines the regular and contingent worker pool sizes that minimize. The bottleneck resource is mistreated.
The answer of the fourth question will be given in the last section of this paper. Besides, in this paper it is supposed that the management of human resources is imposed to some regulations, especially the use of overtime hours which is subject to many legal constraints. In our work we consider the French case and precisely we treat the annualised hours which impose to take into account the maximal limits of the global working overtime hours by week, by period of four consecutive weeks and by year.
Moreover, the treatment of the management flexibility under the presence of a bottleneck resource has not been the object of deep investigations. The proposed approach is supposed to answer the following questions: 1. How to well manage the flexibility levers in order to satisfy the costumers demand and so optimise the production planning 2. How many employees are required to meet the induced capacity need in manpower 3. How to vary the production capacity of bottleneck resource and essentially how to increase it 4. In which manner these informations can be helpful to the decision maker
Furthermore, it is very interesting to apply the management of the human resources and the bottleneck resource in the long term, since this management is not only useful for preparing the implementation in the short term, but it also contributes to bring better costs optimization. The most intuitive question is now the achievement of all such goals under all these constraints. Are we looking for the Holy Grail?
To answer to the first question, it is necessary to prepare the use of flexibility levers to really exploit them in an efficient way. So, the use of flexibility levers is integrated to the process of production planning. In others words, the proposed approach will determine the amount of overtime and/or interim hours to use in parallel to the determination of the quantities to produce to satisfy costumers demands.
3. MATHEMATICAL MODEL To formulate the proposed approach, mathematical model is developed in which the production planning is carried out on a horizon discretized in T periods. The elementary period is noted t. Let’s consider:
734
• For products: N : total number of different products i : product identifier Csi: inventory holding cost of product i per period Si0 : initial stock of product i Cri : manufacturing set-up cost of product i ri :set-up time of product i dit :demand of product i at period t pi : processing time of product i in the bottleneck
• For employees: wTotal : total number of needed employee for the whole horizon On the bottleneck resource: wGoulett: total number of employees succeeding on the bottleneck at period t Hsupt: total number of overtime hours at period t for the bottleneck H1t: number of day overtime hours of the first range per employee on the bottleneck resource and at period t H2t: number of day overtime hours of the second range per employee on the bottleneck resource and at period t HJt: total number of day overtime hours at period t on the bottleneck HNt: total number of night overtime hours at period t Hintt: total number of interim hours at period t HintJt: total number of day interim hours at period t HintNt:total number of night interim hours at period t
• For system production: : maximum number of different functioning modes m : functioning mode identifier Capm: bottleneck capacity if the functioning mode is the mode m per period Hregm : number of regular hours of the functioning mode m per period and per employee HregB : number of indebted regular hours per period and per employee Chmodn,k : changing cost from mode n to mode k Rap : the proportion coefficient between bottleneck resource and the rest of production system M
On the rest of production system: wAtelt: number of permanent employee needed on the rest of production system in regular working wAtelSt: number of permanent employee needed on the rest of production system in overtime hours HsupAtelt: total number of overtime hours induced on the rest of production system at period t H1Atelt: number of overtime hours induced on the rest of production system at period t for the first range H2Atelt: number of overtime hours induced on the rest of production system at period t for the second range HintAtelt: number of interim hours induced on the rest of production system at period t
• For employee: HsupA: maximum number of overtime hours allowed par year and per employee HsupM: maximum number of overtime hours allowed per month and per employee HsupL: limit of overtime hours per month of the first range Tb: bonus rate for the first overtime range ≤ HsupL and doing in day Tm: bonus rate for the day overtime above HsupL and doing in day Tn: bonus rate of the night overtime hours wGouletB : number of basic employees needed on the bottleneck resource pI: productivity coefficient of the interim employees Cb: cost of a regular hour CIJ: cost of interim hour in day CIN: cost of interim hour at night TRSG: the rate of service efficiency of permanent employees on the bottleneck resource TRSA: the rate of service efficiency of permanent employees on the rest of the production system Acc: the rate of employees who accept the overtime
