engineering Costs and ~rod~~t~o~ Economics, 20 ( I 990) 1I 3- 113 120 Elsevier MULTI-OBJECTIVE, MULTI-PER100 PLANNING FOR A MANUFACTURING PLANT T. Ttzaskalik iffst;tute of Econometrics, Academy of Economics, ui. Bogucicka 74,40-226 Kato~jce, Poland ABSTRACT This paper aims at introducing a formalized description of work in an industrial enterprise and using it for laying out optimal multi-period economic plans. In that description basic input and output variables, relations and connections with environments all important technological dependencies and derision functions can be taken into consideration. Industrial enterprises differ, for instance in size, assortment of production etc. We want to derive a solution useful for varied types of enterprises and that is why we introduce the concept of the manufacturing plant. A new interactive hierarchical dynamic pro- gramming technique, briefly described in the paper, is applied to the solution of the problem. An example problem is formuIated and solved with the following objectives taken into consideration: (i) minimization of utiIization of deftit materials, {ii) m~~mizat~on of production size, and (iii) maximization of profits. We consider muiti-objective, multi-period planning in the manufacturing plant as a deterministic problem. INTRODUCTION crete dynamic pro~amming, developed by Brown and Strauch [3], Klatzler (41, Mine and Fukushima [ 5 1, Yu [ 6 1. A wide review of literature in the field of multi-criteria dynamic programming was made by Li and Haimes [ 71. There are many concepts of describing mathematical aspects of planning for industrial enterprises. Trzaskalik [ I ,2] introduced the concept of the manufacturing plant. In the present article we describe the problem of choosing the optimal schedule for a manufacturing plant in the fixed space of time in the future divided into a finite number of periods. We take into consideration more than one objective and that is why we describe the problem as that of optimal multi-objective, multiperiod planning. Methods of solving problems of that sort are naturally connected with multi-criterion dis0167-l 88X/90/$03.50 0 1990-Elsevier Science Publishers B.V. MANUFACTURING PLANT We define a manufacturing plant as a set of features describing really existing enterprises. We assume, that: ( 1) The work of the manufacturing plant is considered in the fixed space of time in the future divided into T basic periods. During the considered space of time the manufacturing 114 plant leads the activities aimed at producing products of specified sorts. (2 ) Production technologies are used for laying out products. Production technology is a logical and usefully well-ordered set of operations, as a result of which we obtain a particular product. It is possible to fix the set of different kinds of products which can be manufactured and the set of production technologies which can be used in the production process in the fixed space of time. (3) Interaction between the manufacturing plant and the environment takes place. The manufacturing plant receives new production technologies, materials used in the production process and manpower from the environment and delivers manufactured products to the environment. (4) Production possibilities of the manufacturing plant are determined by an activity condition vector in each period t (t= 1, . ... T). Components of this vector concern possibilities of: using production technologies, storage of materials and products, putting new production technologies into service, buying materials, utilization of labour resources, selling manufactured products, values of technological coefficients, prices of materials and products, costs per unit of production. We consider the simplified version of the model. We assume that materials bought in a fixed period are utilized for production in the same period and products manufactured in a fixed period are sold in this period. According to these assumptions we can omit the inventory problem and reduce dimensions of the problem. In this simplified model, components of the state vector at the beginning of each period are accumulated amounts of products manufactured by the use of separate technologies before the beginning of the fixed period. Coml l l l l l l l l ponents of the decision vector are lots of products manufactured by the use of particular production technologies in the fixed period. In the fixed space of time the following objectives may be taken into consideration: (i) maximization of profits, (ii) maximization of production size, (iii) maximization of production quality, (iv) minimization of utilization of deficit materials, (v) minimization