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331 lines
10 KiB
331 lines
10 KiB
# PROD, a multiperiod production model
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#
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# References:
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# Robert Fourer, David M. Gay and Brian W. Kernighan, "A Modeling Language
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# for Mathematical Programming." Management Science 36 (1990) 519-554.
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### PRODUCTION SETS AND PARAMETERS ###
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set prd 'products'; # Members of the product group
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param pt 'production time' {prd} > 0;
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# Crew-hours to produce 1000 units
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param pc 'production cost' {prd} > 0;
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# Nominal production cost per 1000, used
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# to compute inventory and shortage costs
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### TIME PERIOD SETS AND PARAMETERS ###
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param first > 0 integer;
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# Index of first production period to be modeled
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param last > first integer;
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# Index of last production period to be modeled
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set time 'planning horizon' := first..last;
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### EMPLOYMENT PARAMETERS ###
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param cs 'crew size' > 0 integer;
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# Workers per crew
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param sl 'shift length' > 0;
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# Regular-time hours per shift
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param rtr 'regular time rate' > 0;
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# Wage per hour for regular-time labor
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param otr 'overtime rate' > rtr;
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# Wage per hour for overtime labor
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param iw 'initial workforce' >= 0 integer;
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# Crews employed at start of first period
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param dpp 'days per period' {time} > 0;
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# Regular working days in a production period
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param ol 'overtime limit' {time} >= 0;
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# Maximum crew-hours of overtime in a period
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param cmin 'crew minimum' {time} >= 0;
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# Lower limit on average employment in a period
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param cmax 'crew maximum' {t in time} >= cmin[t];
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# Upper limit on average employment in a period
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param hc 'hiring cost' {time} >= 0;
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# Penalty cost of hiring a crew
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param lc 'layoff cost' {time} >= 0;
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# Penalty cost of laying off a crew
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### DEMAND PARAMETERS ###
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param dem 'demand' {prd,first..last+1} >= 0;
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# Requirements (in 1000s)
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# to be met from current production and inventory
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param pro 'promoted' {prd,first..last+1} logical;
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# true if product will be the subject
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# of a special promotion in the period
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### INVENTORY AND SHORTAGE PARAMETERS ###
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param rir 'regular inventory ratio' >= 0;
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# Proportion of non-promoted demand
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# that must be in inventory the previous period
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param pir 'promotional inventory ratio' >= 0;
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# Proportion of promoted demand
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# that must be in inventory the previous period
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param life 'inventory lifetime' > 0 integer;
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# Upper limit on number of periods that
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# any product may sit in inventory
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param cri 'inventory cost ratio' {prd} > 0;
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# Inventory cost per 1000 units is
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# cri times nominal production cost
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param crs 'shortage cost ratio' {prd} > 0;
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# Shortage cost per 1000 units is
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# crs times nominal production cost
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param iinv 'initial inventory' {prd} >= 0;
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# Inventory at start of first period; age unknown
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param iil 'initial inventory left' {p in prd, t in time}
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:= iinv[p] less sum {v in first..t} dem[p,v];
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# Initial inventory still available for allocation
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# at end of period t
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param minv 'minimum inventory' {p in prd, t in time}
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:= dem[p,t+1] * (if pro[p,t+1] then pir else rir);
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# Lower limit on inventory at end of period t
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### VARIABLES ###
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var Crews{first-1..last} >= 0;
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# Average number of crews employed in each period
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var Hire{time} >= 0; # Crews hired from previous to current period
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var Layoff{time} >= 0; # Crews laid off from previous to current period
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var Rprd 'regular production' {prd,time} >= 0;
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# Production using regular-time labor, in 1000s
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var Oprd 'overtime production' {prd,time} >= 0;
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# Production using overtime labor, in 1000s
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var Inv 'inventory' {prd,time,1..life} >= 0;
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# Inv[p,t,a] is the amount of product p that is
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# a periods old -- produced in period (t+1)-a --
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# and still in storage at the end of period t
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var Short 'shortage' {prd,time} >= 0;
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# Accumulated unsatisfied demand at the end of period t
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### OBJECTIVE ###
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minimize cost:
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sum {t in time} rtr * sl * dpp[t] * cs * Crews[t] +
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sum {t in time} hc[t] * Hire[t] +
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sum {t in time} lc[t] * Layoff[t] +
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sum {t in time, p in prd} otr * cs * pt[p] * Oprd[p,t] +
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sum {t in time, p in prd, a in 1..life} cri[p] * pc[p] * Inv[p,t,a] +
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sum {t in time, p in prd} crs[p] * pc[p] * Short[p,t];
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# Full regular wages for all crews employed, plus
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# penalties for hiring and layoffs, plus
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# wages for any overtime worked, plus
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# inventory and shortage costs
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# (All other production costs are assumed
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# to depend on initial inventory and on demands,
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# and so are not included explicitly.)
