Modeling Lean and Agile Approaches: A Western Canadian Forest Company Case Study

Table A2. Operational and inventory costs.

Table A3. Yields, capacities and productivities.

Table A4. Allowable quantities to penalize lumber production, capacity usage, and economic penalties.

%O_1kqor, %B_1kqor	Max. % for over/below production defined in function of the demand of lumber product species k, product q, produced/sold by sawmill o, period r.
%O_LOr, %B_LOr	Max. over/below hours usage for logging in period r, defined as percentage of the hours available in period r.
%O_SYlr, %B_SYlr	Max. over/below hours usage for sort-yarding in sort-yard l, period r, defined as percentage of the hours available in period r.
%O_SWor, %B_SWor	Max. over/below hours usage for sawmilling in sawmill o, period r, defined as percentage of the hours available in period r.
PO_1kqor, PB_1kqor	Pct. of the lumber price of lumber product species k, product q, produced-sold by sawmill o, period r, to penalize 1 m3 produced over/below the lumber demand.
PB_2knlr	Pct. of the log price species k, log grade n, produced-sold by sort-yard l, period r, to penalize 1 m3 produced below the log demand.
PO_LOr, PB_LOr	Pct. of the logging cost charged for every over/below hour usage in logging in period r.
PO_SYl r, PB_SYlr	Pct. of the sort-yarding cost charged for every over/below hour usage in sort-yard l period r.
PO_SWor, PB_SWor	Pct. of the sawing cost charged for every over/below hour usage in sawing in sawmill o, period r.

Table A5. Decision variables.

Hir	Land harvested (ha.) at stand i, in period r.
Tijk r	Harvested volume (m3) at stand i, growth type j, species k in period r.
ISCBIjkr	Inventory of stems (m3) at cub-block i, of growth j, species k, in period r.
Uijklr	Vol. of stems (m3) of growth j, species k sent from stand i to sort-yard l, in period r.
Vjklmr	Sort-yard l input of stems (m3) of growth j, species k, bucked with policy m, period r.
ISYSjklr	Inventory of stems (m3) of growth j, species k, at sort-yard l, in period r.
Wknlr	Vol. of logs (m3) of species k, grade n, produced in sort-yard l in period r.
Yknopr	Vol. of logs (m3) of species k, grade n, sawn with sawing policy p, at sawmill o, in period r.
YOknopr	Vol. of logs (m3) of species k, grade n, sawn with sawing policy p, at the outsourcing sawmill of sawmill o, in period r.
ISALknor, ISYLknlr	Inventory of logs (m3) of species k, grade n at sawmill o and at the sort yard l, in period r.
Xknlor	Vol. of logs (m3) of species k, grade n, sent from sort-yard l to sawmill o, in period r.
Zkqor	Vol. of lumber (m3) of species k, product q, produced-sold by sawmill o, in period r.
ZOkqor	Vol. of lumber (m3) of species k, product q, produced-sold by outsourced sawmill of sawmill o, in r.
ZI O_OOkqor	Vol. of lumber (m3) of species k, product q, arrived in interchanges to sawmill o, in period r.
ZI OO_Okqor	Vol. of lumber (m3) of species k, product q, sent in interchanges to sawmill o, in period r.
DO_1kqor, DB_1kqor	Vol. (m3) of over, below demand lumber production of species k, product q, produced-sold by sawmill o, in period r.
DB1knlr	Vol. (m3) of below demand log production of species k, log grade n, produced-sold by sort yard l, in period r.
DO_LOr, DB_LOr	Quantity of over/below cap. Usage (hours) in logging operation in period r.
DO_SYlr, DB_SYlr	Quantity of over/below cap. Usage (hours) in sort-yarding operation at sort-yard l in period r.
DO_SWor, DB_SWor	Quantity of over/below cap. Usage (hours) in sawmilling operation at sawmill o in period r.
BAkqor	Lumber balance (m3) for species k, product q, produced-sold by sawmill o, in period r.
Ckr	Produced chips (m3) of species k in period r.
ISAPkqor	Inventory of sawn-wood products (m3) of species k, sawn-wood products q, in period r.
LOTr, SYTlr,SWTor	Logging time, Sort yard l processing time, and Sawmill o processing time in hours for period r.
BYknor	Binary variable, 1 if sawmill o consumes logs species k, grade n, in period r, 0 otherwise.

