# EGYPT, a static model of fertilizer production
#
# References:
# Robert Fourer, David M. Gay and Brian W. Kernighan, "A Modeling Language
# for Mathematical Programming." Management Science 36 (1990) 519-554.
### SETS ###
set center; # Locations from which final product may be shipped
set port within center; # Locations at which imports can be received
set plant within center; # Locations of plants
set region; # Demand regions
set unit; # Productive units
set proc; # Processes
set nutr; # Nutrients
set c_final; # Final products (fertilizers)
set c_inter; # Intermediate products
set c_ship within c_inter; # Intermediates for shipment
set c_raw; # Domestic raw materials and miscellaneous inputs
set commod := c_final union c_inter union c_raw;
# All commodities
### PARAMETERS ###
param cf75 {region,c_final} >= 0;
# Consumption of fertilizer 1974-75 (1000 tpy)
param fn {c_final,nutr} >= 0;
# Nutrient content of fertilizers
param cn75 {r in region, n in nutr} := sum {c in c_final} cf75[r,c] * fn[c,n];
# Consumption of nutrients 1974-75 (1000 tpy)
param road {region,center} >= 0;
# Road distances
param rail_half {plant,plant} >= 0;
param rail {p1 in plant, p2 in plant} :=
if rail_half[p1,p2] > 0 then rail_half[p1,p2] else rail_half[p2,p1];
# Interplant rail distances (kms)
param impd_barg {plant} >= 0;
param impd_road {plant} >= 0;
# Import distances (kms) by barge and road
param tran_final {pl in plant, r in region} :=
if road[r,pl] > 0 then .5 + .0144 * road[r,pl] else 0;
param tran_import {r in region, po in port} :=
if road[r,po] > 0 then .5 + .0144 * road[r,po] else 0;
param tran_inter {p1 in plant, p2 in plant} :=
if rail[p1,p2] > 0 then 3.5 + .03 * rail[p1,p2] else 0;
param tran_raw {pl in plant} :=
(if impd_barg[pl] > 0 then 1.0 + .0030 * impd_barg[pl] else 0)
+ (if impd_road[pl] > 0 then 0.5 + .0144 * impd_road[pl] else 0);
# Transport cost (le per ton) for:
# final products, imported final products,
# interplant shipment, imported raw materials
param io {commod,proc}; # Input-output coefficients
param util {unit,proc} >= 0;
# Capacity utilization coefficients
param p_imp {commod} >= 0; # Import Price (cif US$ per ton 1975)
param p_r {c_raw} >= 0;
param p_pr {plant,c_raw} >= 0;
param p_dom {pl in plant, c in c_raw} :=
if p_r[c] > 0 then p_r[c] else p_pr[pl,c];
# Domestic raw material prices
param dcap {plant,unit} >= 0;
# Design capacity of plants (t/day)
param icap {u in unit, pl in plant} := 0.33 * dcap[pl,u];
# Initial capacity of plants (t/day)
param exch := 0.4; # Exchange rate
param util_pct := 0.85; # Utilization percent for initial capacity
### DERIVED SETS OF "POSSIBILITIES" ###
set m_pos {pl in plant} := {u in unit: icap[u,pl] > 0};
# At each plant, set of units for which there is
# initial capacity
set p_cap {pl in plant} :=
{pr in proc: forall {u in unit: util[u,pr] > 0} u in m_pos[pl] };
# At each plant, set of processes for which
# all necessary units have some initial capacity
set p_except {plant} within proc;