of costs. It is also possible to consider other objectives. In further considerations we accept for simplicity that each criterion for the fixed space of time is a sum of corresponding criteria for the following periods. MULTI-CRITERIA, PROGRAMMING: APPROACH DISCRETE DYNAMIC HIERARCHICAL We consider a decision process which consists of zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO T periods. Let Y, be the set of all possible state variables in period t, and X,( yt) the set of all permissible decision variables for period t and state yt~ Y,, t = zyxwvutsrqponmlkjihgfedcb 1,...,T. We assume that all these sets are finite. Let D,, the set of all period realizations in period t, be defined as o,={d,=(y,,x,):y,EY~,XrEXr(yr)) (1) We assume that for t= l,...,T there are given transformations Q,: D,p Y,. A sequence of period realizations d= (d, ,...dr) = (Y, 7x1 ,...,YT,XT) (2) is said to be a process realization if Y,+1=Q(yrJ,) Vt= l,...,T (3) Let D be the set of all process realizations. We assume that in each period t there are defined K period criteria functions F f : D, --+R (t= 1,..., T, k= l,..., K). F is the vector-valued criteria function for the whole process and its components F ’ ,...,FK are understood as Fk=Fk(F:,...,Fk,), k= l,...,K (4) 115 period structure of the problem is not taken into consideration. We describe interactive modification of this approach excluding these shortcomings. There are given: the quasi-hierarchy of multiP(d)= i FF(d,) (5) period criteria, the multi-period tolerance I= I coefficients ek for k= I,...,K and the one-peWe postulate maximization of all components riod tolerance coefficients pf for k= 1,...,K, Fk. t=l ,..., T. Let be D’=D. At the kth step Realization p is called ef~cient if zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB (k= l,...,K) the set D“ is known. We find the value 3&D F(d)&F(d*) (6) Components Fk are said to be rnu~t~-periodcriteria functions. The simplest and most important form for applications is the additive function Let D* be the set ofaIi efficient realizations of the process. Let us determine &D. The set of ail efficient realizations better than d, written as D*(d), is defined as D*(d)={d*cD*:F(d*)aF(d)) (7) One of the methods used for multi-criteria problems is the hierarchical approach, which we apply to the problem of multi-criteria, discrete dynamic programming. We assume that multi-period criteria are numerated in such a way that a more important criterion has a lower number than a less important one. The method of solution is sequential. At the first step we maximize the most important criterion using the one-criterion dynamic programming method and we get a set of realizations. Then, from the set received at the previous step, we choose those realizations which maximize a criterion of lower impo~ance, using the method of the full review. On the Kth step we get the solution of the problem. We can modify this approach introducing quasi-hierarchy of criteria. Transition from the more important multi-period criterion number k to the criterion of lower importance number k+ 1 is possible, when the value of the kth criterion reaches the satisfactory (not necessarily optimal) level calculated by using the maximum value Ek and the given tolerance coefficient ek for the kth criterion. There are two shortcomings in the approach described above. Firstly, the necessity of using the method of the full review and secondly, the P”=max{Fk(d):dcDk) (8) and the set of realizations Dk (c k, defined as b”(tk)={dkEDk:Fk(dk) > ( 1-tk)Ek} (9) From this set the decision maker chooses the model realization dk= (df,...,&) which is characterized by the best period values Ff(df). Then we generate the set Dkf ’ consisting of these realizations which are the most similar to the model realization. The set Dk+ ’ is defined as (10) Usually dK( eK) c D*, so we propose the decision maker to choose the final realization among the realizations belonging to B”(eK) and all efftcient realizations better than realizations belonging to BK( rK). If he is not satisfied, he can repeat the procedure taking other model realizations at following steps. EXAMPLE - FORMULATION FORMALIZATION AND We are interested in the work of the manufacturing plant in the fixed space of time divided into two periods. It is possible to manufacture two sorts of products: P, and PZ. Product P, can be manufactured using technologies Ti and T2 and product P2 by using technologies T3 and T+ At the beginning of the present space of time, technologies T1 and T, can be implemented and the remaining tech- 116 (iii) maximization of profits. Each multi-period criterion is the sum of the values of the