Notation	Description
Minimization of costs (1l)	=Logging costs + sort yarding costs + sawmilling costs + cost to penalize below demand satisfaction + logging penalization cost + sort yarding penalization cost + sawmilling penalizations costs
Maximization of profit (1a)	=Lumber and chips incomes-logging costs − sort yarding costs − sawmilling costs − cost to penalize below demand satisfaction − logging penalization cost − sort yarding penalization cost − sawmilling penalizations costs
Minimization of timber procurement costs (1T)	=Logging costs + sort yarding costs + cost to penalize below log demand satisfaction + logging penalization cost + sortyarding penalization cost
Maximization of lumber profits (1W)	=Lumber and chips incomes − cost of buying logs − sawmilling costs − cost to penalize below lumber demand satisfaction − sawmilling penalizations costs
${\displaystyle \sum _{\mathbf{ijkr}}}{\mathbf{T}}_{\mathbf{ijkr}}{\mathbf{C}}_{\mathbf{jk}}^{\mathbf{STU}}+{\displaystyle \sum _{\mathbf{ijkr}}}{\mathbf{T}}_{\mathbf{ijkr}}{\mathbf{C}}_{\mathbf{i}}^{\mathbf{H}}+{\displaystyle \sum _{\mathbf{ijklr}}}{\mathbf{U}}_{\mathbf{ijklr}}{\mathbf{T}}_{\mathbf{il}}^{\mathbf{T}1}+{\displaystyle \sum _{\mathbf{ijkr}}}{\mathbf{I}}_{\mathbf{ijkr}}^{\mathbf{SCB}}{\mathbf{C}}_{\mathbf{i}}^{\mathbf{SCB}}$	=Logging costs: harvesting costs, stumpage costs, primary transportation cost, and stems inventory costs
${\displaystyle \sum _{\mathbf{knlr}}}{\mathbf{W}}_{\mathbf{knlr}}{\mathbf{C}}_{\mathbf{l}}^{\mathbf{SY}}+{\displaystyle \sum _{\mathbf{knlor}}}{\mathbf{X}}_{\mathbf{knlor}}{\mathbf{C}}_{\mathbf{lo}}^{\mathbf{T}2}+{\displaystyle \sum _{\mathbf{jklr}}}{\mathbf{I}}_{\mathbf{jklr}}^{\mathbf{SYS}}{\mathbf{C}}_{\mathbf{l}}^{\mathbf{SYS}}+{\displaystyle \sum _{\mathbf{knlr}}}{\mathbf{I}}_{\mathbf{knlr}}^{\mathbf{SYL}}{\mathbf{C}}_{\mathbf{l}}^{\mathbf{SYL}}$	Sort yarding costs: sort yard processing cost, transportation cost, and stems and logs costs
$\begin{array}{l}{\displaystyle \sum _{\mathbf{kqor}}}{\mathbf{Z}}_{\mathbf{kqor}}{\mathbf{C}}_{\mathbf{o}}^{\mathbf{SW}}+{\displaystyle \sum _{\mathbf{kqo}\_\mathbf{oor}}}\mathbf{Z}{\mathbf{I}}_{\mathbf{kqo}\_\mathbf{oor}}{\mathbf{C}}_{\mathbf{o}\_\mathbf{oor}}^{\mathbf{T}3}+{\displaystyle \sum _{\mathbf{kqor}}}\mathbf{Z}{\mathbf{O}}_{\mathbf{kqor}}{\mathbf{C}}_{\mathbf{o}}^{\mathbf{SWAT}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \sum _{\mathbf{knor}}}{\mathbf{I}}_{\mathbf{knor}}^{\mathbf{SAL}}{\mathbf{C}}_{\mathbf{o}}^{\mathbf{SAL}}+{\displaystyle \sum _{\mathbf{kqor}}}{\mathbf{I}}_{\mathbf{kqor}}^{\mathbf{SAP}}{\mathbf{C}}_{\mathbf{o}}^{\mathbf{SAP}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \sum _{\mathbf{kqor}}}\mathbf{B}{\mathbf{Y}}_{\mathbf{kqor}}{\mathbf{C}}_{\mathbf{kno}}^{\mathbf{SETUP}}\end{array}$	Sawmilling costs, including processing cost at internal and outsourced sawmill, lumber production swat, and logs and lumber inventory costs
${\displaystyle \sum _{\mathbf{kqor}}}\mathbf{DB}\_1_{\mathbf{kqor}}\mathbf{P}{\mathbf{B}}_{\mathbf{kqor}}\mathbf{L}\_{\mathbf{P}}_{\mathbf{kqor}}$	Cost to penalize below lumber demand satisfaction
${\displaystyle \sum _{\mathbf{ir}}}\mathbf{DO}\_\mathbf{L}{\mathbf{O}}_{\mathbf{r}}[\mathbf{PO}\_\mathbf{L}{\mathbf{O}}_{\mathbf{r}}]{\mathbf{C}}_{\mathbf{i}}^{\mathbf{H}}+{\displaystyle \sum _{\mathbf{ir}}}\mathbf{DB}\_\mathbf{L}{\mathbf{O}}_{\mathbf{r}}[\mathbf{PB}\_\mathbf{L}{\mathbf{O}}_{\mathbf{r}}]{\mathbf{C}}_{\mathbf{i}}^{\mathbf{H}}$	Logging penalization costs
${\displaystyle \sum _{\mathbf{lr}}}\mathbf{DO}\_\mathbf{S}{\mathbf{Y}}_{\mathbf{lr}}[\mathbf{PO}\_\mathbf{S}{\mathbf{Y}}_{\mathbf{lr}}]{\mathbf{C}}_{\mathbf{lr}}^{\mathbf{SY}}+{\displaystyle \sum _{\mathbf{lr}}}\mathbf{DB}\_\mathbf{S}{\mathbf{Y}}_{\mathbf{lr}}[\mathbf{PB}\_\mathbf{S}{\mathbf{Y}}_{\mathbf{r}}]{\mathbf{C}}_{\mathbf{lr}}^{\mathbf{SY}}$	Sort yarding penalization costs
${\displaystyle \sum _{\mathbf{or}}}\mathbf{DO}\_\mathbf{S}{\mathbf{W}}_{\mathbf{or}}[\mathbf{PO}\_\mathbf{S}{\mathbf{W}}_{\mathbf{or}}]{\mathbf{C}}_{\mathbf{or}}^{\mathbf{SW}}+{\displaystyle \sum _{\mathbf{or}}}\mathbf{DB}\_\mathbf{S}{\mathbf{W}}_{\mathbf{or}}[\mathbf{PB}\_\mathbf{S}{\mathbf{W}}_{\mathbf{or}}]{\mathbf{C}}_{\mathbf{or}}^{\mathbf{SW}}$	Sawmilling penalization costs
${\displaystyle \sum _{\mathbf{kqor}}}{\mathbf{Z}}_{\mathbf{kqor}}\mathbf{L}\_{\mathbf{P}}_{\mathbf{kqor}}+{\displaystyle \sum _{\mathbf{kr}}}{\mathbf{C}}_{\mathbf{kr}}{\mathbf{C}}_{\_}{\mathbf{P}}_{\mathbf{kr}}+{\displaystyle \sum _{\mathbf{kqor}}}\mathbf{DO}\_1_{\mathbf{kqor}}\mathbf{P}{\mathbf{O}}_{\mathbf{kqor}}\mathbf{L}\_{\mathbf{P}}_{\mathbf{kqor}}$	Lumber and chips incomes: includes lumber incomes, chips incomes, and over demand production incomes
${\displaystyle \sum _{\mathbf{knlr}}}\mathbf{DB}{1}_{\mathbf{knlr}}\mathbf{PB}\_2_{\mathbf{knlr}}\mathbf{l}\_{\mathbf{p}}_{\mathbf{knlr}}$	Cost to penalize below log demand satisfaction
${\displaystyle \sum _{\mathbf{kqor}}}\mathbf{DB}{1}_{\mathbf{kqor}}\mathbf{P}{\mathbf{B}}_{\mathbf{kqor}}\mathbf{L}\_{\mathbf{P}}_{\mathbf{kqor}}$	Cost to penalize below lumber demand satisfaction
${\displaystyle \sum _{\mathit{k}\mathit{n}\mathit{l}\mathit{r}}}{\mathit{W}}_{\mathit{k}\mathit{n}\mathit{l}\mathit{r}}\mathit{l}{\mathit{p}}_{\mathit{k}\mathit{n}\mathit{l}\mathit{r}}$	Costs of buying logs

