Manufacturing Planning by Network Flow Model with Side Constraints

Copyright © IFAC CIM in Process and Manufacturing Industries. Espoo. Finland. 1992

MANUFACTURING PLANNING BY NETWORK FLOW MODEL WITH SIDE CONSTRAINTS M.F. CARVALHO, C.A.O. FERNANDES and 0.5. SILVA FR. Fundaqao Centro Tecnoldgico para Informatica. Instituto de Automaqao. Hodelos Hatematicos . C. P. 6162 . CEP 13081. Campinas. SP. BRAZIL

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Abstract . This paper deals wi th planning and schedul ing problems of mul tiproduct. multiperiod. multistage in a production or assembly lines of a flexible manufacturing systems . Here it is considered that decisions of weekly goals in terms of demand and been taken by a higher planning level as Material raw material have alread y Requirement Planning system . The aim of the model is to disaggregate the weekly production requirements to daily production requirements taken into account the availability of material . machine resource. storage capacity. attempting to meet the daily demand requirement and maximizing the revenue. It will be shown that one part of the problem is represented by a very special structure called " replicated network and another part is represented as linear constraints . These particularities are exploited by the solution technique that use a network flow with addi tional side constraints algorithm . An example illustrates the modeling and solution of a manufacturing planning problem by means of this approach . Key words . Manufacturing Process. Production Optimization. Network Flow. Operations Research. Transportation. Manufacturing Planning. Flexible Manufacturing System .

The middle term encompasses decisions typically made by the FHS supervisor over a time horizon of several days or week. Since the FMS is a part of a large manufacturing environment. the primary objective here is the refinement of the end-part plan established by Master Production Schedule (MPS). In this sense. the Material Requirement Planning (HR?) occupies a very important function. For each end-part. the HR? determines. working in a backward from due date. times and quantities to be produced or ordered of the components . Some criticisms about this procedure are not taken into account the current resource capacity of the real process. the uncertainties of demand requirements. random events (e.g. machine failure) and scarcities of raw materials . In other words. for the HR? to became an efficient plan is necessary that the actual demand. production resources. and suppliers occur. as close as possible. with the inputs used to establish the HR? In fact. this is very difficult to occur in daily operation.

1. INTRODUCTION The modern production processes try to follow the different demand requirements by implementing Flexible Manufacturing System (FMS)' An FMS consists of a set of work stations capable of performing a number of different operations. interconnected by a transporta t ion mechani srn and controlled by a network of computers (Kimemia and Gershwin. 1983). The parts to be produced by the FMS have similar operational requirements or belong to the same final assemble. Any decision to allocate resources for one workpart affects the resources availability to produce all other workpart. Then. in this type of production system. an important problem is the best utilization of resources. during the planning horizon. in such way to meet the demand requirement with competitive cost . quality. and in accordance with due date . It can be assured by an appropriated planning. scheduling. control and monitoring strategies (Kusiak. 1986). The resources can be divided in production resources and raw material resources. both of them have different availability through the horizon . As pointed out above the production resources as well as the raw material resources can be disputed by more than one workpart with totally different characteristic, as for example. the quanti ty per parts. processing time. production cost. revenues. etc. The simultaneous consideration of all these factors brings complexity for the adequate manageaent of the production . The quanti ty of inforaation to be processed . the level of detail necessary for representing the actual systems and the tiae scale over which the decisions must be made. require a multi-level decision hierarchy (Gershwin. 1986) . The long term decision involves capital expenditure and are taken at the top of decision hierarchy The uncertainties of inforaation are very high and the decisions provide "rough-cut" capacity requirements which are used by the top manager in order to determine the impact of production plans on the plant capacity.

Another way to analyze the manufacturing system is working forward from the resources availability (which can be known exactly or by forecasting) to the demand . taking into account costs . storage capacities and level of backorders . In this approach. the manufacturing process can be seen as the flow of parts. entering into loading station following into the machines. conveyors etc . and leaving the system at an unloading station after undergoing a specific sequence of operations . This problell can be modeled as network flow (Gaimon. 1986). If the line is flexible enough to process more than one part. the production resources have to be shared among the parts . One way to plan this type of prodUction system is as mul tiflow model (Gaimon.1992)' In addition. if the temporal components requirements that compose the end-part. are considered. then the resulting model is a multiflow with additional side constraints . In this paper the last case will be analyzed. In the following sections are described the important characteristics of multlproduct process .

169

The required decisions at this level are:

. In the section 3 the mathematical model is formulated, while in section 4 the algori thm of solution is presented . An illustrative example is solved in section 5 and finally some thoughts and remarks are offered as a conclusion.