corresponding one-period criteria. Characteristics Technology The problem of laying out an optimal, multi2 3 4 5 1 6(b) 7 6(a) period plan for a manufacturing plant can be described as a multi-criteria, discrete decision 0,3,5 2 1 2 0 7 8 5 T, process. Decision vector x, (called also produc0,2,4 1 1 1 10 7 8 4 T2 0,4,7 1 2 2 0 10 11 6 tion vector) for the period t ( t= 1,2 ) consists T3 0,2,4 3 2 1 10 10 11 5 T, of4 components: (1) component xlt designs lots of products P, manufactured in the period t by the use nologies must be put into service. Values of of technology T,, characteristics for technologies T, to T, are component xzI designs lots of products P1 (2) shown in Table 1. The respective columns of manufactured in the period t by the use this table describe: ( 1) lots of products which of technology T,, can be manufactured in one period; (2) lots of (3) component x3, designs lots of products P2 material M 1 per unit of production; ( 3 ) lots of manufactured in the period t by the use material M2 per unit of production; (4) lots of of technology T3. labour per unit of production; ( 5 ) costs of put(4) component x41designs lots of products P2 ting technologies into service; (6) income from manufactured in the period t by the use selling production (per unit): (a) in the first of technology T4. period; zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA (b ) in the second period; and (7 ) costs State vector yI for the period t (t= 1,2,3) conof manufacturing per unit of production. sists also of 4 components: Values of all characteristics, except character(1) component yll designs lots of products P1 istic number 6, are the same in the first and the manufactured from the beginning of the second period. process to the end of the period t - 1 by It is impossible to manufacture in one pethe use of technology T 1. riod more than 9 units of products using tech(2) component y2t designs lots of products P, nologies T1 and Tz, more than I1 units using manufactured from the beginning of the technologies T3 and T, and more than 14 units process to the end of the period t- 1 by using all technologies. Moreover, it is imposthe use of technology Pz. sible to manufacture both in the first and seccomponent y,, designs lots of products P2 (3) ond period more than 6 units of products using manufactured from the beginning of the technology T1, more than 7 units using techprocess to the end of the period t- 1 by nology TZ, more than 12 units using technolthe use of technology T3. ogy T3 and more than 10 units using technol(4) component y41designs lots of products P2 ogy T,. Minimum production of product P, in manufactured from the beginning of the one period is 4 units; the same holds for the process to the end of the period t - 1 by product PZ. In each period material M,, matethe use of technology T4. rial MZ, resources of labour and funds for preWe accept y1 = [0,zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR zyxwvutsrqponmlkjihgfedcbaZ 0, 0, O]=. Transition jiuncparing technologies are 18, 17,30 and 10 units, tions for t= 1,2 have the form: respectively. (11) Yr+,=Yr+& We consider the following objectives: For each ae[Wwe define an auxiliary function zyxwvutsr (i) minimization of utilization of deficit material MI. 1 fora>O A(a)= (12) (ii ) maximization of production size, { 0 fora< TABLE 1 117 According to the accepted assumptions we have to take into account the following constraints: l constrains for lots of products which can be manufactured in one period by the use of particular technologies: (29) Now we consider the criteria functions. Utilization of deficit material 1 for the period t is calculated according to the formula: X,,E{O,3,5) (13) ~+,2,4) ~r(x,)=2x,,+X2r+Xjr+3X4, (30) (14) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH X3*E {0,4,7) (15) ~~,~p,2,4~ (16) The multi-period utilization function for material 1 is calculated as @(x,7x2) = l additional constraints for technologies: x,t+x2,<7 (17) X3, +x42, G 11 (18) @, (XI I+ (31) @*(x2) Because we have postulated maximization of all multi-period components of the vector objective function, a new function ~‘f-u,,x2)=-@(X,,%) (32) is defined. Production size for period t is calculated according to the formula F:(x,)=5 xi, r=, XI,+x,z~6 (20) <7 (21) x21 +x22 x3, +x32 =s 12 (22) (23) .x41+x42 < 10 l constraints production: for the minimum size of 4ei,,+.x2,, 4<xx,,+x4, (24) (25) constraints for the maximum utilization of materials: l 2x,,+.x~,+x,,+3~,,,<18 (26) x,,+x~,+2.x~,+2x,,~17 (27) l constraints concerning the maximum lization of labour resources: 2x,,+.r,,+2xj,+x4,,<30 uti(28) l constraints concerning the possibility of putting new