Reliable Optimal Production Control with Cobb-Douglas Model

Transcription

1 Relible Computing 4: 63 69, c 998 Kluwer Acdemic Publishers. Printed in the Netherlnds. Relible Optiml Production Control with Cobb-Dougls Model ZHIHUI HUEY HU Texs A&M University, College Sttion, TX 7784, USA, e-mil: zhuey@tmu.edu (Received: 3 December 995; ccepted: 6 Februry 997) Abstrct. Production is the most fundmentl ctivity in our economy. In this pper, Cobb-Dougls production function is used s the mthemticl model to describe the reltionship mong production, lbor nd cpitl. Two relible production optiml control problems re studied. Algorithms to find dynmic optiml control intervls re provided with intervl prmeter presenttions nd intervl computtions.. Optiml Production Control with Cobb-Dougls Model: Trditionl (Non-Intervl) Cse Production. Every dy, vrious products re produced to meet different demnds in our society. Production is truly one of the most fundmentl ctivities in our economy. Every producing firm wnts to mximize its profits: In n equilibrium mrket, where the mount produced is more or less fixed (by the demnd nd by the firm s mrket shre), in order to mximize profits, the firm needs to minimize production costs. In seller s mrket, in which the supply of product is smller thn the demnd for it, mximizing profits mens producing the mximum mount within the vilble production costs. Production function. In both cses, to optimize production, we must know how the output Q depends on the production costs. The dependence of Q on controllble prmeters is clled production function. One of the most widely used production functions ws proposed by Cobb nd Dougls nd hs the following form (see, e.g., [3]): Q = A L α K β, () where L is lbor (mesured in certin units), K is cpitl, nda, α, ndβre (constnt) prmeters; these prmeters depend on the firm, on the produced unit, etc., nd hve to be determined experimentlly. Comment. In mny rel-life situtions, α + β =. This equlity hs simple economic interprettion: if we increse both lbor nd cpitl twofold, we will thus

2 64 ZHIHUI HUEY HU produce twice s mny units of the product. This dditionl ssumption mkes the corresponding formuls simpler. Cost function. The production cost C consists of the cost of lbor + the cost of cpitl: C = L + b K, (2) where is the cost of unit of lbor, nd b is the cost of cquiring unit of cpitl. Depending on the mrket, we hve one of the following two optimiztion problems: Optimiztion problem for the equilibrium mrket: Given: A, α, β,, b, ndq. Minimize: C = L + b K. Subject to: Q = A L α K β. Optimiztion problem for the seller s mrket: Given: A, α, β,, b, ndc. Mximize: Q = A L α K β. Subject to: C = L + b K. How we cn solve these problems. Both conditionl optimiztion problems cn be solved by using the Lgrnge multiplier method (see, e.g., [7]). In both cses, we cn even get n explicit expression for the solution. Lgrnge Multiplier method reduces both problems to the sme (unconditionl) optimiztion problem: A L α K β λ ( L + b K) mx, where λ is the (unknown) Lgrnge multiplier. We cn simplify the problem by dividing the objective function by (positive) constnt A: L α K β µ ( L + b K) mx, where we denoted µ = λ / A. Differentiting the new objective function w.r.t. ech of the vribles L nd K, nd equting these derivtives to 0, we get the following equtions: α L α K β = µ ; (3) β L α K β = µ b. (4) To eliminte the uxiliry prmeter µ from this system, we divide (3) by (4), thus getting: α β K (5) b nd

3 RELIABLE OPTIMAL PRODUCTION CONTROL WITH COBB-DOUGLAS MODEL 65 L b β α. (6) Substituting this expression for K in terms of L into the corresponding condition, we get, in both cses, n (esily solvble) eqution from which we cn determine the optiml mount of lbor L. From this expression, using (6), we cn get n explicit expression for the optiml mount of cpitl K: Optiml control for the equilibrium mrket: ( ) Q / (α +β) ( α A β b ) β / (α +β) ; (7) ( ) Q /(α+β) ( β A α ) α /(α+β). (8) b Comment. As we hve mentioned, when α + β =, these equtions tke n even simpler form: ( Q A ) ( Q A) ( α β b ) β ; (7) ( ) β α. (8) α b Optiml control for the seller s mrket: α α + β C ; (9) β α +β C b. (0) These formuls cn lso be simplified if α + β =: αc ; (9) β C b. (0) 2. A More Relistic (Intervl) Cse Why intervls? If we know ll the prmeters exctly, then it mkes sense to try to follow recommendtions (7) (8) (or (9) (0)) precisely. In rel life, the prmeters A, α, nd βof the Cobb-Dougls model, the perunit costs nd b, the required production level Q nd the vilble totl cost C re only pproximtely known. At best, we know intervls of possible vlues of these prmeters: A =[A,A], α =[α,α], β =[β,β], =[,], b =[b,b], nd Q =[Q,Q](orC=[C,C]).