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### CONSTRAINTS ###
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rlim 'regular-time limit' {t in time}:
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sum {p in prd} pt[p] * Rprd[p,t] <= sl * dpp[t] * Crews[t];
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# Hours needed to accomplish all regular-time
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# production in a period must not exceed
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# hours available on all shifts
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olim 'overtime limit' {t in time}:
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sum {p in prd} pt[p] * Oprd[p,t] <= ol[t];
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# Hours needed to accomplish all overtime
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# production in a period must not exceed
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# the specified overtime limit
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empl0 'initial crew level': Crews[first-1] = iw;
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# Use given initial workforce
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empl 'crew levels' {t in time}: Crews[t] = Crews[t-1] + Hire[t] - Layoff[t];
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# Workforce changes by hiring or layoffs
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emplbnd 'crew limits' {t in time}: cmin[t] <= Crews[t] <= cmax[t];
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# Workforce must remain within specified bounds
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dreq1 'first demand requirement' {p in prd}:
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Rprd[p,first] + Oprd[p,first] + Short[p,first]
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- Inv[p,first,1] = dem[p,first] less iinv[p];
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dreq 'demand requirements' {p in prd, t in first+1..last}:
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Rprd[p,t] + Oprd[p,t] + Short[p,t] - Short[p,t-1]
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+ sum {a in 1..life} (Inv[p,t-1,a] - Inv[p,t,a])
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= dem[p,t] less iil[p,t-1];
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# Production plus increase in shortage plus
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# decrease in inventory must equal demand
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ireq 'inventory requirements' {p in prd, t in time}:
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sum {a in 1..life} Inv[p,t,a] + iil[p,t] >= minv[p,t];
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# Inventory in storage at end of period t
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# must meet specified minimum
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izero 'impossible inventories' {p in prd, v in 1..life-1, a in v+1..life}:
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Inv[p,first+v-1,a] = 0;
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# In the vth period (starting from first)
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# no inventory may be more than v periods old
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# (initial inventories are handled separately)
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ilim1 'new-inventory limits' {p in prd, t in time}:
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Inv[p,t,1] <= Rprd[p,t] + Oprd[p,t];
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# New inventory cannot exceed
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# production in the most recent period
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ilim 'inventory limits' {p in prd, t in first+1..last, a in 2..life}:
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Inv[p,t,a] <= Inv[p,t-1,a-1];
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# Inventory left from period (t+1)-p
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# can only decrease as time goes on
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### DATA ###
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data;
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set prd := 18REG 24REG 24PRO ;
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param first := 1 ;
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param last := 13 ;
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param life := 2 ;
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param cs := 18 ;
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param sl := 8 ;
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param iw := 8 ;
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param rtr := 16.00 ;
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param otr := 43.85 ;
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param rir := 0.75 ;
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param pir := 0.80 ;
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param : pt pc cri crs iinv :=
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18REG 1.194 2304. 0.015 1.100 82.0
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24REG 1.509 2920. 0.015 1.100 792.2
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24PRO 1.509 2910. 0.015 1.100 0.0 ;
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param : dpp ol cmin cmax hc lc :=
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1 19.5 96.0 0.0 8.0 7500 7500
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2 19.0 96.0 0.0 8.0 7500 7500
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3 20.0 96.0 0.0 8.0 7500 7500
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4 19.0 96.0 0.0 8.0 7500 7500
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5 19.5 96.0 0.0 8.0 15000 15000
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6 19.0 96.0 0.0 8.0 15000 15000
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7 19.0 96.0 0.0 8.0 15000 15000
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8 20.0 96.0 0.0 8.0 15000 15000
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9 19.0 96.0 0.0 8.0 15000 15000
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10 20.0 96.0 0.0 8.0 15000 15000
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11 20.0 96.0 0.0 8.0 7500 7500
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12 18.0 96.0 0.0 8.0 7500 7500
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13 18.0 96.0 0.0 8.0 7500 7500 ;
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param dem (tr) :
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18REG 24REG 24PRO :=
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1 63.8 1212.0 0.0
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2 76.0 306.2 0.0
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3 88.4 319.0 0.0
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4 913.8 208.4 0.0
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5 115.0 298.0 0.0
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6 133.8 328.2 0.0
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7 79.6 959.6 0.0
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8 111.0 257.6 0.0
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9 121.6 335.6 0.0
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10 470.0 118.0 1102.0
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11 78.4 284.8 0.0
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12 99.4 970.0 0.0
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13 140.4 343.8 0.0
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14 63.8 1212.0 0.0 ;
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param pro (tr) :
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18REG 24REG 24PRO :=
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1 0 1 0
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2 0 0 0
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3 0 0 0
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4 1 0 0
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5 0 0 0
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6 0 0 0
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7 0 1 0
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8 0 0 0
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9 0 0 0
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10 1 0 1
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11 0 0 0
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12 0 0 0
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13 0 1 0
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14 0 1 0 ;
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end;
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