production technologies into service (33) The multi-period function of production size is calculated as follows: F’(x,,xz)=F:(x,)+F:(“~~?cz) (34) Income q(q) for one period is calculated according to the formula: (y,(~,)=7(~,, +x21)+10(*~31+x4,) (35) ~~2(~2)=8(~12 +x22) (36) + 11 (X32 +x42) Costs of production pf(xt) in the period t are calculated according to the formula: p,(X,)= 5X,,+4X,, +6X,,+ %X4,, (37) Costs of putting technologies into service y,(y,,x, ) are calculated according to the formula: +lon[~(y,l+X,,)-~(Y41)1 (38) Profit for period t can be expressed in the following terms: F:QJ,J,) =%(&)-P,(4) -V,(Y&) (39) where (xI,pt and yt are defined according to the 118 3682 (_oooo Fig. 1.The graph of the example problem. formulae (35)-( 38). The zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA multi-period profit 6) minimization of utilization of deficit mafunction is calculated as follows: terial 1, (ii) maximization of production size, F3(yl,X1,y2,XZ)=F:(y,,X,)+F:(yz,X*) (40) (iii) maximization of profit. The example problem of multi-objective, We get the following sequence of problems: multi-period planning for the manufacturing plant can be formulated as Problem 1 “Max”{[F’(d),F2(d),F3(d)lT:deD} (41) Max(F’(d):dED’} where F ‘, F2, F 3 are defined according to where (32)) (34), (40) and D is the set of realizations defined according to ( 1 1zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA )- (29 ) . D'=D The graph of the example problem is shown The graph of the process with one-period in Fig. 1. Nodes of the graph represent states, values of the first criterion is shown in Fig. 2. arcs are decisions. We obtain P ’ = 20. There are two plans corresponding to this value: APPLICATION OF THE HIERARCHICAL x,=[044O]T ~~=[3240]~ METHOD (42) Let us assume that the decision maker accepts the following quasi-hierarchy of criteria: and xz=[32401T X*=[O44O]T (43) 119 c 0000 Fig. 2. The graph of the process with one-period values of the first criterion. Fig. 3. The reduced graph of the process with one-period values of the second criterion. Fig. 4. The reduced graph of the process with one-period values of the third criterion. 120 Let us assume that the decision maker accepts the following values of coefficients: e’=O, ,ui =0.5, pi =0.25 and takes plan (42) as the model plan. Hence F 1(dt ) = - 8, F;(d:)=-12. If the decision maker accepts e3=O then this plan is the solution of the problem. It is an efficient plan. CONCLUSIONS Problem 2 Max{F2(d):deD2\ where D’={dcD’:F;(d,)>-12,F:(d,)>-15} (44) The reduced graph of the process (according to (43 ) ) with one-period values of the second criterion is shown in Fig. 3. We obtain F2 = 20. There is only one plan corresponding to this value: x,=[32401T xZ=[3440]T (45) Let us assume that the decision maker accepts the following values of coefficients: e2= 0, PU:= 0.2, $ = 0.2. It means that the plan (45) becomes the model plan. Hence Ff(d:)=9,F:(6;)=11. Problem 3 MaxjF3(d):dcD3} where D3={dcD2:F:(d,)>8.1, F:(d,)>,9.8} (46) The reduced graph of the process (according to (46) ) with one-period values of the criterion 3 is shown in Fig. 4. We obtain E3=63. The only plan corresponding to this value is _x,= [ 3 2 4 01’ xz=[34401T (47) Application of the modified hierarchical approach to multi-criteria, discrete dynamic programming is the way for effective solution of problems of this sort. This approach has been used for multi-criteria, multi-period planning for the manufacturing plant and the optimal plan corresponding to the chosen quasi-hierarchy of criteria has been obtained. REFERENCES Ttzaskalik, T., 1980. Mathematical Models of Multi-Stage Planning for Industrial Enterprise, Doctoral Dissertation, Academy of Economics Katowice. Trzaskalik. T., 1984. Optimal multi-period planning for manufacturing plant - general problems. In: A. Straszak (Ed.), Large Scale Systems - Theory and Applications, Pergamon Press. Brown, T.A. and Strauch, R.E., 1965. Dynamic programming in multiplicative lattices. Math. Anal. Appl., 12: 364370. Klotzler, R., 1978. Multi-objective dynamic programming. Math. Operationsforsch. Statist. Ser. Optimization, 9(3): 423-426. Mine, H. and Fukushima, M., 1979. Decomposition of multiple criteria dynamic programming problem by dynamic programming. Int. J. Syst. Sci., lO(5): 557-566. Yu, P.L.. 1978. Dynamic programming in finite stage multi-criteria decision problems. The University of Kansas, School of Business, Working Paper No 118, 1978. Li. D. and Haimes, V., 1988. Multi-objective Dynamic Programming: The State of The Art. VII International MDCM Conference, Manchester, August 2 l-26. (Received 1990) Augusr 1989; accepred in revised form May