# Model PROD from June 1989 version of CSTR 133
# This model determines a series of workforce levels that will most
#economically meet demands and inventory requirements over time. The
#formulation is motivated by the experiences of a large producer in the
#United States. The data are for three products and 13 periods.
#### 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