# DIST, a product distribution model

#

# References:

# Robert Fourer, David M. Gay and Brian W. Kernighan, "A Modeling Language

# for Mathematical Programming." Management Science 36 (1990) 519-554.

###  SHIPPING SETS AND PARAMETERS  ###

set whse 'warehouses';  # Locations from which demand is satisfied

set dctr 'distribution centers' within whse;

                        # Locations from which product may be shipped

param sc 'shipping cost' {dctr,whse} >= 0;

                        # Shipping costs, to whse from dctr, in $ / 100 lb

param huge 'largest shipping cost' > 0;

                        # Largest cost allowed for a usable shipping route

param msr 'minimum size restriction' {dctr,whse} logical;

                        # True indicates a minimum-size restriction on

                        # direct shipments using this dctr --> whse route

param dsr 'direct shipment requirement' {dctr} >= 0;

                        # Minimum total demand, in pallets, needed to

                        # allow shipment on routes subject to the

                        # minimum size restriction

###  PLANT SETS AND PARAMETERS  ###

set fact 'factories' within dctr;

                        # Locations where product is manufactured

param rtmin 'regular-time total minimum' >= 0;

                        # Lower limit on (average) total regular-time

                        # crews employed at all factories

param rtmax 'regular-time total maximum' >= rtmin;

                        # Upper limit on (average) total regular-time

                        # crews employed at all factories

param otmin 'overtime total minimum' >= 0;

                        # Lower limit on total overtime hours at all factories

param otmax 'overtime total maximum' >= otmin;

                        # Upper limit on total overtime hours at all factories

param rmin 'regular-time minimums' {fact} >= 0;

                        # Lower limits on (average) regular-time crews

param rmax 'regular-time maximums' {f in fact} >= rmin[f];

                        # Upper limits on (average) regular-time crews

param omin 'overtime minimums' {fact} >= 0;

                        # Lower limits on overtime hours

param omax 'overtime maximums' {f in fact} >= omin[f];

                        # Upper limits on overtime hours

param hd 'hours per day' {fact} >= 0;

                        # Regular-time hours per working day

param dp 'days in period' {fact} > 0;

                        # Working days in the current planning period

###  PRODUCT SETS AND PARAMETERS  ###

set prd 'products';     # Elements of the product group

param wt 'weight' {prd} > 0;

                        # Weight in 100 lb / 1000 cases

param cpp 'cases per pallet' {prd} > 0;

                        # Cases of product per shipping pallet

param tc 'transshipment cost' {prd} >= 0;

                        # Transshipment cost in $ / 1000 cases

param pt 'production time' {prd,fact} >= 0;

                        # Crew-hours to produce 1000 cases

param rpc 'regular-time production cost' {prd,fact} >= 0;

                        # Cost of production on regular time,

                        # in $ / 1000 cases

param opc 'overtime production cost' {prd,fact} >= 0;

                        # Cost of production on overtime, in $ / 1000 cases

###  DEMAND SETS AND PARAMETERS  ###

param dt 'total demand' {prd} >= 0;

                        # Total demands for products, in 1000s

param ds 'demand shares' {prd,whse} >= 0.0, <= 1.0;

                        # Historical demand data, from which each

                        # warehouse's share of total demand is deduced

param dstot {p in prd} := sum {w in whse} ds[p,w];

                        # Total of demand shares; should be 1, but often isn't

param dem 'demand' {p in prd, w in whse} := dt[p] * ds[p,w] / dstot[p];

                        # Projected demands to be satisfied, in 1000s

set rt 'shipping routes available' :=

 {d in dctr, w in whse:

         d <> w  and  sc[d,w] < huge  and

         (w in dctr or sum {p in prd} dem[p,w] > 0)  and

         not (msr[d,w] and sum {p in prd} 1000*dem[p,w]/cpp[p] < dsr[d]) };

                        # List of ordered pairs that represent routes

                        # on which shipments are allowed

###  VARIABLES  ###

var Rprd 'regular-time production' {prd,fact} >= 0;