# At each plant, list of processes that are
# arbitrarily ruled out
set p_pos {pl in plant} := p_cap[pl] diff p_except[pl];
# At each plant, set of possible processes
set cp_pos {c in commod} := {pl in plant: sum {pr in p_pos[pl]} io[c,pr] > 0};
set cc_pos {c in commod} := {pl in plant: sum {pr in p_pos[pl]} io[c,pr] < 0};
set c_pos {c in commod} := cp_pos[c] union cc_pos[c];
# For each commodity, set of plants that can
# produce it (cp_pos) or consume it (cc_pos),
# and their union (c_pos)
### VARIABLES ###
var Z {pl in plant, p_pos[pl]} >= 0;
# Z[pl,pr] is level of process pr at plant pl
var Xf {c in c_final, cp_pos[c], region} >= 0;
# Xf[c,pl,r] is amount of final product c
# shipped from plant pl to region r
var Xi {c in c_ship, cp_pos[c], cc_pos[c]} >= 0;
# Xi[c,p1,p2] is amount of intermediate c
# shipped from plant p1 to plant p2
var Vf {c_final,region,port} >= 0;
# Vf[c,r,po] is amount of final product c
# imported by region r from port po
var Vr {c in c_raw, cc_pos[c]} >= 0;
# Vr[c,pl] is amount of raw material c
# imported for use at plant pl
var U {c in c_raw, cc_pos[c]} >= 0;
# U[c,pl] is amount of raw material c
# purchased domestically for use at plant pl
var Psip; # Domestic recurrent cost
var Psil; # Transport cost
var Psii; # Import cost
### OBJECTIVE ###
minimize Psi: Psip + Psil + Psii;
### CONSTRAINTS ###
subject to mbd {n in nutr, r in region}:
sum {c in c_final} fn[c,n] *
(sum {po in port} Vf[c,r,po] +
sum {pl in cp_pos[c]} Xf[c,pl,r]) >= cn75[r,n];
# Total nutrients supplied to a region by all
# final products (sum of imports plus internal
# shipments from plants) must meet requirements
subject to mbdb {c in c_final, r in region: cf75[r,c] > 0}:
sum {po in port} Vf[c,r,po] +
sum {pl in cp_pos[c]} Xf[c,pl,r] >= cf75[r,c];
# Total of each final product supplied to each
# region (as in previous constraint) must meet
# requirements
subject to mb {c in commod, pl in plant}:
sum {pr in p_pos[pl]} io[c,pr] * Z[pl,pr]
+ ( if c in c_ship then
( if pl in cp_pos[c] then sum {p2 in cc_pos[c]} Xi[c,pl,p2] )
- ( if pl in cc_pos[c] then sum {p2 in cp_pos[c]} Xi[c,p2,pl] ))
+ ( if (c in c_raw and pl in cc_pos[c]) then
(( if p_imp[c] > 0 then Vr[c,pl] )
+ ( if p_dom[pl,c] > 0 then U[c,pl] )))
>= if (c in c_final and pl in cp_pos[c]) then sum {r in region} Xf[c,pl,r];
# For each commodity at each plant: sum of
# (1) production or consumption at plant,
# (2) inter-plant shipments in or out,
# (3) import and domestic purchases (raw only)
# is >= 0 for raw materials and intermediates;
# is >= the total shipped for final products
subject to cc {pl in plant, u in m_pos[pl]}:
sum {pr in p_pos[pl]} util[u,pr] * Z[pl,pr] <= util_pct * icap[u,pl];
# For each productive unit at each plant,
# total utilization by all processes
# may not exceed the unit's capacity
subject to ap:
Psip = sum {c in c_raw, pl in cc_pos[c]} p_dom[pl,c] * U[c,pl];
# Psip is the cost of domestic raw materials,
# summed over all plants that consume them
subject to al:
