Economic Models - Convex Optimization

Health Service Management 157
• Variables X 32 ...X 35 refer to the average cost per patient, as it is stated
in the first four rows of Table 2.
— Variable X 36 denotes total per patient expenses.
— Variables X 37 and X 38 refer to the equipment purchase and
replacement.
• Variable X 39 stands for the personnel insurance expenses.
— Variable X 40 refers to the depreciation, laboratory, fixed and other
expenses.
• Variable X 41 is used to sum up all the above expenses (i.e., X 37 + X 38 +
X 39 + X 40 )
• X 42 , which also is a definition variable, is used to denote total operating
cost for the final year t f (i.e., t 5 ) of the planning period.
• Finally, variable X 43 denotes the average charge per patient, so that the
desired gross profit would not be less than 750,000.
3.2. Formulation of the Constraints (Goals)
To facilitate the presentation, I classified the goals into the following seven
categories.
3.2.1. Personnel Employed
The operation pattern of the clinic, regarding the personnel employed is
presented in Table 1. Given that, the decision maker does not want to alter
this pattern, I must first introduce the following 10 constraints (goals)
X i + d − i
− d + i
= b i (i = 1,...,10) (5)
where b 1 = 10, b 2 = 1, b 3 = 3, b 4 = 14, b 5 = 3, b 6 = 1, b 7 = 3, b 8 = 4,
b 9 = 4 and b 10 = 10, are the desired targets, regarding the members of
each group as shown in Tables 1, 3 and 4.
It is useful to recall that di − and d
i + denote the so-called deviational
variables from the goals, which are complementary to each other, so that
di − × d
i + = 0. If the ith goal is not completely achieved, then the observed
slack will be expressed by di − , which represents the negative deviation from
the goal. On the other hand, if the ith goal is over achieved, then d
i
+ will
take a non-zero value. Finally, if this particular goal is exactly achieved,
then both di − and d
i + will be zero.