In order to facilitate the model, we define the following variables: ChGoulett: load on the bottleneck resource at period t ChAtelt: load on the rest of production system at period t SChAtelt: overload on the rest of production system at period t The objective function to optimize (minimize) is: Min
The set of decision variables is:
N
T
∑∑
• For products: Nli: number of production orders of product i at [1..T] qit: quantity to produce of product i at period t σ it : 1 if qit >0 0 otherwise Sit: stock of product i at the end of period t
Inventory holding cost
Cs i * S it
t =1 i =1
N
+ ∑ Cri * Nl i
Manufacturing set-up cost
i =1 T
+ ∑ HregB * wTotal * Cb t =1
Permanent employee cost + T
• δmt : 1 0
∑ ((1 + Tb) * H1 + (1 + Tm) * H2
For system production: if the functioning mode at period t is m otherwise
t
t
+ (1 + Tn) * HN t )) * Cb * wGouletB
t =1
Bottleneck overtime cost + 735
T
∑ ((1 + Tb) * H1Atel
t
(13) ∀t H1t ≤ HsupL The overtime hours of the first range on the bottleneck at a period is less than the permitted limit t + 12 (14) ∀t Hsup ≤ Hsup A
+ (1 + Tm) * H2Atel t ) * Cb
t =1
Overtime cost on the rest of production system
∑
+ T
∑ (HintJt * CIJ + HintNt * CIN)* wGouletB
The sum of the overtime hours throughout the year must be less than the amount authorised by the regulation (15) ∀t ChAtelt= Rap *ChGoulett The load induced on the rest of production system is proportional of that of bottleneck ∀t SChAtelt=max(0,ChAteltTRSA*HregB*wAtelt) (16) This constraint calculate the overload on the rest of the manufacturing system ∀t SChAtelt= TRSA*HsupAtelt + pI*HintAtelt (17) The overload on the rest of production system will be covered by the use of overtime and interim hours (18) ∀t HsupAtelt=H1Atelt+H2Atelt The total number of overtime hours on the rest of the production system is the sum of the amount of the first and the second range (19) ∀t WatelSt= Acc * wAtel t
t =1
Bottleneck interim cost T
+ ∑ HintAtel * CIJ t t =1
Interim cost on the rest of production system T −1 M
+∑
M
∑ ∑ Chmod
* δ nt * δ k(t +1) )
nk
t =1 n =1 k =1
Changing-mode cost The following constraints are considered: (1) For t=1, Si1 = Si0+ qi1 – di1 (2) For t>1, Sit = Si(t-1)+ qit – dtit These constraints express the relation between stocks at successive periods of the product i M
∑δ
∀t
=1
mt
(3)
m =1
This constraint expresses the number of employees who accept overtime (20) ∀t HsupAtelt ≤ HsupM*wAtelSt The total number of overtime hours on the rest of production system at a period is less than the permitted amount by month multiplied per the number of employee accepting the overtime (21) ∀t H1Atelt ≤ HsupL*wAtelSt The total number of overtime hours of the first range on the rest of production system at a period is less than the permitted amount multiplied per the number of employee accepting the overtime t + 12 ∀t HsupAtel /wAtelS ≤ Hsup A (22)
Only one mode should be taken at a period T
∑σ
∀i = 1..N
it
(4)
= Nl i
t =1
The total number of production orders for a product i is calculated from its state. N
∀t ChGoulett=
∑( p * q i
it
+ σ it * ri )
(5)
i =1
This constraint calculate the load on the bottleneck resource M ChGoulet ≤ TRSG* ( Hreg + Hsup ) + pI * Hint (6) t
∑
m
t
t
m=1
∑
The load induced on the bottleneck at a period by products must be lower than its capacity (7) ∀t Hsupt = HJt +HNt The total number of overtime hours on the bottleneck is the sum of those doing in day and those doing at night (8) ∀t HJt = H1t +H2t The total number of overtime hours on the bottleneck doing in day is the sum of the amount of the first and the second range (9) ∀t Hintt=HintJt + HintNt The total number of interim hours on the bottleneck is the sum of those doing in day and those doing at night ∀t
M
HJt+HintJt≤max(0,416- ∑ Hreg
m
t
The total number of overtime hours of the first range on the rest of production system at a period is less than the permitted amount multiplied per the number of employee accepting the overtime ∀t
wGoulett=wGouletB*
M
∑
m *δmt
(23)
m =1
This constraint expresses the number of employee on the bottleneck (24) ∀t wTotal= wGoulett+WAtelt This constraint expresses the total number of needed employee As it is noticeable, this model is non-linear. Its nonlinearity comes from the changing mode cost and from constraints. Besides, this model deals with the case of multi-product system, the case of monoproduct is obtained from this model by considering N=i=1. Then, we assume that is not necessary to consider either a set-up time (i.e ri=0) and or a manufacturing set-up cost (i.e Cri=0).
