The SFU Wellness Center is an organization providing help to distressed students

Question

The SFU Wellness Center is an organization providing help to distressed students on campus during the two weeks exam period at the end of every semester. You’ve been hired as a business consultant to help the Center develop a hiring policy so that it can provide the most meaningful student service (S) possible. You’ve determined that service can be described as a function of Medical (M) and Counseling (C) staff input as follows:

                                                          S = M + 0.5C + 0.5MC – C2

         The staff budget for the SFU Wellness Center for the coming semester is $1,200.00. Each member of the medical staff costs $60 and each member of the counseling staff costs $30.

(a) Determine the optimal combination of medical and counseling staff for the SFU Wellness Center.

(b) Solve for and interpret the Lagrangian multiplier.

(c)           If the Center were required to break even, what fee per service would you recommend that they charge?

Explanation / Answer

Solution:- Now we understand that M & C are independent variables

so we can derive the objective function as S(M,C) = M+0.5C+0.5MC–C²

60M+30C1200 is an inequality constraint since 1200 is the budget and the cost of the medical staff is $60 and that of the counseling staff is $30 thus reducing the same we get G(M,C) = 2M+C400


(a) To derive the optima combination we will have to maximise S subject to G0 … (i)
Thus it means that we should get G=0
now in (i) we can replace G0 by G=0 and use the method of Lagrange Multipliers.

Lagrangian is L = S–G where is a Lagrange Multiplier

L = M+0.5C+0.5MC–C² (2M+C40)

Optimality conditions are : L/M=0, L/C=0 and G=0

1+0.5C2 = 0 … (ii)
0.5+0.5M2C– = 0 … (iii)
2M+C = 40 … (iv)

Now From (ii) we get C = 42 and from (iii) M = 189

Using these in (iv) : 3618+42 = 40 = 3/2

C = 4 and M = 18

(b) Now we can interpret as the “marginal cost of the constraint”. This is possible by relaxing the constraint by an amount from g=0 to g=, the optimal objective value will increase by . With this the limits can further be changed as follows

40 to 40+ or the limit of 1200 to 1200+30.

Hence we get /30=0.05 is the increase in optimal service level.

(c) In order to break even we can arrive at it by inverting the result of (b). From this we derive that 20$/unit service as a break-even charge.

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