                        # Regular-time production of each product

                        # at each factory, in 1000s of cases

var Oprd 'overtime production' {prd,fact} >= 0;

                        # Overtime production of each product

                        # at each factory, in 1000s of cases

var Ship 'shipments' {prd,rt} >= 0;

                        # Shipments of each product on each allowed route,

                        # in 1000s of cases

var Trans 'transshipments' {prd,dctr} >= 0;

                        # Transshipments of each product at each

                        # distribution center, in 1000s of cases

###  OBJECTIVE  ###

minimize cost:  sum {p in prd, f in fact} rpc[p,f] * Rprd[p,f] +

                sum {p in prd, f in fact} opc[p,f] * Oprd[p,f] +

                sum {p in prd, (d,w) in rt} sc[d,w] * wt[p] * Ship[p,d,w] +

                sum {p in prd, d in dctr} tc[p] * Trans[p,d];

                        # Total cost:  regular production, overtime

                        # production, shipping, and transshipment

###  CONSTRAINTS  ###

rtlim 'regular-time total limits':

    rtmin <= sum {p in prd, f in fact}

                        (pt[p,f] * Rprd[p,f]) / (dp[f] * hd[f]) <= rtmax;

                        # Total crews must lie between limits

otlim 'overtime total limits':

    otmin <= sum {p in prd, f in fact} pt[p,f] * Oprd[p,f] <= otmax;

                        # Total overtime must lie between limits

rlim 'regular-time limits' {f in fact}:

    rmin[f] <= sum {p in prd}

                        (pt[p,f] * Rprd[p,f]) / (dp[f] * hd[f]) <= rmax[f];

                        # Crews at each factory must lie between limits

olim 'overtime limits' {f in fact}:

    omin[f] <= sum {p in prd} pt[p,f] * Oprd[p,f] <= omax[f];

                        # Overtime at each factory must lie between limits

noRprd 'no regular production' {p in prd, f in fact: rpc[p,f] = 0}:

    Rprd[p,f] = 0;

noOprd 'no overtime production' {p in prd, f in fact: opc[p,f] = 0}:

    Oprd[p,f] = 0;      # Do not produce where specified cost is zero

bal 'material balance' {p in prd, w in whse}:

    sum {(v,w) in rt}

       Ship [p,v,w] + (if w in fact then Rprd[p,w] + Oprd[p,w]) =

    dem[p,w] + (if w in dctr then sum {(w,v) in rt} Ship[p,w,v]);

                        # Demand is satisfied by shipment into warehouse

                        # plus production (if it is a factory)

                        # minus shipment out (if it is a distn. center)

trdef 'transshipment definition' {p in prd, d in dctr}:

    Trans[p,d] >= sum {(d,w) in rt} Ship [p,d,w] -

                  (if d in fact then Rprd[p,d] + Oprd[p,d]);

                        # Transshipment at a distribution center is

                        # shipments out less production (if any)

###  DATA -- 3 PRODUCTS  ###

data;

set prd := 18REG 24REG 24PRO ;

set whse := w01 w02 w03 w04 w05 w06 w08 w09 w12 w14 w15 w17

            w18 w19 w20 w21 w24 w25 w26 w27 w28 w29 w30 w31

            w32 w33 w34 w35 w36 w37 w38 w39 w40 w41 w42 w43

            w44 w45 w46 w47 w48 w49 w50 w51 w53 w54 w55 w56

            w57 w59 w60 w61 w62 w63 w64 w65 w66 w68 w69 w71

            w72 w73 w74 w75 w76 w77 w78 w79 w80 w81 w82 w83

            w84 w85 w86 w87 w89 w90 w91 w92 w93 w94 w95 w96

            w98 x22 x23 ;

set dctr := w01 w02 w03 w04 w05 w62 w76 w96 ;

set fact := w01 w05 w96 ;

param huge := 99. ;

param rtmin := 0.0 ;

param rtmax := 8.0 ;

param otmin :=  0.0 ;

param otmax := 96.0 ;

