Students may work in groups of up to three people. You are strongly encouraged to work in groups; if you’re unable to find a partner, please let me know. You may be able to use piazza to find teammates. You may consult your textbooks, your notes, online information, and me. You may not solicit help from other sources.

1 Introduction

Province Healthcare operates several hospitals. A pandemic has started to affect their operations, with multiple patients hospitalized in intensive care. Province Healthcare has forecast the expected additional demand for intensive care in the next week, and this will exhaust the current supply of IC beds. They plan to create extra IC beds by converting current regular beds into IC beds and/or creating extra wards of IC beds. Each IC patient needs one dedicated nurse at all times.

Province Healthcare wants to set up a linear programming problem to model its operations and t minimize the cost of meeting all demand for IC beds.

The data for the problem is contained in patients.dat, an ampl file on LMS. Also on LMS is the start of an AMPL model file patients.mod.

(The solution to your linear program may be fractional. This is OK, you don’t need to require the variables to be integer. We will return to this issue in the third part of the project.)

2 Set of Hosptials

Province Healthcare operates one hospital in each of a set of cities, each with its own catchment area. Patients can be transported between any two hospitals. Each patient requires an IC bed and also nursing care. For nursing, the hospitals are divided into two zones, and nurses from one zone can only work at hospitals in the same zone. The zones are given by the sets ZONE1 and ZONE2 in the data file. The set of cities is the union of the two zones.

3 Demand

The forecasted additional demand for IC beds over the coming week is given by the parameter demand. You can assume this forecast is exactly accurate.

Each hospital has a current set of patients in intesive care, given by the parameter used_IC. They also have a current set of available beds in IC, given by the parameter avail_IC.

Demand at any hospital for IC beds can be satisfied by a mixture of two strategies:

• convert existing regular beds into IC beds, or create new sets of IC beds. For a given hospital, the cost using either method is the same and is given by the parameter cost.
• moving patients to other hospitals. The cost to move each patient is given by the parameter transp_costs.

4 Nursing

Each hospital currently has enough IC-trained nurses to fully staff each of their currently used IC beds and their available IC beds. If they expand capacity then they will need additional nurses.

Each IC patient needs one dedicated nurse at all times, so it needs 3 additional nurses over the course of 24 hours for each additional available IC bed. This number 3 is given by the parameter nurse_per_pat.

Some of the regular nurses at each hospital can be coverted to IC nurses in time for the forecasted increase in demand. The number of such nurses available is given by the parameter extra_nurses.

IC nurses can also move between hospitals in their own zone. The cost to transfer a nurse from one hospital to another is given by the parameter nurse_cost.

5 Balance

Province Healthcare wants to balance the availability of IC beds between the hospitals, based on the populations of their catchment areas. In particular, for each city they want the value of the ratio

to be at least the product of minratio and the average value of this ratio. The population (in thousands) is given by the parameter pop.

AMPL note: You can express the number of elements in the set CITIES using the expression card{CITIES}.

6 Variables

You will probably need several sets of variables. The principal decision variables are the number of extra IC beds at each hospital, the number of patients moved between hospitals, and the number of nurses moved between hospitals. You may also find it useful to define other (state) variables which are linear functions of these (decision) variables.

If you want your variables to be nonnegative, you have to explicitly state they are nonnegative.

7 Constraints

You will need several sets of constraints in your model.

8 The data

All the data can be found on LMS, and also on the course webpage at

The parameter definitions in the model file can be found on LMS and at

9 Questions

1. Due: Friday, October 27, 11.59pm, on LMS. (30 points)
   Please submit a written report, your problem files, and any AMPL output used. Choose a name for your model file that is distinctive to your group.

   Formulate and solve the problem as a linear optimization problem. Clearly define every variable in your formulation. Explain every constraint including coefficients.

   Hint: your optimal value should be between $250,000 and $300,000.

   Only one submission is needed from each group, but ensure that every team member’s name appears at the top of the report.

2. Sensitivity analysis (40 points): To be distributed later. Due on Friday November 17.
3. Extra conditions (30 points): To be distributed later. Due on Tuesday December 5.

10 AMPL notes

• AMPL comes with an IDE, which is fairly user friendly. I believe it is integrated into the single downloads for all 3 operating systems.
• If you are working in a Windows operating system and don’t use the IDE, you can edit your model and data files in Office, for example. Save the files as plain text files. Windows may append the suffix txt to the file names, in which case you would need to include that suffix when asking ampl to read the file.
• If you want to read in a new model file chips.mod and data file chips.dat, you can use the reset command:

  ampl: reset;
  ampl: model chips.mod;
  ampl: data chips.dat;

• If you want to reset the whole data file and read in a new data file chips.dat, but you want to keep the model file as before, you can type

  ampl: reset data;
  ampl: data chips.dat;

• More hints are available online from my page about AMPL:

  http://homepages.rpi.edu/~mitchj/ampldetails.html

• This project is available from LMS and also from the course webpage, at

  http://homepages.rpi.edu/~mitchj/matp4700/#project

  where you will also find more information about AMPL.