Multi-role Workload Allocation

Model introduction

Higher-education teaching teams often need to allocate multiple types of work across the same pool of staff or graduate students. A common setting involves three roles:

1. Teaching-assistant work (TA), such as tutorials, laboratory sessions, and other forms of direct teaching support.
2. Grading work (GR), such as marking assignments, tests, and examinations.
3. A third role, E, which captures lighter or more intermittent duties such as exam invigilation, consultation support, and routine administrative tasks.

This model is useful when a department must satisfy exact demand for each role while also accounting for role-specific preferences, priority rules for lighter duties, and protections for selected cohorts. It can also incorporate previous workload and balance total workload over a longer planning horizon.

Let there be $N_s$ individuals indexed by $i \in \{1,\ldots,N_s\}$ and $N_j$ courses or tasks indexed by $j \in \{1,\ldots,N_j\}$. For each task-role pair, where $r \in \{\mathrm{TA}, \mathrm{GR}, \mathrm{E}\}$, the required demand $d_{j,r}$ must be fully assigned.

For each individual-task pair, $P^{\mathrm{TA}}_{i,j}$ and $P^{\mathrm{GR}}_{i,j}$ denote the user-supplied TA and grading preference scores. Higher values indicate stronger preferences. Year of study is denoted by $y_i$ (capped within 1-4), and $s_i$ is a user-configurable score that guides E allocation. The model also tracks past semester TA and GR workload, denoted by $t_i^{(1)}$ and $g_i^{(1)}$.

TA and GR may protect different cohorts. Let $y^\ast_{\mathrm{TA}}, y^\ast_{\mathrm{GR}} \in \{1,2,3,4\}$ denote the protected years for the two roles. Each role’s fairness spread excludes its own protected cohort. The model supports one protected cohort per role, but the two roles may protect the same or different cohorts.

The generalized model balances:

1. annual TA and GR workload fairness,
2. role-specific TA and grading preferences,
3. score-guided allocation of lighter E duties, and
4. separate TA and GR workload protection for selected cohorts.

Model formulation

Objective function

$$\begin{align*} X_{i,j,r} &\in \mathbb{Z}_{\ge 0} && \text{units of role } r \text{ in course } j \text{ assigned to student } i \\ t_i^{(2)} &= \sum_{j=1}^{N_j} X_{i,j,\mathrm{TA}} && \text{current-semester TA workload} \\ g_i^{(2)} &= \sum_{j=1}^{N_j} X_{i,j,\mathrm{GR}} && \text{current-semester GR workload} \\ e_i^{(2)} &= \sum_{j=1}^{N_j} X_{i,j,\mathrm{E}} && \text{current-semester E workload} \\ T_i &= t_i^{(1)} + t_i^{(2)} && \text{yearly TA workload} \\ G_i &= g_i^{(1)} + g_i^{(2)} && \text{yearly GR workload} \\ w_i^{\mathrm{TA}} &\ge 0 && \text{slack for the protected cohort's TA soft bound} \\ w_i^{\mathrm{GR}} &\ge 0 && \text{slack for the protected cohort's GR soft bound} \\ T_{\max},T_{\min} &\ge 0 && \text{TA workload bounds outside } y^\ast_{\mathrm{TA}} \\ G_{\max},G_{\min} &\ge 0 && \text{GR workload bounds outside } y^\ast_{\mathrm{GR}} \end{align*}$$

$$\begin{aligned} \min\quad &\alpha_{\mathrm{TA}}(T_{\max}-T_{\min}) +\alpha_{\mathrm{GR}}(G_{\max}-G_{\min})\\ &-\beta_{\mathrm{TA}} \sum_{i=1}^{N_s}\sum_{j=1}^{N_j} P^{\mathrm{TA}}_{i,j}X_{i,j,\mathrm{TA}}\\ &-\beta_{\mathrm{GR}} \sum_{i=1}^{N_s}\sum_{j=1}^{N_j} P^{\mathrm{GR}}_{i,j}X_{i,j,\mathrm{GR}}\\ &-\phi \sum_{i=1}^{N_s}\sum_{j=1}^{N_j} s_iX_{i,j,\mathrm{E}}\\ &+\rho_{\mathrm{TA}} \sum_{i:y_i=y^\ast_{\mathrm{TA}}}w_i^{\mathrm{TA}} +\rho_{\mathrm{GR}} \sum_{i:y_i=y^\ast_{\mathrm{GR}}}w_i^{\mathrm{GR}}. \end{aligned}$$

