A hospital has three departments and three work shifts, each with a different le

Question

A hospital has three departments and three work shifts, each with a different levels of necessary staffing for nurses. The hospital would like to identify the minimum number of nurses required to meet the following constraints: the hospital must allocate at least 13, 32 and 22 nurses to the three departments (total over all shifts), the hospital must assign at least 26, 24 and 19 nurses to the three shifts (total over all departments), and the minimum and maximum number of nurses allocated to a each department in a specific shift must satisty the following limits: Shift Department 1 Department 2 Department 3 16, 8 11, 12] (7.12 17, 12] (5, 7) 17. 12] Shift 2 (4,6 Shift 3 [2, 4 [11, 12] [10, 12] Considering that this problem can be modeled as a kind of maximum-flow model, answer the tollowing questions to determine the minimum number of nurses required to satisfy all the constraints. (a) 13 points] Define nodes, arcs, and arc properties in words. Calculate all arc parameters and show your calculations in detail. Draw the network graph showing the relevant parameter values on the network. oints] Write the maximum-flow formulation for this problem using the given information/parameter values. (Reminder: Generic formulations will not receive any points.) 110 points

Explanation / Answer

(a) Nodes: There are nine nodes each representing each shift and department.

Arcs: Nurses assigned to each department for each shift represent the arc.

Arc properties: arc properties are the minimum and maximum nurses that can be assigned to each department in each shift.

(b) Objective: Min X11+x12+x13+x21+x22+x23+x31+x32+x33 , where X11, x12,... represent the number of nurses assigned to department 1 and shift 1 and 2 respectively and so on.. Xij, where i represent dept and j represent shift

s.t.

6 <= X11 <= 8

4 <= X12 <= 6

2 <= X13 <= 4

11 <= X21 <= 12

11 <= X22 <= 12

10 <= X23 <= 12

7 <= X31 <= 12

7 <= X32 <= 12

5 <= X33 <= 7

X11 + X12 + X13 >= 13

X21 + X22 + X23 >= 32

X31 + X32 + X33 >= 22

X11 + X21 + X31 >= 26

X12 + X22 + X32 >= 24

X13 + X23 + X33 >= 19

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