Abstract: In this paper the concepts of physical system theory and mathematical programming are jointly considered to model multi-stage manufacturing systems.
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EuropeanJournal of Operational Research47 (1990) 248-261 North-Holland
Technology selection models for multi-stage production systems: Joint application of physical system theory and mathematical programming N. S I N G H
Department of Industrial Engineering, University of Windsor, Windsor, Ontario, Canada N9B 3P4 SUSHIL
Centre for Management Studies, H T Delhi, Hauz Khas, New Delhi 110016, India
Abstract: In this paper the concepts of physical system theory and mathematical programming are jointly considered to model multi-stage manufacturing systems. Each stage has a number of alternative technologies. The objective of the present modelling approach is to select an appropriate technology at each stage to minimize the total cost of production subject to continuity, and budget constraints. The resulting non-linear 0-1 programming model is linearized and illustrated by a simple three stage (having alternative technologies at each stage) manufacturing example. A goal programming model is also developed and solved. Keywords: Technology selection, mathematical programming, production systems, physical system theory
1. Introduction Manufacturing is one of the most important issues in the world today. The modelling of manufacturing systems helps in analyzing various planning and operating decisions. Most manufacturing activities invariably involve multi-stages. And each stage constitutes a technology. For example, items requiring turning operations can be manufactured on any of the following machines: engine lathe, turret lathe, single spindle automatic lathe, multiple spindle automatic lathe, numerically controlled turning machine, etc. Each machine represents a technology with its distinct characteristics in terms of quality, cost of production, purchasing cost, etc. Since each stage in a multi-stage manufacturing situation has alternative technologies to choose from, the problem of selecting a technology at each stage becomes complex because of interactions between stages and budgetary constraints. Some attempts have been made in the literature to address the problem of technology selection. In the case of manufacturing, Satin and Chen [11] develop a mixed integer programming model for technology selection considering technology on an aggregate basis. Singh and Rajoria [13] addressed the problem of technology selection in the case of the Marble Industry. However, these models do not capture the realities of the multi-stage production where each stage poses a technology selection problem and where there is interaction between stages as output from the previous stage becomes input to the successive stage. Also each technology at every stage has different scrap rates, different compatibilities with other technologies and certainly different budget requirements besides continuity constraints. ReceivedOctober 1989 0377-2217/90/$3.50 © 1990 - ElsevierSciencePublishers B.V.(North-Holland)
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Also a large number of papers has appeared in literature on modelling of multi-stage manufacturing systems. Davis and Kennedy [2] presented an interesting modelling approach using Markov methodology, Singh and Sushil [17] used physical system theory to model manufacturing systems. However, none of the existing papers considers the selection of alternative technologies at each stage for multi-stage production systems considering interactions among the stages. In this paper we develop simple models based on the concept of physical system theory and mathematical programming that may be used for technology selection in multi-stage manufacturing systems. The concepts of physical system are outlined. A non-linear 0-1 programming model to minimize the cost of production treating the technology as 0-1 variable is developed subject to the constraints on budget and continuity. The 0-1 non-linear model is linearized and illustrated by a simple three stage serial manufacturing system having three alternative technologies at the first stage, two at the second and two at the third using hypothetical data. A goal programming formulation is also proposed and solved for the illustrative example.
2. Technology selection problem In this section we discuss the technology selection problem in the context of a multi-stage manufacturing system by considering a small but representative example. A generalized model is then developed in subsequent sections using the concepts of physical system theory and mathematical programming. Consider, for example, a three stage serial manufacturing system designed to manufacture a part requiring three operations, that is, turning, milling and drilling at the first, second and third stage respectively. At each stage, the parts not meeting the specifications are scrapped. Suppose, three alternate technologies were available for turning at the first stage: Engine Lathe (EL), Turret Lathe (TL), and Single Spindle Automatic Lathe (SSAL); two for milling at the second stage: Universal Milling Machine-Ordinary Grade I (UMOG) and Universal Milling Machine-Supergrade II (UMSG); two for drilling at the third stage: Pillar Drilling Machine (PDM) and Radial Drilling Machine (RDM). The process capabilities, processing costs and budget requirements for machines at each stage are different. Moreover, in a serial manufacturing system the output from a stage becomes input to a subsequent stage. Accordingly, the parts manufactured in sequence on EL, UMOG and PDM will have different yield (production output), rejects and unit output cost than being manufactured on TL, UMOG and PDM or SSAL, UMSG and RDM. There are twelve such possibilities for this problem. Further, there are constraints on the total investment that can be made to acquire these technologies as well as on the raw material availability. Therefore, the management is faced with the problem of selecting an appropriate set of technologies. The selection criteria could either be based on unit cost minimization or a multi-objective approach. For this purpose, we develop generalized non-linear 0-1 programming and goal programming models using the concepts of physical system theory (PST). PST allows to model elemental components separately and computes the interaction which logically results from network theoretic considerations. Thus, a system theory model is custom tailored to fit each individual system modelled [12]. The PST modelling approach can, therefore, be usefully applied to model multi-stage production systems since each production stage can be conceived as a component. Upon joining the components (stages) in a one-to-one correspondence to form a system network, the production system characteristics can be studied considering the interaction between stages.