Psil = sum {c in c_final} (
sum {pl in cp_pos[c], r in region}
tran_final[pl,r] * Xf[c,pl,r]
+ sum {po in port, r in region} tran_import[r,po] * Vf[c,r,po] )
+ sum {c in c_ship, p1 in cp_pos[c], p2 in cc_pos[c]}
tran_inter[p1,p2] * Xi[c,p1,p2]
+ sum {c in c_raw, pl in cc_pos[c]: p_imp[c] > 0}
tran_raw[pl] * Vr[c,pl];
# Total transport cost is sum of shipping costs for
# (1) all final products from all plants,
# (2) all imports of final products,
# (3) all intermediates shipped between plants,
# (4) all imports of raw materials
subject to ai:
Psii / exch = sum {c in c_final, r in region, po in port}
p_imp[c] * Vf[c,r,po]
+ sum {c in c_raw, pl in cc_pos[c]} p_imp[c] * Vr[c,pl];
# Total import cost -- at exchange rate --
# is sum of import costs for final products
# in each region and raw materials at each plant
### DATA ###
data;
set center := ASWAN HELWAN ASSIOUT KAFR_EL_ZT ABU_ZAABAL ABU_KIR TALKHA SUEZ ;
set port := ABU_KIR ;
set plant := ASWAN HELWAN ASSIOUT KAFR_EL_ZT ABU_ZAABAL ;
set region := ALEXANDRIA BEHERA GHARBIA KAFR_EL_SH DAKAHLIA DAMIETTA
SHARKIA ISMAILIA SUEZ MENOUFIA KALUBIA GIZA BENI_SUEF FAYOUM
MINIA ASSIOUT NEW_VALLEY SOHAG QUENA ASWAN ;
set unit := SULF_A_S SULF_A_P NITR_ACID AMM_ELEC AMM_C_GAS C_AMM_NITR
AMM_SULF SSP ;
set proc := SULF_A_S SULF_A_P NITR_ACID AMM_ELEC AMM_C_GAS CAN_310 CAN_335
AMM_SULF SSP_155 ;
set nutr := N P205 ;
set c_final := UREA CAN_260 CAN_310 CAN_335 AMM_SULF DAP SSP_155 C_250_55
C_300_100 ;
set c_inter := AMMONIA NITR_ACID SULF_ACID ;
set c_ship := AMMONIA SULF_ACID ;
set c_raw := EL_ASWAN COKE_GAS PHOS_ROCK LIMESTONE EL_SULFUR PYRITES
ELECTRIC BF_GAS WATER STEAM BAGS ;
set p_except[ASWAN] := CAN_335 ;
set p_except[HELWAN] := CAN_310 ;
set p_except[ASSIOUT] := ;
set p_except[KAFR_EL_ZT] := ;
set p_except[ABU_ZAABAL] := ;
param cf75 default 0.0 :
CAN_260 CAN_310 CAN_335 AMM_SULF UREA :=
ALEXANDRIA . . 5.0 3.0 1.0
ASSIOUT 1.0 20.0 26.0 1.0 27.0
ASWAN . 40.0 . . .
BEHERA 1.0 . 25.0 90.0 35.0
BENI_SUEF 1.0 . 15.0 1.0 20.0
DAKAHLIA 1.0 . 26.0 60.0 20.0
DAMIETTA . . 2.0 15.0 8.0
FAYOUM 1.0 . 20.0 6.0 20.0
GHARBIA . . 17.0 60.0 28.0
GIZA . . 40.0 6.0 2.0
ISMAILIA . . 4.0 6.0 2.0
KAFR_EL_SH 1.0 . 10.0 45.0 22.0
KALUBIA . . 25.0 16.0 7.0
MENOUFIA 1.0 . 24.0 21.0 30.0
MINIA 2.0 15.0 35.0 1.0 41.0
NEW_VALLEY . . . . 1.0
QUENA . 95.0 2.0 . 3.0
SHARKIA 1.0 . 31.0 50.0 28.0
SOHAG . 65.0 3.0 . 7.0
SUEZ . . 1.0 . .
: SSP_155 C_250_55 C_300_100 DAP :=
ALEXANDRIA 8.0 . . .
ASSIOUT 35.0 5.0 .1 .
ASWAN 8.0 . . .
BEHERA 64.0 1.0 .1 .1
BENI_SUEF 13.0 3.0 . .
DAKAHLIA 52.0 1.0 . .
DAMIETTA 5.0 . . .
FAYOUM 17.0 1.0 . .
GHARBIA 57.0 1.0 .2 .1
GIZA 14.0 1.0 .1 .
ISMAILIA 4.0 . . .
KAFR_EL_SH 25.0 2.0 .1 .
KALUBIA 22.0 1.0 . .1
MENOUFIA 33.0 2.0 .1 .1
MINIA 50.0 3.0 .2 .1
NEW_VALLEY 1.0 . . .
QUENA 8.0 . . .
SHARKIA 43.0 1.0 .1 .
SOHAG 20.0 1.0 . .
SUEZ 1.0 . . . ;
param fn default 0.0 : N P205 :=
AMM_SULF .206 .
CAN_260 .26 .
CAN_310 .31 .
CAN_335 .335 .