* δ mt ) (10)
This constraint determines the amount of hours which can be done in day ∀t M HNt+HintNt≤min(304,720Hreg * δ ) (11) m
t
t
m =1
∑
t
t
mt
m =1
This constraint determines the amount of hours which can be done at night (12) ∀t Hsupt ≤ HsupM The overtime hours on the bottleneck at a period is less than the permitted amount by month 736
4. EXPERIMENTAL RESULTS
For each demand profile, the influence of every parameter of the model is analysed by varying the considered parameter and keeping constant the other parameters. The detailed results of these experimentations are furnished in ElBedoui-Maktouf 2007. A part of these results are shown in Fig.2. Notice that D1, D2, D3 and D4 correspond respectively to the fourth demand profiles presented in the beginning of this section.
The proposed approach was tested on four demand profiles. Three of them deal with the demand fluctuation. These profiles are: • Seasonal (Fig. 1-a), as f(t) = a sin (w t+ b), • Seasonal with amplitude and average perturbation (Fig. 1-b), as f(t) = (a+ d* random()) sin (w t+ b) + random() *c, • Seasonal with amplitude and average perturbation and with tendency (Fig. 1-c), as f(t) = (a+ d* random()) sin (w t+ b) + random() *c +e * t, • Arbitrary (Fig. 1-d), as f(t) = b*random(). N.B. random () is a random function. 3 30
800
3 20
700
The validation of this model is proved through its right behaviour and its coherence. The results show that: • The increase of inventory cost leads to the decrease of the stock level and to increase the production capacity of bottleneck resource by the increase of functioning mode, overtime and interim hours. Also the number of the permanent employee is increased. • the increase of the interim efficiency leads to reduce the rate of interim hours. • The increase of basic cost leads to the diminishing of the functioning mode (i. e. the regular hours) and the overtime and to the increase of the interim hours. The number of permanent employee is decreased. • The increase of the proportionality coefficient leads to the increase of the total number of permanent employees and to the increase of the induced hours on the rest of the production system and so the average of overtime and interim hours on.
600
310
500
3 00
400
2 90
300
2 80
200
2 70
100 0
2 60 1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
(b)
(a) 700 600 500 400 300 200 100 0
700 600 500 400 300 200 100 0 1
2
3
4
5
6
7
8
9
10 11 12
1
2
3
4
5
6
7
8
9
10 11 12
(d) (c) Fig.1. Types of demand profiles The software used to solve the mathematical model is LINGO, which has four different solvers. These solvers are: a direct solver, a linear solver, a nonlinear solver and a branch-and-bound manager. The demand profiles are generated by EXCEL, and are sent to LINGO to optimize the proposed mathematical model. The results are obtained then as an EXCEL table.