(a) daily productions that meet the weekly demand requirement; (b) loading machine at each time interval of each part ; (c) The level of work-in-process (WIP); (d) The level of storage for each time , each stage and each part; (e) The level of backorder. (fl The level of storage of each part, for each stage; (g) The needs of raw material and components for each period of time, at each stage; (h) and the level of demand attainment for each stage of each part .

2. SYSTEM DESCRIPTION The multiproduct, multistage and multiperiod planning problem of a production system of a discrete manufacturing process will be presented . In this problem, each fabrication or assemble stage is composed of one or more machine groups where the capacity of each group is determined by the sum of individual machine production rates. The production rates of each machine group can be different over the planning horizon, in reason of contemplating the failure or repair of the individual machines. Each item or end-part has a previous defined process sequence. Two or more i terns can share the same process sequence or can have their own sequence . The time horizon is divided in discrete unit period of time, like hour, turn or day, and for each period is established a demand goal, based on the demand expectation that can be or not totally met . Backorder is allowed until a specified level.

3. TIlE MODEL

The manufacturing planning problem is modeled by a network flow with additional linear constraints. The network models the flow of parts through the production lines and storage facilities. The additional linear constraints denotes the bundle capacity constraints and the balance equations relating components and parts . The notation used here is given as follow:

processing cost is assoc i ated to each machine, for each product , each stage and sometimes each time period. The raw material has also a cost while the demand has sales price . If an item leaves the stage 5 it can follow immediately to the stage 5+1 or can be stored for processing in the subsequent period . In the last case is associated a inventory carrying cost and a storage capacity.

K

For the production of each final item or end-part, a main production line and adjacent productions lines are considered . Through the main line flow the parts while into the adjacent lines flow the components to be attached to each part, at specific points of the main line to form a final item . It is allowed to define the number of components per parts . The representation of adjacent lines requirements is necessary to prevent the creation of superfluous interstage stock, due to unbalanced f low of sets of parts wi th sets of components . In this way, one of the most important manufacturing production planning function is to coordinate the flow between the main line and the adjacent lines in order to synchronize the production and permi t eff icient and timely assembly of the end-part. The Fig. 1 exhibit the features, discussed above .

1

A

T S Q r

a,t

w

k,l.q

C

k,l,_

a,t

~,t,q d

a,t

Ya, t,q Pa,t.,q

V

k.l._

z a,t n

a,q

Xa,_

set of indices for the itens or end-parts; set of indices for the periods ; set of indices for productions stages; set of indices for components ; per unit sales price for the part k at period t; per unit storage cost for the part k at period t and injection point q; per unit processing cost for the part k at period t and stage s ; per unit backorder cost for the part k at period t; per unit purchase cost of part (or component) k at period t and injection point q ; demand of end-part k at period t; storage level of component q at period t; purchase level for the component k at period t and injection point q; processing level for the part k at period t and stage s; storage level for the part k at period and stage s; backorder level of part k at period t; number of components q per part k maximum production capacity of stage s at tilae t; weekly demand of product k; identity matrix

3.1. Objective Function It Is composed of two main terms : the revenue due to demand .et and the production cost . The last cost is represented by the purchase cost, the processing cost, the storage cost. and the backorder cost all over the time horizon.

o o

Fd(k)-F (k)-F (k)-F (k)-Fb(k) } p c w

Operation

(1 )

The first term of (1) is associated with demand attainaent . Although the goal is to meet the daily demand, It can be expressed In two forms :

Queue

• total demand attainment; and • optimal demand attainaent . In the first case if there exist enough raw material. components and production capacity. the demand can be completely met by setting a sales

Fig . 1: Flow of parts and components

170

price "r", large enough as big-M method of linear programming . In the last case the level of demand supplied will be determined by the sum of individual demand supply at a profit . If in a given period of time the sales price is less than the total production cost (purchase plus production cost) then it is allowed demand shedding. The general form of this term is: T

I

r

Kt t

t=1

o

1

I

p

0

K, t

Q

T

I

I h

t =l

q=1

k ,t ,q

P

~

:s

x

~

:s

v

~

:s

d

S

T

I

I

8=1

t=1

(k)

c

k , t , q

X

Kt t , s k

t

t. s

1< . t 1< . t

:s

X

:s

v

:s

d

:s

2

(11 )

s=l ... .. 5 ; and k=1. . ... K

The matrix form of equations (7-10) can be written as :

(3 )

k,t,s

(10)

t ••

t=1 . .... T;

A

1

x

0

A

0

2

0

(4)