All objective weights are non-negative and user-specified. $\alpha_{\mathrm{TA}}$ and $\alpha_{\mathrm{GR}}$ control the importance of annual workload balance for the two main roles. $\beta_{\mathrm{TA}}$ and $\beta_{\mathrm{GR}}$ reward role-specific preference satisfaction. The E term rewards allocation to individuals with larger $s_i$ when $\phi > 0$. Finally, $\rho_{\mathrm{TA}}$ and $\rho_{\mathrm{GR}}$ penalize violations of the protected cohorts’ soft workload limits.

Demand satisfaction

For every task and role, assigned units must match demand:

$$\sum_{i=1}^{N_s}X_{i,j,r}=d_{j,r}, \quad \forall j,\quad r\in\{\mathrm{TA},\mathrm{GR},\mathrm{E}\}.$$

Role-specific workload spread

Annual TA spread is measured outside the TA-protected cohort:

$$T_{\min}\le T_i\le T_{\max}, \quad \forall i:y_i\ne y^\ast_{\mathrm{TA}}.$$

Annual GR spread is measured outside the GR-protected cohort:

$$G_{\min}\le G_i\le G_{\max}, \quad \forall i:y_i\ne y^\ast_{\mathrm{GR}}.$$

The two protected years may be the same or different. If a role’s protection penalty is disabled, no cohort is protected for that role and all individuals enter its fairness spread.

Annual workload equality

Let $C$ denote semester workload capacity per individual. The model fixes each individual’s annual workload total at $2C$:

$$T_i+G_i+e_i^{(2)}=2C,\quad \forall i.$$

Protected-cohort soft upper bounds

Current-semester TA workload is softly capped for the TA-protected cohort:

$$t_i^{(2)} \le t_{\max}^{(P)}+w_i^{\mathrm{TA}}, \quad \forall i:y_i=y^\ast_{\mathrm{TA}}.$$

Current-semester grading workload is separately capped for the GR-protected cohort:

$$g_i^{(2)} \le g_{\max}^{(P)}+w_i^{\mathrm{GR}}, \quad \forall i:y_i=y^\ast_{\mathrm{GR}}.$$

The slack variables preserve feasibility while making excess workload costly through the corresponding $\rho$ terms.

Optional current-semester workload bounds

If supplied, role-wide lower and upper bounds are also imposed:

$$\begin{align} t_{\min}^{(2)} \le t_i^{(2)} \le t_{\max}^{(2)}, \quad &\forall i, \\ g_{\min}^{(2)} \le g_i^{(2)} \le g_{\max}^{(2)}, \quad &\forall i, \\ e_{\min}^{(2)} \le e_i^{(2)} \le e_{\max}^{(2)}, \quad &\forall i. \end{align}$$

If a bound is omitted, its corresponding constraint is not added.

Package interface

The multi-role workflow consists of:

1. extract_multirole_info() or extract_info(assignment = "multirole"),
2. prepare_multirole_model() or prepare_model(assignment = "multirole"),
3. solve_assignment(assignment = "multirole") or the lower-level ompr::solve_model() and assign_job() functions.

The bundled multi-role example data demonstrate the generalized interface. Here the same matrix is supplied for TA and GR preferences only to keep the example compact. In practice, the matrices can contain different user-defined scores.