param rmin  :=  w01 0.00   w05 0.00   w96 0.00 ;

param rmax  :=  w01 3.00   w05 2.00   w96 3.00 ;

param omin  :=  w01  0.0   w05  0.0   w96  0.0 ;

param omax  :=  w01 48.0   w05  0.0   w96 48.0 ;

param hd    :=  w01  8.0   w05  8.0   w96  8.0 ;

param dp    :=  w01 19.0   w05 19.0   w96 19.0 ;

param wt  :=  18REG  47.3    24REG  63.0    24PRO  63.0 ;

param tc  :=  18REG  40.00   24REG  45.00   24PRO  45.00 ;

param dt  :=  18REG 376.0    24REG 172.4    24PRO 316.3 ;

param cpp :=  18REG 102.     24REG  91.     24PRO  91. ;

param dsr := w01 96.  w02 96.  w03 96.  w04 96.  w05 96.

             w62 96.  w76 96.  w96 96. ;

param pt (tr) :

       18REG     24REG     24PRO    :=

w01    1.194     1.429     1.429

w05    1.194     1.509     1.509

w96    0.000     1.600     1.600    ;

param rpc (tr) :

       18REG     24REG     24PRO    :=

w01    2119.       2653.    2617.

w05    2489.       3182.    3176.

w96       0.       2925.    2918.   ;

param opc (tr) :

       18REG     24REG     24PRO    :=

w01    2903.     3585.     3579.

w05       0.        0.        0.

w96       0.     3629.     3622.    ;

param sc  default 99.99 (tr) :

         w01     w02     w03     w04     w05     w62     w76     w96  :=

w01      .      2.97    1.14    2.08    2.37    1.26    2.42    1.43

w02     4.74     .      4.17    6.12    7.41    3.78    7.04    5.21

w03     2.45    4.74     .      3.67    2.84    0.90    2.41    2.55

w04     1.74    5.03    2.43     .      3.19    2.45    2.69    0.58

w05     2.70    5.16    2.84    2.85     .      3.26    3.34    2.71

w06     1.99    4.17    2.13    2.19    2.52    2.06    2.00    1.51

w08     0.21    2.92    1.24    2.07    2.29    1.25    2.32    1.55

w09     0.66    3.76    1.41    2.47    1.82    1.66     .      1.87

w12     1.38    3.83    1.68    2.53    2.39     .      1.96    1.94

w14     2.47    1.58    2.40    3.59    3.85    2.25     .      3.05

w15     1.06    4.95    2.48    1.39    3.41    1.96     .      1.02

w17     0.88    3.39    1.46    2.00    2.67    1.45     .      1.46

w18     7.90    6.57    7.79    9.59    10.81    .       .      6.70

w19     1.42    4.12    1.96    1.99    3.52    1.88     .      1.26

w20     3.03    1.59    2.34    4.76    3.98    1.88     .      3.73

w24     1.58    2.80    2.27    2.87    3.19    1.31     .      2.05

w25     1.51    5.05    2.74    0.57    2.98     .      2.95    0.27

w26     1.75    3.61    2.70    1.54    4.07    3.52     .      1.03

w27     2.48    6.87    3.17    1.59    2.08    3.45     .      0.99

w28     2.05    6.83    2.97    1.13    2.91     .       .      1.26

w29     4.03    3.68    4.46    3.20    5.50     .       .      3.20

w30     2.48    5.78    2.99    2.24    1.79    3.10     .      1.39

w31     2.34    5.41    2.87    1.67    1.66     .       .      1.39

w32     14.36    .       .       .       .       .       .       .