N

N

N

U

U

U

1

k , t , s

1

b

1

....-

2

1

2

1

b

1

v

A

---.-~, --

c

X

the capacity constrain t s per product :

Z

The third term in (1 ) is as s oci ated wit h t he production cost (ene r gy + work- fo rce + return t o the investment + maintenance ) . I ts general form is: F

:s

k. t, s

t=l •... • T s=l • ... • S

The second term in (1) i s associated with the purchase cost of part or components that belongs to the final product (end-part) . Remark that part denotes a component flow through the main line . On the other hand, components flow through the adjacent line until joint to the part at main line (see Fig. 1). Its general form is:

F (k)

X

1<=1

( 2)

d

the bundle capacity con s traints :

2

Y

N

~ +1 \

Z

b

3

0 (12)

The fourth term in (1) is associated with the inventory cost of parts or components . The model has considered the parts belonging to the main line as q = 1, and letting the q = 2,3, ... Q for the adjacent lines . Its general form is : T

Q

F

.

(k)

I q=1

I w t=1

(S)

Y k,t,q

The dimension of each matr ix A is ((S+1l-T+2. (2-T-1)(S+1l+T] and U is (T-S~ T-S] matrix that contemplate the bundle constraints. The N1matrix considers the relation between part and components and the dimension of each one depends on the application. It is expected not to be large when comparing to dimension of A. As an example. for one microel e c tronics. the main line has twelve operations and for adjacent lines . In other words. there are four points where the different lines match each other .

k , t , q

The last term of (1) is related to backorder level . The total demand lateness for each part k is defined as :

the cost

4. SOLUTION (6)

The final problem (1)-(12) has a especial structure call ed Network F low With Addi tional Side Constraints (NFASC) (Kenn ington. 1980 ) that must be exploited by the solut ion technique . Any base for this problem can be expressed by:

3.2. Constraints : The constra i nts are composed of two main groups : the balance constraints which is formed by balance equations of components, demand and parts . The other group is concerned with bundle capacity constraints and individual machine capa c it y constraints per parts . Each one of them can be mathematically written as follow :

B

n

-

k,t,q - l

k,q

x

k,t,q

- y

+

k. l ,q

P

k , l,q

= 0

( 13)

ii G

where B is a basic matrix corresponding to a K trees each one related to the network of each product . The variables associated with Bare called key variables . The remaining variables are related with C and F and are called non-key 1 variables (Kennington. 1980) . The iiinverse of matrix ii is given by :

• the raw material and components constraints : y

TECHNIQUE

(7)

the demand constraints :

o

T

I tel

d

It,t

:s

D

It

, k=l, ... , K

(8)

(14)

• the balance equations : where the work base W is defined as W = F - G x

k,t.a-t

+v

k,t,a-l

-x

k.t , &

-v

t . t,s

a- 1 c.

0 The aatrix denoted by C has at most two non-zero elements by column and the elements of matrix G are placed on well defined positions . These aatrices do not need to be stored explicitly .

(9)

k=l, . . .. K t=l . .... T s=l .... . 5

171

PRODUCT 1

The updating of the a-lcan be performed from 8 and W. The efficiency of the algorithm lies precisely in how to gen~[ate, store, and work the necessary elements of B . This paper is using the algorithm NETS I DE, a special code for network with side constraints developed by Kennington (1990), but the authors are working in a dedicated algorithm that will exploit the special structure of manufacturing system . S. APPLICATION

To illustrate the basic characteristics of these manufacturing systems, Fig . 2 presents a system composed of two stages, two products that share the same production resources and production sequence within planning horizon of f i ve days . The availability of raw material for the time horizon comes from a higher level planning (such as MRP) and is denoted by RH, the daily maximum purchase is p where t is associated with time period (t=1~2~ : ~S), k is associated with the final product (k = 1, 2) and q is associated with the entrance points of the components int o the production (or assembly) system . For ea ch stage there exist a bundle production capac it y constraint that can be different for each period of time , simulating the failur e or repair of machines . As an example , Fi g . 2 shows that the availability of machines at period 4 and stage one , X. 1 is equal to 1, while at stage 2 and period 4 is equal to 4 . The weekly demand, D can be disaggregated in daily goals . The d . i~ the maximum demand of part k, to be suppli~Ot at time t, at sale price rk,t

2

..

.4 ·

.4 ,

-

2

4 '

-

2

4 .

c-

MATERIAl (AM)

.4 ' - - 9I- - 1 4I \ \, j I 1"",,\

---

f-

~~

13

-:,:,. 16-

/' ./

2_- 7 - U

!

