\$title Design of Just-in-Time Flowshops (JIT,SEQ=250) \$onText This just-in-time flowshop problem involves P products and S stages. Each stage contains identical equipment performing the same type of operation on different products. The objective is to minimize the total equipment related cost. Gunasekaran, A, Goyal, S K, Martikainen T, and Yli-Olli, P, Equipment Selection Problems in just-in-time Manufacturing Systems. Journal of the Operational Research Society 44 (1993), 345-353 Gutierrez, R A, and Sahinidis, N, A Lagrangian Approach to the Pooling Problem. Interantional J Production Research 34 (1996), 797-818. Keywords: mixed integer nonlinear programming, equipment selection, just in time manufacturing, capacity planning \$offText Set p 'products' s 'stages'; Parameter alpha(p,s) 'processing cost' beta(p,s) 'production inbalance cost' delta(p) 'demand' mu(s) 'machine cost' M 'maximum amount of money for investment' pi(p,s) 'priority weight' sigma(p,s) 'product cycles' q(p,s) 'batch size' tau(p,s) 'processing time' omega(s) 'resource requirements' bigomega 'max resource available for all machines'; Variable n(s) 'number of machines' pr(p,s) 'production rate' dpr(p,s) 'absolute value differences' obj; Integer Variable n; Equation objdef prdef(p,s) budget resource abs1 abs2; objdef.. obj =e= sum((p,s), alpha(p,s)*sigma(p,s)/pr(p,s)) + sum((p,s)\$(ord(s) < card(s)), beta(p,s)*dpr(p,s)) + sum(s, mu(s)*n(s)); prdef(p,s).. pr(p,s) =e= pi(p,s)/tau(p,s)*n(s); budget.. sum(s, mu(s)*n(s)) =l= M; resource.. sum(s, omega(s)*n(s)) =l= bigomega; abs1(p,s+1).. dpr(p,s) =g= pr(p,s) - pr(p,s+1); abs2(p,s+1).. dpr(p,s) =g= pr(p,s+1) - pr(p,s); Model jit / all /; Set p / p1*p3 / s / s1*s4 /; Parameter delta(p) / p1 3000, p2 2000, p3 4000 / mubase(s) / s1 5000, s2 5500, s3 4000, s4 6000 / omega(s) / s1 60, s2 50, s3 80, s4 40 / M / 6E6 / bigomega / 3000 /; Table dat(*,p,s) 'data from table 10.16' p1.s1 p1.s2 p1.s3 p1.s4 p2.s1 p2.s2 p2.s3 p2.s4 p3.s1 p3.s2 p3.s3 p3.s4 alpha 2.0 1.5 3.0 2.0 3.0 2.5 1.0 2.0 2.0 2.0 2.0 1.0 beta 60 90 60 80 90 80 80 70 80 100 80 90 kappa 0.2 0.3 0.4 0.3 0.1 0.4 0.3 0.2 0.2 0.3 0.2 0.2 pi 0.2 0.4 0.5 0.5 0.6 0.3 0.3 0.2 0.2 0.3 0.2 0.3 tau 1 1 1 1 1 1 1 1 1 1 1 1 q 800 800 800 800 700 700 700 700 900 900 900 900; Set c 'cases' / case1*case9 /; Parameter multq(c) / case1 1, case2 1, case3 1 case4 1, case5 1, case6 0.5 case7 1.5, case8 2, case9 2.5 / multmu(c) / case1 1, case2 0.5, case3 1.5 case4 2, case5 2.5, case6 1 case7 1, case8 1, case9 1 / rep(c,*) 'summary report'; n.lo(s) = 1; pr.lo(p,s) = .01; pi(p,s) = dat('pi',p,s); alpha(p,s) = dat('alpha',p,s); beta(p,s) = dat('beta',p,s)*1e+5; abort\$sum(s, abs(1 - sum(p, pi(p,s))) > 1e-10) 'weight do not add to 1', pi; option optCr = 0; loop(c, q(p,s) = dat('q',p,s)*multq(c); mu(s) = mubase(s)*multmu(c); tau(p,s) = q(p,s)*(dat('tau',p,s) - dat('kappa',p,s)*1e-5*mu(s)); sigma(p,s) = delta(p)/q(p,s); pr.lo(p,s) = pi(p,s)/tau(p,s); solve jit using minlp min obj; option limRow = 0, limCol = 0, solPrint = off; rep(c,s) = n.l(s); rep(c,'ModelStat') = jit.modelStat; if(jit.modelStat = %modelStat.optimal% or jit.modelStat = %modelStat.integerSolution%, rep(c,'Processing') = sum((p,s), alpha(p,s)*sigma(p,s)/pr.l(p,s)); rep(c,'Imbalance') = sum((p,s)\$(ord(s) < card(s)), beta(p,s)*abs(pr.l(p,s)- pr.l(p,s+1))); rep(c,'Investment') = sum(s, mu(s)*n.l(s)); ); ); display rep;