w33     3.87    4.27    5.11    3.48    5.66    4.03     .      3.05

w34     3.26    4.80    3.21    2.70    4.14     .       .      1.77

w35     2.34    2.84    2.89    3.35    3.78    2.68     .      2.52

w36     2.43    5.69    2.96    2.95    1.02    2.61    1.07    2.54

w37     2.23    4.64    2.41    1.99    4.30    2.61     .      1.44

w38     4.66    4.36    5.23    3.04    4.46     .       .      3.82

w39     1.11    3.51    1.10    2.53    3.07    1.12     .      2.23

w40     2.99    4.78    4.23    1.57    3.92     .       .      1.80

w41     4.93    4.00    5.43    4.45    6.31     .       .      3.81

w42     3.86    6.55    5.03    2.11    4.41     .       .      2.63

w43     4.61    4.45    3.77    1.22    4.31     .       .      2.35

w44     2.05    4.48    1.06    3.70    3.46    1.10     .      3.21

w45     0.92    3.42    1.58    3.04    1.82    1.94     .      2.52

w46     1.36    2.44    0.95    3.08    2.78    0.39    2.16    2.37

w47     1.30    3.39    1.60    2.49    4.29    2.04     .      1.68

w48     1.65    3.78    1.03    2.97    2.21    1.31     .      2.74

w49     1.96    3.00    1.50    3.24    3.68    1.00     .      2.99

w50     0.90    4.14    1.60    1.95    3.61    1.61     .      1.52

w51     1.59    3.95    0.25    2.96    2.58    1.00    2.41    2.71

w53     1.59    3.79    1.28    3.12    3.10    0.89     .      2.98

w54     1.72    4.36    1.61    2.92    2.34    1.91    1.97    3.05

w55     2.45    2.73    2.21    4.47    4.30    2.57     .      4.48

w56     1.10    3.73    1.59    2.74    2.33    1.45     .      2.44

w57     0.95    3.39    1.37    2.30    2.47    1.15     .      1.95

w59     3.29    5.35    3.32    3.81    1.52    3.38    1.34    4.08

w60     2.41    6.12    2.46    3.65    2.35     .      1.37    4.06

w61     3.32    5.50    3.41    3.38    1.23     .      0.99    4.28

w62     1.12    3.00    0.82    3.22    2.95     .      3.33    2.53

w63     3.59    6.36    3.25    4.12    1.84    3.59    1.46    4.03

w64     1.85    4.45    2.17    3.43    2.13    2.03     .      4.02

w65     2.78    4.79    2.81    2.94    1.54    2.90    1.07    2.94

w66     3.90    5.79    3.05    3.65    1.36    3.39    1.22    3.57

w68     2.61    5.20    2.90    2.34    1.68    3.19    1.48    2.31

w69     2.94    5.21    2.78    3.43    0.21    3.26    0.68    2.54

w71     2.06    4.98    2.38    2.44    1.59    2.97    1.05    2.55

w72     2.61    5.50    2.83    3.12    1.35    3.23    0.88    2.99

w73     8.52    6.16    8.03    8.83    10.44   7.38    10.26    .

w74     6.11    5.46    9.07    9.38    10.80    .       .      8.25

w75     2.66    4.94    2.87    3.69    1.52    3.15    1.24    4.00

w76     1.99    5.26    2.23    3.36    0.58    3.17     .      2.50

w77     4.32    3.07    5.05    3.88    6.04     .       .      4.15

w78     5.60    2.59    5.78    5.56    7.10     .       .      5.60

w79     4.25    2.32    4.93    4.57    6.04     .       .      4.58

w80     5.94    4.00    5.60    7.02    9.46     .       .      7.51

w81     5.39    2.21    5.10    6.22    6.46     .       .      6.58

w82     8.80    5.69    9.29    9.88    11.69   8.63    11.52    .

w83     4.40     .      5.24    5.21    5.81    3.91    7.04    5.33

w84     5.87    5.43    6.17    5.70    7.63     .       .      5.70

w85     3.90    3.65    3.38    4.57    5.64    3.05     .      5.04

w86     5.48    2.10    5.70    6.37    7.33     .       .      6.19

w87     8.88    5.54    9.50    9.71    11.64   8.85    11.68    .