I

r- 02

1.·"/

1 - - 6 - - 11,'

Fig . 3 : Network scheme of the Fig . 2 Tables 1 and 2 present the data input for the system of Fig . 2 and time horizon of three periods . The bundle capacity constraint of the machine group at stage 1. is of 4 units per time period and have to be shared between the two product . The availability of components at the injections points (bold arrows at Fig . 3) are :

.

5

~l,l.l 8 ~1,2.1 "' Pl , 3.1

"'

6

Each end-part needs two components and each end-part 2 needs one component . The unit production cost at stage 1 and 2 are equal to 1 and the unit bacltorder cost is 1. 2 for the time horizon and each end part . The results are presented at Table 3 where the column t is related to the time interval. The column s is related to the stages and stage 0 corresponds to the raw material. the stage 1 and two to the productions stages while stage 3 represents demand attainment . Due to bundle capacity constraints, the results show one unit of bacltorder of part 1 at time t-1, supplied at tl.e t-2 .

...

4

4

4 '

4

4

[)oN

4

,-r- I

43

:4

;2

RAW

.. j --10--1:5\ I

i

MG2

MG1

2.

PRODUCT 2

Product 2

TABLE 1: Input Data of product 1

' 4

t

-1

Product 1 Fig . 2: Two items Process

Figure 3 shows the network representation associated with Fig . 2, where the daily demand and the daily raw aaterial(RH) availability are represented by arcs . Each horizontal arc represents the production rate for each stage (aachine group) and ti.e interval, while vertical arc represents the storage availability . The arrows indicate the injection points of components that participate of the two end-part, that occurs only at first stage . The demands of the optimization period Dw are considered as a negative injections and are disaggregated for each time interval.

product 1 raw purchase IIaX material cost demand

sale cost 10 .

3

1.

2

2

1.

1.

10 .

3

3

1.2

3.

10.

2.

TABLE 2 : Input Data of product 2 product 2 t

-1

172

raw purchase aax aaterial cost demand

sale cost

4

1.

2.

8.

2

3

1.

3.

8.

3

2

1.5

2.

8.

TABLE 3: Production Scheduling product 2

product 1 t

s

production storage backorder

0 1

2

3

1 2 3

production storage backorder 3. 3. 2. 2.

1. 1. 1.

2.

3.

0 1 2 3

3. 3. 3. 1.

2. 2.

0

2. 2. 2. 3.

O. 2. 2. 2.

1

2 3

1. 1.

1.

6. CONCLUSION This paper has presented a new modeling and solution technique for planning and scheduling problem of a multiproduct, multiperiod, multistage of a production or assembly lines of a flexible manufacturing system. The aim of the model is to disaggregate the weekly goals in term of demand and raw materials in daily requirements (hour, turn, day). It was shown that a set of the constraint of the problem has a network structure while another set is represented by additional linear constraints with elements located in well-defined positions . The solution algorithm used was a Netside, a generative technique that exploit the special structure of the problem . An example shows the particularities of the modeling and solution technique. Acknowledgment : The authors would I ike to thank Dr. J. L. Kennington and Dr . Alan Whisman by permission to use the program Netside . This work was supported by CNPq-Brazil under Grant 501237/91 7. REFERENCES

Gaimon, C. Optimal Inventory, Backlogging and Machine Loading in a Serial Multi-stage, Multiperiod Production Envirorunent , Int. J. Prod. Res. NQ 3, 647-662 . Gershwin, S. F.; Hlldebrant, R. R. ; Suri, R.; Mitter. S.K . (1986) - Control Perspective on Recent Trends in Manufacturing Systems. IEEE Control System Magazine. Vol . 6, NQ 2, 3-15. Ghosh. S. ; and Gaimon. C. (1992) Routing Flexibility and Production Scheduling in a Flexible Manufacturing System With Setup, European J . of Cp . Research. 60/2. 344-366 . Kennington. J . L. and Helgason. R. V. (1980)Algorithms for network Programming. John Wiley & Sons. New York. Kennington. J.L. and Whisman A. (1990) Netside User's Guide. Department of Computer Science and Engineering. Southern Methodist University. Dallas . Kiaenia. J . ; Gershwin. S. S. (1983) - An Algorithm for the Computer Control of Production in Flexible Manufacturing System. lEE Trans . • Vol . 15. NQ 4. 353-362. Kusiak. A. (1986) - Application of Operational Research Models and Techniques in Flexible Manufacturing Systems. European Journal of Operation Research. 24. 336-345 . Rodaamer. F. A. and Write. K. P. (1988) - A Recent Survey of Production Scheduling. IEEE TRANS. on Syst . Man and Cybernet .• Vol . 18. NQ 6. 841-851 .

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1.

2.