Question

A rural clinic hires its staff from nearby cities and towns on a part-time basis. The clinic attempts to have a general practitioner (GP), a nurse, and an internist on duty during at least a portion of each week. The clinic has a weekly budget of $1,200. A GP charges the clinic $40 per hour, a nurse charges $20 per hour, and an internist charges $150 per hour. The clinic has established the following goals, in order of priority: (1) A nurse should be available at least 30 hours per week. (2) The weekly budget of $1,200 should not be exceeded. (3) A GP or an internist should be available at least 20 hours per week. (4) An internist should be available at least 6 hours per week.

A). IN ADDITION TO QUESTIONS BELOW MOST IMPORTANTLY LOOKING FOR SOLUTIONS AND FULL PICTURE OF HOW TO INSERT THE CONSTRAINTS AND VARIABLES UNDER THE QM FOR WINDOWS V4 SOFTWARE (COMPUTER MODEL FORMULATION)

B. Formulate a goal programming model to determine the number of hours to hire each staff member to satisfy the various goals.

C. What goals are not satisfied? By how much?

D. What goals are not satisfied? By how much?

E. Model formulation, please follow the three steps: define variables; objective function, and constraints.For model formulation, please follow the three steps: define variables; objective function, and constraints.

Answer 1

Decision variables

Let Xg, Xn, and Xi be the availability (in hours) for the GP, nurse, and internist respectively.

Also, let Uj and Oj be the underachievement and overachievement of the j-th goal. j=1,2,3,4

Objective function

Minimize Z = P1 U1; P2 O2; P3 U3; P4 U4

Constraints

Xn + U1 - O1 = 30 (Goal 1)

40 Xg + 20 Xn + 150 Xi + U2 - O2 = 1200 (Goal 2)

Xg + Xi + U3 - O3 = 20 (Goal 3)

Xi + U4 - O4 = 6 (Goal 4)

Xi, Xg, Xn, Uj, Oj >= 0

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Implementation in QM for Windows

(a) To start with

(b) Data entry

Note that d+ and d- stands for O and U in our formulation.

(c) Solution

Note that based on priority, the first two goals were achieved and the next two are underachieved by 5 and 6 units respectively.