Objective Function & Cost Structure¶

Objective function¶

The model minimizes yy (total cost plus penalties):

totalcostE..  yy =e= y
              + Sum(lakecatch, Penalty2 × exceed_p(lakecatch))
              + Sum(k, exceed(k) × Penalty1)
              + (exceed_lav × Penalty1)

Where y = actual total cost:

totalcost..  y =e= Total_cost_all = Sum(k, Total_cost_k(k))

Penalty values¶

Variable	Penalty	Meaning
`exceed(k)`	`Penalty1 = 9.999×10¹²` DKK/ton N	Unmet N target in catchment k
`exceed_p(lakecatch)`	`Penalty2 = 9.999×10¹³` DKK/kg P	Unmet P target in lake catchment
`exceed_lav`	`Penalty1`	Unmet tripartite lavbund floor

The penalty values are set so high that the solver will always prefer meeting targets to using the penalty — effectively making targets "hard" while still maintaining a feasible model.

Total cost per coastal catchment (`Total_cost_k_eq`)¶

Total_cost_k(k) =e=
    Sum(i$setki(k,i), Sum(j$nit(j),  CostM(i,j) × x(i,j) × PotV(i,j)))   [N field measures]
  + Sum(i$setki(k,i), Sum(j$pho(j),  CostM(i,j) × PotV(i,j) × x(i,j)))   [P field measures]
  + Sum(i$setki(k,i), Sum(j$comb(j), CostM(i,j) × x(i,j) × PotV(i,j)))   [Combined measures]
  + Sum(i$setki(k,i), Sum(j$comb2(j),CostM(i,j) × x(i,j)))                [Combined BZ: cost not multiplied by PotV!]
  + sum(ret$kret(k,ret), MWCost_ret(ret))                                    [Mini-wetlands]
  + sum(p$kp(k,p),  sum(j, WWTCost(p,j) × WWTx(p,j)))                     [WWT upgrades]
  + sum(pp$kpp(k,pp), ofcost(pp) × OFx(pp))                                [Overflow treatment]
  + Sum(w$kw(k,w),  Sum(w_m, Cost_W(w,w_m) × v_w(w,w_m)))                 [Stream measures]
  + Sum(ero_stretch$kero(k,ero_stretch), Sum(str_m, Cost_S(ero_stretch,str_m) × v_str(ero_stretch,str_m))) [Trees]

Note on comb2: For combined measures with a BZ component (NPB10_BZ10, etc.), CostM(i,j) is already a total cost for the field (not per ha), so it is NOT multiplied by PotV(i,j).

Field-level cost structure (`CostM`)¶

Field costs have two components for most measures:

CostM(i,j) = Implementation cost + Opportunity cost (lost gross margin)

Opportunity cost: `prodcost(i)`¶

Average annual gross margin lost per ha, computed from 5 years of crop data:

prodcost(i) = Sum(cy, CostE(i,cy)) / Countcy(i)

Where CostE(i,cy) = gross margin of the crop in year cy on field i (from Cost_new_avg.inc, indexed by oeko status, livestock level, soil type, and crop code). Fields with "unacceptable" crops in a year (fallow, non-cropped, etc.) are excluded from the average.

CostM formulas by measure¶

Measure	Formula (DKK/ha/yr)
CCS	315–396 (by soil and livestock)
CCW	1,980–3,168 (by soil and livestock)
EC	prodcost(i) ± 599–1,676 (by soil)
WL	3,486 + prodcost(i) [+200 if livestock ≥ 0.8]
IC	325
SA	prodcost(i) + 250 [+200 if livestock ≥ 0.8]
FO	prodcost(i) − Adj_Ann_Jordv_2pro(i) [+200 if livestock]
EW	200
BZ10	prodcost(i)
BZ20	prodcost(i)
LRl	prodcost(i) + 1,016 [+200 if livestock]
LRh	prodcost(i) [+200 if livestock]
N20	178
N10	44.5
PPC	80 + 0.1 × prodcost(i)
NPB10	50
NPB20	100
OT	200
PWET (wtype 0,1,2)	prodcost(i) + 9,733
PWET (wtype 3)	prodcost(i) + 13,200
IBZ	7,938

Livestock surcharge: 200 DKK/ha/yr added for fields with livestock(i) ≥ 0.8 for measures WL, SA, FO, LRl, LRh. Added 2021-11-05.

Negative costs set to zero: CostM(i,j)$(CostM(i,j)<0) = 0

`Adj_Ann_Jordv_2pro(i)` — land value annuity adjustment¶

Used in the FO cost formula. Represents an annual payment farmers receive for afforestation (kompensation). This reduces the net cost of forestation. Comes from Adj_Ann_Jordv_2pro.inc.

Mini-wetland costs (`MWCost_ret_eq`)¶

MWCost_ret(ret) =e=
    sum(mw1$MW1Ret(ret,mw1), mw1x(mw1) × (mw1cost + CostMW_ha_N(ret) × 0.20))
  + sum(mw2$MW2Ret(ret,mw2), mw2x(mw2) × (mw2cost + CostMW_ha_N(ret) × 0.50))
  + sum(mw3$MW3Ret(ret,mw3), mw3x(mw3) × (mw3cost + CostMW_ha_N(ret) × 1.00))

Component	MW1	MW2	MW3
Fixed construction cost	27,270 DKK	36,983 DKK	53,171 DKK
Land area (ha)	0.20 ha	0.50 ha	1.00 ha
Land opportunity cost	`CostMW_ha_N(ret) × 0.20`	`× 0.50`	`× 1.00`

CostMW_ha_N(ret) = area-weighted average prodcost(i) across all fields in the retention area. (Updated Dec 2024 to include gross margin loss for the wetland's own area.)

Stream measure costs (`Cost_W`)¶

Measure	Class 1	Class 2	Class 3
Sand trap	9,328 DKK	12,460 DKK	— (excluded)
Ochre trap	121,537 DKK	121,537 DKK	—
Re-meandering	7,000 DKK/km + xcost	25,000 DKK/km + xcost	25,000 DKK/km + xcost
Raising	5,000 DKK/km	8,000 DKK/km	19,000 DKK/km

re_meandering_xcost(w) is an additional cost loaded from re_meandering_xcost.inc (extra costs for re-meandering, calculated outside the model).

Trees on erosion stretches: 290.8 DKK/yr per stretch.

Open questions¶

✅ wtype(i): Watercourse type code per field (integer), from P_waterways.inc (confirmed 2026-04-05). Not wetland type or water table — it reflects the width/character of the adjacent watercourse. See PWET.
✅ Adj_Ann_Jordv_2pro(i): Annuitized income from forest grant scheme (skovrejsning compensation), from Adj_Ann_Jordv_2pro.inc. Reduces net cost of FO. Based on Lundhede (2020) / Meilby et al. (2014).
Sand traps: the code suggests class 3 costs were "made up" — is class 3 sand trap still excluded? ⚠️ Open — confirm in model.
Is the 200 DKK/ha livestock surcharge still the correct figure or has it been revised?

Concepts: - Cost concepts — opportunity cost, prodcost, livestock surcharge explained - N Reduction — target constraints consuming the cost - P Reduction — P target constraints - Constraints — tripartite floors and mutual exclusions

Individual measures (cost table): - N measures: CCS, CCW, EC, WL, IC, SA, FO, EW, BZ10, BZ20, LRl, LRh, N10, N20 - P measures: PPC, NPB10, NPB20, OT, PWET, IBZ - Infrastructure: MW, Stream measures, WWT

Objective Function & Cost Structure¶