4 66 ZHIHUI HUEY HU Intervl control. Different vlues of the prmeters A,,from the given intervls A,,led to different optiml vlues of lbor nd cpitl. In such situtions, it is resonble to supply the decision-mker not with pir of numbers (L, K), but with intervls [K,K] nd[l,l] of possibly optiml vlues of K nd L. The decision mker will then use his experience nd intuition to select the vlues K nd L from these intervls. To estimte the desired intervls K nd L, we suggest to use (nive) intervl computtions [], [5], [6], i.e., to replce ech opertion with rel numbers by the corresponding opertion with intervls. To implement intervl computtion on computer, we used the INTLIB librry [2]. Optiml intervl control for the seller s mrket. Formuls (9) nd (0) tht correspond to the seller s mrket cn be implemented in strightforwrd wy: α C ; (9 ) β C b. (0 ) In these formuls, every vrible occurs only once nd therefore, the results of intervl computtions coincides with the exct intervls of possible vlues of K nd L (see, e.g., [], [6]). In the formuls (9) nd (0), the vrible α occurs both in the numertor nd in the denomintor; this cn cuse n overestimtion. To void it, we rewrite α /(α +β) s / ( + β / α). If we pply nive intervl computtions to thus re-written formul, we get the following expressions: +(β /α ) C ; (9 ) +(α /β ) C b. (0 ) In these formuls, every vrible occurs only once nd therefore, the results of intervl computtions lso coincides with the exct intervls of possible vlues of K nd L. Optiml intervl control for the equilibrium mrket. The implementtion of the formuls (7), (8), (7), nd (8) requires some extr work: Nmely, since INTLIB does not contin the intervl function to-the-power x y, we first hve to represent expressions of this type s exp(y ln(x)): { exp α+β ( ln(q) ln(a) ) β + α + β ( ln(α) ln(β)+ln(b) ln() ) } ; (7 )

5 RELIABLE OPTIMAL PRODUCTION CONTROL WITH COBB-DOUGLAS MODEL 67 { exp α+β ( ln(q) ln(a) ) + α α + β ( ln(β) ln(α)+ln() ln(b) ) } ; (8 ) nd only then pply intervl computtions: { exp α +β ( ln(q) ln(a) ) { exp + (α / β )+ ( ln(α ) ln(β )+ln(b) ln() ) } ; (7 ) α +β ( ln(q) ln(a) ) + +(β /α ) ( ln(β ) ln(α )+ln() ln(b) ) }. (8 ) These formuls cn lso be simplified if α + β =: Q A exp {β ( ln(α ) ln(β )+ln(b) ln() ) }; Q A exp {α ( ln(β ) ln(α )+ln() ln(b) ) }. (7 ) (8 ) 3. Optiml Control in Dynmiclly Chnging Environment Formultion of the problem. In rel life, decisions re mde in dynmiclly chnging environment. For exmple, innovtions cn mke the production process less lbor-intensive; cpitl cn be become more esily vilble, etc. As result, the prmeters A, α, β, etc., cn chnge. If we hve been using some vlues L nd K from the optiml intervls, nd the sitution hs chnged, then we wnt to check whether these vlues L nd K re still possibly optiml (i.e., whether they still belong to the new optiml intervls L nd K). If both vlues L nd K re still possibly optiml, we do not need to chnge nything in our production. If t lest one of the vlues L nd K is outside the corresponding new optiml intervl, then we need to choose new vlues for L nd K. Seller s mrket: nive intervl computtions re sufficient. For the seller s mrket, the bove formul (9 ), (0 ), (9 ), nd (0 ) givetheexct intervl of possibly optiml vlues. So, for the seller s mrket, to check whether the old prmeters L old nd K old re still possibly optiml, it is sufficient to: use the new vlues of A,, to compute the new optiml intervls L new nd K new ;nd