3. Physical system theory concepts Physical system theory [7,8] is based on the notion of linear graphs. The fundamental postulates of physical system theory are as follows:
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3I. Singh and Sushil / Technology selection models for multi-stage production systems
(i) Component postulate: The performance characteristics of a N-terminal component can be completely and uniquely defined by a component terminal graph of N - 1 edges and a set of N - 1 equations relating N - 1 'across' and N - 1 'through' variables. (ii) System postulate: On uniting component terminal graphs in a 1 : 1 correspondence with actual inter-relations of the components, the system graph is obtained. (iii) Interconnection postulate. (a) Continuity law: The law states that material is conserved in the system, i.e., the algebraic sum of flow rates into vertices vanishes identically. (b) Compatibility law: The cost evaluated of whatever path is followed between any two vertices will be the same, i.e., the algebraic sum of the cost vectors for a closed circuit vanishes. Further, the sum of the scalar product of the across and through variables into the vertices vanishes identically. This is known as the principle of conservation. The product of across and through variables represent the dollar flow rate. The different components of the system under consideration are conceptualized to perform any one, or a combination, of the following basic processes: (i) Transformation process: The transformation process can be defined as a transformation of resources to achieve a well-defined change in their physical, chemical, technological, biological or functional characteristics. (ii) Transportation process: This can be viewed as a special type of transformation process wherein material is simply moved from one geographic location to another at a cost. This includes collection, translocation and distribution of various kinds of materials. (iii) Storage process: This can be viewed as a special type of transformation process in which the input and output are identical in form, and the resources are carried over time. The focus of the present study is on the material transformation process taking place within multi-stage manufacturing systems. The multi-stage production system is identified as a collection of interconnected components and the approach taken is that of component to system construction. Each component is modelled in terms of two complementary variables; a through variable Y, the flow of parts; an across variable X, the unit cost of parts production. The theoretical model of technology selection based on the three postulates of physical system theory is developed in the next section. It includes cost equations, flow equations, continuity equations and compatibility equations. The concepts and principles of physical system theory have been applied at macro level for modelling of national economic systems (Sushil [19]) and (Satsangi and Ellis [12]) and at micro level to project crashing (Kanda [6]) and productivity analysis in manufacturing systems (Sushil, Singh and Jain [20]).
4. Physical system theory model We develop a technology selection model for multi-stage manufacturing systems. Consider a serial production system with i stages, i = 1, 2 . . . . . m, and each stage having j number of alternate technologies, j = 1, 2 . . . . . n i, for the i-th stage. We first use the component postulate of PST to model the transformation process, Pij, corresponding the j-th alternative at the i-th stage. We define technological coefficients to represent the input requirements (or waste generated) per unit of output. The units not meeting the specifications are scrapped and considered as waste. We assume that there is one type of input, one type of output and one type of waste generated at the transformation process P~j. Accordingly: Technological coefficient of input at P~j: k,j = (Input at
Pij)/(Output from P,j).
(1)
Technological coefficient of waste at P~j: k,'~ = (Waste generated at P , j ) / ( O u t p u t from
eij).
(2)
N. Singh and Sushil / Technology selection models for multi-stage production systems
Input ~ C
~
~
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Output ~
0
aste
Figure 1. Free-body diagram of the transformation process
The free-body diagram of the transformation process P,j is shown in Figure 1. Accordingly, the component terminal equations for P~j are: (i) Material flow equations: These equations follow from the very definition of technological coefficients.
Yi,= kijYi~,
(3) (4)
Y,.; = k,'~.~, where ~ • material output of process P~j (superscript o designates output), Y,j, Y,~ : input required and waste generated respectively at P~j, k,j, k,'~ : technological coefficients of input and waste respectively at P,j.
(ii) Cost equation: This follows from the principal of conservation that the dollar inflow rate equals dollar outflow rate at the transformation process Pij. That is, (Dollar value of inputs) + (Dollar value of processing costs) = (Dollar value of output) + (Dollar value of waste). Therefore,
or
x', 3 = k . X .
-
k,3 x,y
(5)
where X,~ Xij X,~ f~j(Y,~)
: : : :
unit cost of output of process Pit cost per unit material input at Pij, salvage value per unit waste at Pij, processing cost per unit output as a function of output for process P,j.
~j(Y,~) accounts for labour, capital, overheads and also takes care of scale of economies, f,j can, however, be assumed to be constant at P~j and independent of Y~ if the scale of economies is not considered. We make this assumption in this paper. Using the second and third postulate of PST, the component of various transformation processes P~j (i = 1, 2 . . . . . m; j = 1, 2 . . . . . n,) can be united to obtain the system graph which will be governed by the constraints following the principle of conservation with respect to unit flow rates and dollar flow rates. However, to simplify the treatment, we start with a three stage system having only one technology at each stage. The component models for this system are given in Appendix A. Note that the edges in the component models are assigned unique numbers for ease of representation. These component models are then united in a one-to-one correspondence to obtain a system graph given in Appendix A with accompanying governing equations. The generalized equations for unit output cost and input material requirements as a function of technology variable A,j are developed in Appendix B and Appendix C using the results obtained in Appendix A. The technology variable A~j
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(i = 1, 2 ..... m; j = 1, 2 ..... ni) is defined as a 0-1 variable to represent the j-th technology at the i-th stage of process P,j. Its value is one if j-th technology is selected, zero otherwise. In the technology selection model, our objective is to select only one technology at each stage such that the unit output cost is minimized. Accordingly, the unit output cost as a function of A,j is given (for details, see Appendix A and B) below: m
nl
m
m
ni
hi- 1
z : I1 E A,+k,+X,+ E FI E A,/.,j E A,-1,j(f,-1,j - k w,-l,jx, w-,.j) i~l j=l
1=2 i=l j=l
j=l
~m
+ E Amj(/.j- ~Lx~j),
(6)
j=l
where X 1 is the incoming material cost per unit at the first stage ( X 1 = X11 = X12 . . . . .
Xln,).
The incoming material requirement for given finished product is given by 'through' variables at different stages solving one equation at a time and moving backwards (for derivation, see Appendix C). Accordingly, we have,
0