Semester history and capacity

By default, extract_multirole_info() expects student_id, year, past_ta, and past_gr as the first four columns of student_df and uses the supplied prior workload. Semester capacity C is also supplied at extraction and stored in the returned input list. prepare_multirole_model() reads this stored value when enforcing annual workload of 2 * C; it does not take a separate capacity argument.

For an allocation with no prior-semester workload data, use single_semester = TRUE. In this mode, only student_id and year are required as the first two columns, and any supplied past-workload columns are ignored. Extraction generates t1 = 0 and g1 = C for every individual:

single_semester_students <- grouper::multirole_students_ex001[
  , c("student_id", "year", "Name")
]

single_semester_inputs <- grouper::extract_multirole_info(
  student_df = single_semester_students,
  d_mat = grouper::multirole_demand_ex001,
  p_ta_mat = grouper::multirole_prefmat_ex001,
  e_mode = "rr",
  C = 4,
  single_semester = TRUE
)

cbind(t1 = single_semester_inputs$t1, g1 = single_semester_inputs$g1)
#>      t1 g1
#> [1,]  0  4
#> [2,]  0  4
#> [3,]  0  4
#> [4,]  0  4

The uniform synthetic g1 = C workload adds the same constant to every individual’s annual GR workload, so it does not change the GR spread. It fills the prior-semester half of the annual equality and leaves C units per individual for current TA, GR, and E allocation.

E-allocation scoring

extract_multirole_info() accepts a four-value s vector ordered by Years 1 to 4. The default is set to c(-1, 0, 1, 2), which encourages E allocation to more senior students.

default_inputs <- grouper::extract_multirole_info(
  student_df = grouper::multirole_students_ex001,
  d_mat = grouper::multirole_demand_ex001,
  p_ta_mat = grouper::multirole_prefmat_ex001,
  p_gr_mat = grouper::multirole_prefmat_ex001,
  e_mode = "none"
)

default_inputs$s
#> [1] -1  0  1  2

Users can provide a different encoding to control the E-allocation objective:

custom_inputs <- grouper::extract_multirole_info(
  student_df = grouper::multirole_students_ex001,
  d_mat = grouper::multirole_demand_ex001,
  p_ta_mat = grouper::multirole_prefmat_ex001,
  p_gr_mat = grouper::multirole_prefmat_ex001,
  e_mode = "none",
  s = c(0, 1, 3, 6)
)

custom_inputs$s
#> [1] 0 1 3 6

The score vector affects only the E objective. Protection and TA fairness use student_df$year. Both preference matrices are used exactly as supplied, so users can choose their own numeric scoring schemes during preprocessing rather than using the example 3/2/1/-99 encoding.

Role-specific terms

The new GR terms are disabled by default. Set their weights to positive values to enable GR workload spread, grading preferences, and GR cohort protection:

multi_role_model <- grouper::prepare_model(
  default_inputs,
  assignment = "multirole",
  alpha_ta = 2,
  alpha_gr = 2,
  beta_ta = 1,
  beta_gr = 1,
  phi = 1,
  rho_ta = 10,
  rho_gr = 10,
  protected_year_ta = 1,
  protected_year_gr = 3,
  ta_protected_max = 1,
  gr_protected_max = 1
)

The TA and GR protected years must each be one value from 1 to 4. The selected cohort receives that role’s soft upper bound and slack penalty, and is excluded from that role’s fairness spread.

Keeping the model small

An objective weight set to NULL or zero is disabled during model construction. The corresponding objective expression is not added. For spread and protection terms, their supporting variables and constraints are also omitted. For example:

ta_only_model <- grouper::prepare_multirole_model(
  default_inputs,
  alpha_gr = NULL,
  beta_gr = NULL,
  rho_gr = NULL
)

This conditional construction is useful for larger allocation problems because the solver receives only the variables and constraints needed for the selected formulation. When rho_ta or rho_gr is disabled, that role has no protected cohort and its fairness spread, if active, includes every individual.