6 68 ZHIHUI HUEY HU check whether L old L new nd K old K new. Equilibrium mrket: we need new lgorithm. For equilibrium mrket, the corresponding formuls (7 ) nd(8 ) cn overestimte. So: if L old L new or K old K new, then we definitely need to chnge the production prmeters; but if L old L new nd K old K new, it does not necessrily men tht the old prmeters re still possibly optiml: it cn be tht they re no longer possibly optiml, but they nevertheless belong to the overestimted intervls (7 ) nd (8 ). To check whether chnge is needed or not, we, therefore, need new lgorithm. Equilibrium mrket: towrds new lgorithm. We re given the intervls A, α,,q, nd the (old) vlues K nd L. We need to check whether these vlues re still possibly optiml, i.e., where there exist A A, α α,,q Q for which these K nd L re optiml, or, equivlently, for which the equtions () nd (5) re both true. Let us describe these conditions () nd (5) in terms of α nd β. If we pply logrithm to both sides of the eqution (), we conclude tht ln(l) α +ln(k)β =ln(q) ln(a). The coefficients t α nd β in the left-hnd side re known constnts. Since ln(x) is strictly incresing, the right-hnd side tkes its smllest possible vlue when Q tkes its smllest possible vlue nd A tkes the lrgest, i.e., when Q = Q nd A = A. Similrly, the right-hnd side tkes the lrgest possible vlue when Q = Q nd A = A. Thus, the existence of Q nd A for which this equlity is true is equivlent to the following double inequlity: ln(q) ln(a) ln(l) α +ln(k)β ln(q) ln(a). () The eqution (5) cn be re-written s b = α β K L. For given α nd β, stisfying this equlity mens tht its right-hnd side must be within the intervl of possible vlues of the rtio /b,i.e.(since nd b re positive), within the intervl [ / b, / b]: / b α β K L / b. If we multiply both sides of ech inequlity by β, we get two equivlent inequlities tht re liner in α nd β: ( / b) β K α; (2) L

7 RELIABLE OPTIMAL PRODUCTION CONTROL WITH COBB-DOUGLAS MODEL 69 K α ( / b) β. (3) L Finlly, the conditions tht α α nd β β cn lso be represented in terms of liner inequlities: α α α; (4) β β β. (5) Thus, K nd L re possibly optiml if nd only if the system of liner inequlities () (5) hs solution. The existence of such solution cn be esily checked by liner progrmming methods. Hence, we rrive t the following method: Equilibrium mrket: new lgorithm. Given: (old) vlues L nd K, nd new intervls A, α, β,, b,ndq. To check whether the vlues L nd K re possibly optiml, we must pply liner progrmming to check whether system of inequlities () (5) is consistent (i.e., hs solution). Acknowledgements The uthor wishes to cknowledge Professor Vldik Kreinovich of the Computer Science Deprtment t the University of Texs t El Pso. His encourgement nd suggestion of pplying intervl computtions mde this pper possible. The uthor lso wnts to thnk the nonymous referees for their vluble suggestions. References. Alefeld, G. nd Herzberger, J.: Introduction to Intervl Computtions, Acdemic Press, New York, Kerfott, R. B., Dwnde, M., Du, K., nd Hu, C.: Algorithm 737: INTLIB: A Portble Fortrn-77 Intervl Stndrd-Function Librry, ACM Trns. Mth. Softwre 20 (4) (994), pp Lncster, K.: Introduction to Modern Micro Economics, Rnd McNlly & Co., Chicgo, Mings, T.: The Study of Economics: Principles, Concepts & Applictions, Dushkin Publ., Guilford, CT, Moore, R. E.: Intervl Arithmetic nd Automtic Error Anlysis in Digitl Computing, Ph.D. Disserttion, Stnford University, Moore, R. E.: Methods nd Applictions of Intervl Anlysis, SIAM, Phildelphi, Stewrt, J.: Clculus, Brooks/Cole Publ. Co., Monterey, CA, 99.