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