# 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;