w89     4.62    4.01    4.03    6.30    6.30    3.81     .      7.77

w90     4.35    2.72    4.61    4.01    5.60     .       .      3.20

w91     7.61    4.42    7.83    6.85    8.79     .       .      7.66

w92     7.15    2.69    6.91    7.20     .       .       .      7.06

w93     3.17    3.95    4.37    3.74    5.05     .       .      2.40

w94     1.21    3.07    0.90    2.74    3.17     .      2.63    2.39

w95     5.82    3.29    6.55    7.06    11.47    .       .      7.83

w96     1.77    5.20    2.72    0.59    3.47    2.48     .       .

w98     3.04    1.92    3.64    3.70    4.90    3.05     .      3.88

x22     4.08    6.25    4.15    4.30    1.77     .      1.77     .

x23     3.39    5.74    3.55    4.08    1.69     .      1.47     .      ;

param msr (tr) :

         w01     w02     w03     w04     w05     w62     w76     w96    :=

w01        0       0       0       0       0       0       1       0

w02        0       0       0       0       0       0       1       0

w03        0       0       0       0       0       0       1       0

w04        0       0       0       0       0       0       1       0

w05        0       0       0       0       0       0       0       0

w06        0       1       1       1       1       1       1       1

w08        0       1       1       1       1       1       1       1

w09        0       1       1       1       1       1       0       1

w12        0       1       1       1       1       0       1       1

w14        1       1       1       1       1       0       0       1

w15        0       1       1       1       1       1       0       1

w17        0       1       1       1       1       1       0       1

w18        0       1       1       1       1       0       0       1

w19        0       1       1       1       1       0       0       1

w20        1       1       1       1       1       0       0       1

w24        0       1       1       1       1       0       0       1

w25        0       1       1       1       1       0       1       0

w26        1       1       1       0       1       1       0       1

w27        1       1       1       0       1       1       0       1

w28        1       1       1       0       1       0       0       1

w29        0       1       1       1       1       0       0       1

w30        1       1       1       0       1       1       0       1

w31        1       1       1       0       1       0       0       1

w32        0       0       0       0       0       0       0       0

w33        1       0       1       1       1       1       0       1

w34        1       1       1       0       1       0       0       1

w35        1       1       1       1       1       0       0       1

w36        0       1       1       1       0       1       1       1

w37        1       1       1       0       1       1       0       1

w38        1       1       1       0       1       0       0       1

w39        0       1       1       1       1       1       0       1

w40        1       1       1       0       1       0       0       1

w41        1       0       1       1       1       0       0       1

w42        1       1       1       0       1       0       0       1

w43        1       1       1       0       1       0       0       1

w44        1       1       1       1       1       0       0       1

w45        0       1       1       1       1       1       0       1

w46        0       1       1       1       1       0       1       1

w47        0       1       1       1       1       1       0       1

w48        0       1       1       1       1       0       0       1

w49        1       1       1       1       1       0       0       1

w50        0       1       1       1       1       1       0       1

w51        0       1       1       1       1       0       1       1

w53        1       1       1       1       1       0       0       1

w54        0       1       1       1       1       1       1       1

w55        0       1       1       1       1       0       0       1

w56        0       1       1       1       1       1       0       1

w57        0       1       1       1       1       1       0       1

w59        0       1       1       1       0       1       1       1

w60        0       1       1       1       1       0       1       1

w61        0       1       1       1       0       0       1       1

w62        0       0       0       0       0       0       1       0

w63        0       1       1       1       0       1       1       1

w64        0       1       1       1       1       1       0       1

w65        0       1       1       1       0       1       1       1

w66        0       1       1       1       0       1       1       1

w68        0       1       1       1       0       1       1       1

w69        0       1       1       1       0       1       1       1

