### Notations

I: = set of applicants (i = 1,2,….,m)

J: = set of academic programs (j = 1,2,…,n)

K: = set of placement criteria (k = 1,2,…,r)

T: = set of cutoff points (t = 1,2,…,v)

G: = set of component objectives (g = 1,2,…,h)

K_{t}: = {k ϵ K: k = criterion for which cutoff point t ϵ T is applicable}

### Parameters

The parameters in our model are the following:

U_{j}: = maximum intake capacity of program j

L_{j}: = minimum number of applicants required to be placed in program j

s_{ik}: = applicant i’s score on criterion k (all criteria scores are converted to relative scores based on maximum raw criteria scores)

p_{ij}: = applicant i’s preference rating for program j (where the ratings in increasing preference order are p_{ij} = 1,2,…,n)

γ_{jk}: = minimum score on criterion k that is considered to be desirable for entry to program j

ω_{k}: = weight assigned to criterion k in computing overall average score

q_{i}: = overall average score of applicant \mathrm{i}={\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{r}}{\mathrm{\omega}}_{\mathrm{k}}{\mathrm{s}}_{\mathrm{ik}}}

d_{ij}: = a measure of the desirablity of assigning applicant i to program j = p_{ij} q_{i}

β_{tk}: = the value of criterion k ϵ K_{t} corresponding to cutoff point t ϵ T

θ_{tk}: = the preference value corresponding to cutoff point t ϵ T and criterion k ϵ K_{t}

σ_{g}: = weight for component objective g

### Variables

x_{ij}: = equals 1 if applicant i is placed in program j and 0 otherwise.

### Constraints

There are two types of constraints: program capacity constraints and assignment requirements. Given the maximum intake capacity and minimum number of placements required per program we have the following constraints.

\begin{array}{ccc}\hfill {L}_{j}\le {\displaystyle \sum _{i=1}^{m}{x}_{\mathit{ij}}}\le {U}_{j}\hfill & \hfill \mathit{for}\hfill & \hfill j=1,2,\dots ..,n\hfill \end{array}

(1)

Also, it’s required that an applicant gets placement in no more than one of the programs open for enrollment. This requirement is specified as:

\begin{array}{ccc}\hfill {\displaystyle \sum _{j=1}^{n}{x}_{\mathit{ij}}}\le 1\hfill & \hfill \mathit{for}\hfill & \hfill i=1,2,\dots ..,m\hfill \end{array}

(2)

### Objective

Our objective function consists of four components each representing certain requirements specified by the program management office. The first component represents the desire to place applicants in programs of their higher preferences, with applicants’ overall average score determining priorities. This component is thus:

{\mathrm{z}}_{1}={\displaystyle \sum _{\mathrm{i}\in \mathrm{I}}{\displaystyle \sum _{\mathrm{j}\in \mathrm{J}}{\mathrm{d}}_{\mathrm{ij}}{\mathrm{x}}_{\mathrm{ij}}}}

(3a)

The second component of our objective function represents the desire to place applicants in their top rated programs. This component is formulated as:

{\mathrm{z}}_{2}={\displaystyle \sum _{\mathrm{i}\in \mathrm{I}}{\displaystyle \sum _{\begin{array}{c}\hfill \mathrm{j}\in \mathrm{J}\hfill \\ \hfill {\mathrm{p}}_{\mathrm{ij}}=\mathrm{n}\hfill \end{array}}{\mathrm{q}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{ij}}}}

(3b)

In equation (3b), the variables are weighted by the applicants’ overall average score on all criteria. The aim is that assignment of applicants in their first rated programs be done with applicants with higher overall scores given higher priorities.

The third component represents the program placement office’s desire that applicants who have some specified scores on a certain criterion be placed in programs of their higher preference ratings (both the score and the preference rating cutoff points are decided by the office). Thus, for instance, assuming six open programs for enrollment, if it’s required that applicants who scored 350 and above marks on college entrance examination be placed in programs of their first (rating of 6) or second (rating of 5) preference, the criterion cutoff point is 350 and the preference cutoff point is 5.

{\mathrm{z}}_{3}={\displaystyle \sum _{\mathrm{i}\in \mathrm{I}}{\displaystyle \sum _{\mathrm{j}\in \mathrm{J}}{\displaystyle \sum _{\mathrm{t}\in \mathrm{T}}{\displaystyle \sum _{\begin{array}{c}\hfill \mathrm{k}\in {\mathrm{K}}_{\mathrm{t}}\hfill \\ \hfill {\mathrm{s}}_{\mathrm{ik}}\ge {\mathrm{\beta}}_{\mathrm{tk}}\hfill \\ \hfill {\mathrm{p}}_{\mathrm{ij}}\ge {\mathrm{\theta}}_{\mathrm{tk}}\hfill \end{array}}{\mathrm{s}}_{\mathrm{ik}}{\mathrm{x}}_{\mathrm{ij}}}}}}

(3c)

The final and fourth component of our objective function represents the desire to minimize the cost of placements of applicants who fall short of the minimal requirements of entry specified in terms of minimum score on a criterion for a program. This is formulated as:

{\mathrm{z}}_{4}={\displaystyle \sum _{\mathrm{i}\in \mathrm{I}}{\displaystyle \sum _{\mathrm{j}\in \mathrm{J}}{\displaystyle \sum _{\begin{array}{c}\hfill \mathrm{k}\in \mathrm{K}\hfill \\ \hfill {\mathrm{s}}_{\mathrm{ik}}<{\mathrm{\gamma}}_{\mathrm{jk}}\hfill \end{array}}\left({\mathrm{s}}_{\mathrm{ik}}-{\mathrm{\gamma}}_{\mathrm{jk}}\right){\mathrm{x}}_{\mathrm{ij}}}}}

(3d)

We combine these four components into a single objective function.

\begin{array}{cc}\hfill \mathrm{maximize}\hfill & \hfill \mathrm{Z}={\displaystyle \sum _{\mathrm{g}\in \mathrm{G}}{\mathrm{\sigma}}_{\mathrm{g}}{\mathrm{z}}_{\mathrm{g}}}\hfill \end{array}

(3e)

The model presented above is what Pentico (2007) refers to as a semi-assignment model, except for the multiple objective components. More generally, the model is a transportaiton model with applicants being sources and programs being recipients. Thus, our model can be solved as a linear program without explicitly imposing the binary restrictions on the decesion variables.

Our model is easily usable with many alternative placement criteria and weighting schema. Whatever the number of criteria and weighting schema used, however, college administrators should deliberate and agree on a set of objective and measurable criteria that will be used for placement of applicants in programs. The chosen criteria may depend on the objective of placement decisions.

Another contentious issue is the construction of the intake capacity parameters of the programs. Previously the capacity of departments was arbitrarily decided. This used to be the cause of conflicts among departments. Because students’ preference rankings tend to be skewed in favor of one or two programs, some political activity is involved to convince administrators to place applicants in programs even if that is against the interest of applicants. The real gap here is the absence of any standard that justifiably determines the number of students each program can enroll. Defining ex ante the criteria and formula for capacity determination will be of much help in this regard.

At CoBE, there are no formal mechanisms to determine the effective intake capacities of programs. In the case of placements for undergraduate programs reported below, the simple rule applied was that a minimum of a class size of applicants be placed in each of the six programs offered at CoBE. Satisfying this, any number of eligible applicants can be placed in a program. Though this is problematic in principle (for instance, suppose all applicants make one program their first choice), experience suggests this simple rule is usually adequate. Applicants’ preferences, though skewed towards a few programs, tend to be distributed across programs without causing too much imbalance between capacity and enrolment for any one program.