Other interesting points are deduced: • The demands which have seasonality and which have a high standard deviation are less sensitive to the variation of the parameters than those of low standard deviation • So the variation of the equilibrium state is low for the demand of high standard deviation than for those of low standard deviation • The arbitrary demand with high or low standard deviation has the similar comportment. • The number of permanent employee is situated in neighbourhood of the maximum estimated number • The standard deviation between the minimal estimated number and the maximum estimated number of permanent employee high for the demand with a high standard deviation and low for those with low standard deviation. • Despite of the inactivity of permanent employee in some periods, their work in the other periods is very opportune
In order to avoid essentially the problem of local optimum and to reduce the execution time of experiments, the model was linearised. The result of each experiment determines: • the optimal total number of permanent employees and for each period: • the number of permanent employees on the bottleneck • the number of permanent employees on the rest of production system • the functioning mode i.e. the number of shifts on bottleneck • the rate of overtime and interim hours on the bottleneck • the rate of overtime and interim hours on the rest of production system • the quantities to produce and so the stock of each product
Finally, it is important to note that the knowledge of the optimal number of permanent employee is very interesting for the enterprise in order to estimate its future planning. The anticipation is also very useful and helpful to the decision maker.
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5. CONCLUSION
Stock level 8000 7000 6000 5000 4000 3000 2000 1000 0
In the context of contract-order, companies always face demand fluctuations. So, they need to assume a well management of flexibility in order to guarantee their engagements with the lowest costs. The integration of flexibility levers (as overtime, interim and functioning modes) in the planning process becomes a really requirement to anticipate decisions, to offer the possibility of costs and needs evaluation and to advance flexibility exploitation, so as to better prepare their implementation.
D4 D3 D2 D1
0
10
1
50
0 10
0 15
0 20
0 30
0 40
0 00 00 10 10
inventory cost
permanent employees number 20 D4 D3 D2 D1
15 10 5
In this paper, the possibility of the management of some flexibility levers under legal constraints and under the constraint of bottleneck resource was studied. This management was starting from the long term by using an estimate of a rate model and this, for two fundamental reasons: - On the one hand, it is important for the enterprise to analyze in advance its capacity to honour its engagements for contract-orders over important horizons. The enterprise has to check for arrangements to execute these contract orders and to evaluate its risks. - On the other hand, such analysis at long term can be useful to prepare and optimize medium and short term decisions such as the recruiting policy and the operators training.
40 0 10 00 10 00 0
30 0
20 0
15 0
50
10 0
1
10
0
0
inventory cost
Regular hours 7000 6000 5000 4000 3000 2000 1000 0
D1 D2 D3 D4
0
10
1
50
60 100 150 200 300 400 600 660 800 000 1 basic cost
Interim hours 4000 3500 3000 2500 2000 1500 1000 500 0
REFERENCES
D1 D2 D3 D4
0
10
1
50
Chaabouni H., “Etude d’une politique optimale de gestion des ressources humaines dans une entreprise travaillant à capacité infinie”. Mémoire de DEA en Informatique, Faculté des Sciences de Tunis, Octobre 2001. Edieal J. Pinker, Richard C. Larson “Discrete optimisation optimising the use of contingent labor when demand is uncertain”. European Journal of Operational Research. Vol. 144, pp39-55, 2003. Hung R. “ Scheduling a workforce under annualised hours”. The International Journal of Production Research. Vol. 37, No. 11. pp 2419-2427, July 1999. Kane H., Baptiste P. and Barakato O., “Approche mathématique pour l’analyse et l’aménagement du temps de travail : cas des heures supplémentaires et d’intérim”. In : CIP’99, Tanger, November 1999, 542-548. ElBdoui-Maktouf K., “Vers un modèle d pilotage global intégrant différents leviers de flexibilité stock et capacité ”thesis in preparation, FST Oz Shy and Rune Stenbacka, “Strategic outsourcing”. Journal of Economic Behaviour & Organisation , Vol. 50.Issue 2, pp 203-224, February 2003.
60 100 150 200 300 400 600 660 800 000 1 basic cost
Overtime 140 120 100 80 60 40 20 0
80 0 10 00
66 0
60 0
40 0
30 0
20 0
15 0
10 0
60
50
10
1
0
D1 D2 D3 D4
Basic cost
Permanent employees number 20 15
D1 D2 D3 D4
10 5 0
0
1
10
50
60 100 150 200 300 400 600 660 800 000 1 Basic cost
Fig.2. A part of results
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