w71        0       1       1       1       0       1       1       1

w72        0       1       1       1       0       1       1       1

w73        0       1       1       1       0       1       1       0

w74        0       1       1       1       0       0       0       1

w75        0       1       1       1       0       1       1       1

w76        0       0       0       0       0       0       0       0

w77        1       0       1       1       1       0       0       1

w78        1       0       1       1       1       0       0       1

w79        1       0       1       1       1       0       0       1

w80        1       0       1       1       1       0       0       1

w81        1       0       1       1       1       0       0       1

w82        1       0       1       1       1       1       1       0

w83        1       0       1       1       1       0       1       1

w84        1       0       1       1       1       0       0       1

w85        1       1       1       1       1       0       0       1

w86        1       0       1       1       1       0       0       1

w87        1       0       1       1       1       1       1       0

w89        1       0       1       1       1       1       0       1

w90        0       1       1       1       1       0       0       1

w91        1       0       1       1       1       0       0       1

w92        1       0       1       1       1       0       0       1

w93        1       1       1       0       1       0       0       1

w94        0       0       1       1       1       0       1       1

w95        1       0       1       1       1       0       0       1

w96        0       0       0       0       0       0       0       0

w98        1       0       1       1       1       1       0       1

x22        1       1       1       1       0       0       1       0

x23        1       1       1       1       0       0       1       0    ;

param ds default 0.000 (tr) :

         18REG     24REG     24PRO    :=

w01      0.000     0.000     0.008

w02      0.004     0.000     0.000

w03      0.000     0.000     0.000

w04      0.010     0.002     0.000

w05      0.000     0.000     0.000

w06      0.010     0.008     0.008

w08      0.030     0.024     0.024

w09      0.014     0.018     0.020

w12      0.014     0.012     0.010

w14      0.007     0.007     0.012

w15      0.010     0.019     0.018

w17      0.013     0.010     0.011

w19      0.015     0.012     0.009

w20      0.012     0.021     0.022

w21      0.000     0.000     0.000

w24      0.012     0.022     0.018

w25      0.019     0.025     0.020

w26      0.006     0.015     0.021

w27      0.008     0.010     0.015

w28      0.011     0.016     0.019

w29      0.008     0.020     0.013

w30      0.011     0.013     0.015

w31      0.011     0.013     0.017

w32      0.006     0.000     0.000

w33      0.000     0.015     0.014

w34      0.008     0.007     0.005

w35      0.002     0.006     0.014

w36      0.015     0.013     0.005

w37      0.017     0.016     0.015

w38      0.015     0.009     0.012

w39      0.007     0.017     0.022

w40      0.009     0.014     0.020

w41      0.003     0.014     0.011

w42      0.017     0.011     0.012

w43      0.009     0.013     0.011

w44      0.002     0.012     0.012

w45      0.016     0.025     0.028

w46      0.038     0.062     0.040

w47      0.007     0.010     0.010

w48      0.003     0.015     0.016

w49      0.005     0.016     0.017

w50      0.011     0.008     0.007

w51      0.010     0.022     0.021

w53      0.004     0.026     0.020

w54      0.020     0.017     0.025

w55      0.004     0.019     0.028

w56      0.004     0.010     0.008

w57      0.014     0.020     0.018

w59      0.012     0.006     0.007

w60      0.019     0.010     0.009

w61      0.028     0.010     0.012

w62      0.000     0.000     0.000

w63      0.070     0.027     0.037

w64      0.009     0.004     0.005

w65      0.022     0.015     0.016

w66      0.046     0.017     0.020

w68      0.005     0.012     0.016

w69      0.085     0.036     0.039

w71      0.011     0.013     0.010

w72      0.089     0.031     0.034

w75      0.026     0.012     0.010

w77      0.001     0.004     0.002

w78      0.002     0.004     0.002

w79      0.001     0.004     0.002

w80      0.001     0.001     0.002

w81      0.001     0.003     0.002

w83      0.009     0.010     0.008

w84      0.001     0.002     0.002

w85      0.001     0.004     0.005

w86      0.001     0.002     0.002

w87      0.002     0.003     0.000

w89      0.001     0.001     0.002

w90      0.006     0.017     0.013

w91      0.002     0.010     0.013

w92      0.000     0.003     0.002

w93      0.002     0.006     0.007

w95      0.001     0.007     0.007

w96      0.000     0.000     0.000

w98      0.006     0.005     0.002    ;

end;