Mathematical Formulation A Critique On An Optimization Model For Annual Harvest Planning

V. Mathematical Formulation

• First the defined sets used in the model are introduced. T is the number of time periods i I is the set of harvesting areas i J is the set of mills j K is the set of assortments k Q is the set of all harvesting teams q R is the set of roads r connecting the areas to state roads R P r is roads adjacent to road r, i.e., roads connecting r to a higher level in the road hierarchy Ri is the set of roads r that connects area I to the next level of roads or public roads connecting the areas to state roads I r is the set of areas connected to road r • The data needed to formulate the model are given below. • All costs are unit costs when applicable. The penalty costs are used to make sure that a near feasible solution always is attainable see the original journal • The objective is to minimize the total cost. • The first term Σ i єl Σ q єQ Σ t єT C H iq z iqt represents the harvesting cost, including harvesting, forwarding, and traveling cost. There is a specific cost for each combination of harvesting team and area. • The second term Σ r єR Σ t єT C R rt u rt corresponds to the road-opening cost, mainly because of the need of snow removal during the winter periods. • The third term in the objective function Σ j єJ Σ k єK Σ t єT C B kt v jkt corresponds to costs associated with purchased logs, and the next term Σ i єl Σ j єJ Σ k єK Σ t єT C T ijkt x ijkt corresponds to transportation from areas to mills. The following three terms Σ i єl Σ k єK Σ t єT H F kt l F ikt + Σ j єJ Σ k єK Σ t єT H l jk l l jkt + Σ j єJ Σ k єK Σ t єT H T jkl l T jkt represent storage cost in the forest, at terminals, and at mills, respectively. The storage cost at the forest and terminals are due to quality deterioration of timber stored outdoors. • Further, the eight term Σ j єJ Σ k єK Σ t єT P s s jkt represents a large cost for not satisfying the demand. The next two term Σ r єR Σ t єT P R rt u rt + Σ i єl Σ q єQ Σ t єT P A it y l iqt +y 2 iqt-1 .represent the estimated risk for area or roads with low accessibility respectively. • Finally, the last two term Σ i єl Σ q єQ Σ t єT P D y 2 iqt + Σ q єQ Σ t єT P O q w O qt correspond to penalty for harvesting an area in two different adjacent time periods and penalty for assigning more working days to a team than the standard overtime, respectively. • Constraints 2 mean that each harvest area can be harvested at most once. The next constraints 3 specify that total proportion of each harvest area will sum to 1, which means that if an area is harvested, it is fully harvested. • The variable y 1 ikt and y 2 ikt are needed, since harvesting in areas can begin in one period t and be finished in the beginning of the next time period t +1. if an area is harvested, it is always fully harvested. Each area corresponds to at most one time period of harvesting. Given this, no area is harvested in more than two adjacent time periods. The interpretation of y 1 ikt and y 2 ikt-1 are identical, but we have included a low cost corresponding to y 2 ikt , to decrease the risk of obtaining solutions where harvesting of areas are divided into two time periods. • In all their tests, for each team, in each time period, at most one area is not finished. To guarantee that at most one area for each team is divided into two periods, we need twice the number of binary harvesting variables for decisions if harvesting is started in time period t and further if harvesting is finished in a time period. This requirement is not strict, as the resulting annual plan is used as a basis for operational planning on a rolling time horizon, for typically 1.5 months, starting from a list of areas corresponding to 2 months. Universitas Sumatera Utara • The constraints 4 and 5 correspond to the storage balances at harvest areas and mills, respectively. The storage at terminals is directly included in the storage balance at mills. The slack variable in eq. 5 is introduced to guarantee feasible solutions, and the penalty cost for slack assures that the demand will be satisfied if possible. The constraints 6 restrict the total volume of purchased raw material, and constraints 7 and 8 specify the storage capacity at mills and terminals, respectively. • Constraints 9 and 10 require at least one road connected to area i to be open if transportation of logs from i is done or if area i is harvested, respectively. These two constraints are needed, as the logs can be transported from an area i in a later time period than the area was harvested. Constraints 11 specify the precedence relation between roads. The set R P r is the set of roads connecting road r to the next level or higher level, if the state road is the highest in the road hierarchy. Constraints 12 correspond to the restriction that the crews have a limited number of working days each month to use; the use of more working days corresponds to a penalty cost. Finally, the variable restrictions are given in 13 and 14. • The annual harvesting planning problem gives a large-scale, mixed-integer, linear problem. There are a number of binary variables corresponding to harvesting and road opening decisions and continuous variables describing storage and flow. The linear relaxation of this model gives a good estimation of the objective value of the integer problem. For each harvest area, the value of the corresponding binary harvest decision variable gives the proportion of the volume at that area, which is possible to use, to fulfill the demand, constraints 3 and 4. The road network is highly aggregated, which gives that constraints 10 allow small values of u rt . However, the cost corresponding to road opening is small compared with the total cost, so these constraints do not result in a weak lower bound from the linear relaxation.

VI. A single district case study