×

Constraint 1 ensures that all period cut-block harvests do not exceed the cut-block area. Constraint 2 computes the cut-block volume harvested, and constraint 3 limits the time expended in logging. As I constrained the rate of capacity usage between periods, constraint 4 computes the time used for logging per period and constraint 5 adds slack variables to track logging time deviations between periods. The same principle was applied for sort-yards and sawmills (constraints 5–11). Several flow balance constraints were applied on the: logging side, sort-yard side, and internal and outsourced sawmills side (12, 13, 18, and 19). Constraint 14 determined log productions at the sort-yards. Constraint 15 limits the time expended in sort-yarding. Constraint 16 calculated the time used at sort-yards per period, and constraint 17 works in the same fashion as constraint 5. Constraints 20-21 determined the lumber production at internal and outsourced sawmills. A setup cost was applied when sawmills changed log species and grades; thus, a binary variable, which is 1 if log species and grades are changed, and zero otherwise (constraint 22).

The chips produced are determined by constraint 23. Constraint 24 limits the time expended in sawmilling. Constraint 25 limits the time expended in outsourcing sawmilling to 20% of the available period's sawmilling capacity, and constraint 27 ensures that internal sawmill production will not be exceeded by outsourced sawmill production. As I constrained the deviations of capacity usage between periods, constraint 26 computes the time used on sawmilling per period. Constraint 28 limited the sawmilling time deviations between periods by adding slack variables. A market constraint 29 ensured that lumber demand would be satisfied with all sources of lumber production. Balance constraint 30 determined the volume of lumber to be considered for order fulfillment parameter calculations. Constraints 31–32 determined the volume of lumber that arrived in interchanges from other sawmills to sawmill "o" and determined the lumber that is sent to other sawmills from sawmill "o". Constraint 33 ensured that the amount of lumber produced over demand will not exceed a given [%] of the lumber demand. Constraint 34 ensured that the amount of lumber produced below the demand will not exceed a given [%] of the lumber demand. Constraints 35–38 ensured that the final inventories are positive.