C_250_55 .25 .055
C_300_100 .30 .10
DAP .18 .46
SSP_155 . .15
UREA .46 . ;
param road default 0.0 :
ABU_KIR ABU_ZAABAL ASSIOUT ASWAN HELWAN KAFR_EL_ZT SUEZ TALKHA :=
ALEXANDRIA 16 210 607 1135 244 119 362 187
ASSIOUT 616 420 . 518 362 504 527 518
ASWAN 1134 938 518 . 880 1022 1045 1036
BEHERA 76 50 547 1065 184 42 288 120
BENI_SUEF 359 163 257 775 105 248 270 261
DAKAHLIA 208 138 515 1033 152 58 219 3
DAMIETTA 267 216 596 1114 233 131 286 66
FAYOUM 341 145 308 826 88 230 252 243
GHARBIA 150 65 485 1003 122 20 226 55
GIZA 287 48 372 890 .9 133 169 146
ISMAILIA 365 142 536 1054 173 241 89 146
KAFR_EL_SH 145 105 525 1043 162 20 266 35
KALUBIA 190 97 439 957 76 66 180 81
MENOUFIA 157 154 472 990 109 33 213 90
MINIA 384 288 132 650 230 372 394 386
NEW_VALLEY 815 619 199 519 561 703 726 717
QUENA 858 662 242 276 604 746 769 760
SHARKIA 240 60 473 991 110 78 214 58
SOHAG 715 519 99 419 461 603 626 617
SUEZ 370 224 541 1059 178 246 . 298 ;
param rail_half default 0 :
KAFR_EL_ZT ABU_ZAABAL HELWAN ASSIOUT :=
ABU_ZAABAL 85 . . .
HELWAN 142 57 . .
ASSIOUT 504 420 362 .
ASWAN 1022 938 880 518 ;
param : impd_barg impd_road :=
ABU_ZAABAL 210 .1
ASSIOUT 583 0
ASWAN 1087 10
HELWAN 183 0
KAFR_EL_ZT 104 6 ;
param io default 0.0 :=
[*,AMM_C_GAS] AMMONIA 1.0
BF_GAS -609.
COKE_GAS -2.0
ELECTRIC -1960.
STEAM -4.
WATER -700.
[*,AMM_ELEC] AMMONIA 1.0
EL_ASWAN -12.0
[*,AMM_SULF] AMMONIA -.26
AMM_SULF 1.0
BAGS -22.
ELECTRIC -19.
SULF_ACID -.76
WATER -17.
[*,CAN_310] AMMONIA -.20
BAGS -23.
CAN_310 1.0
LIMESTONE -.12
NITR_ACID -.71
STEAM -.4
WATER -49.
[*,CAN_335] AMMONIA -.21
BAGS -23.
CAN_335 1.0
LIMESTONE -.04
NITR_ACID -.76
STEAM -.4
WATER -49.
[*,NITR_ACID] AMMONIA -.292
ELECTRIC -231.
NITR_ACID 1.0
WATER -.6
[*,SSP_155] BAGS -22.
ELECTRIC -14.
PHOS_ROCK -.62
SSP_155 1.0
SULF_ACID -.41
WATER -6.
[*,SULF_A_P] ELECTRIC -75.
PYRITES -.826
SULF_ACID 1.0
WATER -60.
[*,SULF_A_S] ELECTRIC -50.
EL_SULFUR -.334
SULF_ACID 1.0
WATER -20. ;
param util default 0 :=
[*,*] SULF_A_S SULF_A_S 1 SULF_A_P SULF_A_P 1
NITR_ACID NITR_ACID 1 AMM_ELEC AMM_ELEC 1
AMM_C_GAS AMM_C_GAS 1 SSP SSP_155 1
C_AMM_NITR CAN_310 1 C_AMM_NITR CAN_335 1
AMM_SULF AMM_SULF 1 ;
param p_imp default 0.0 :=
PYRITES 17.5 AMM_SULF 75.
EL_SULFUR 55. DAP 175.
UREA 150. SSP_155 80.
CAN_260 75. C_250_55 100.
CAN_310 90. C_300_100 130.
CAN_335 100. ;
param p_r default 0.0 :=
ELECTRIC .007
BF_GAS .007
WATER .031
STEAM 1.25
BAGS .28 ;
param p_pr default 0.0 :=
[HELWAN,COKE_GAS] 16.0
[ASWAN,EL_ASWAN] 1.0
[*,LIMESTONE] ASWAN 1.2
HELWAN 1.2
[*,PHOS_ROCK] ABU_ZAABAL 4.0
ASSIOUT 3.5
KAFR_EL_ZT 5.0 ;
param dcap default 0.0 :=
[ABU_ZAABAL,*] SSP 600
SULF_A_P 227
SULF_A_S 242
[ASSIOUT,*] SSP 600
SULF_A_S 250
[ASWAN,*] AMM_ELEC 450
C_AMM_NITR 1100
NITR_ACID 800
[HELWAN,*] AMM_C_GAS 172
AMM_SULF 24
C_AMM_NITR 364
NITR_ACID 282
[KAFR_EL_ZT,*] SSP 600
SULF_A_P 50
SULF_A_S 200 ;
end;
