MATP4700 Math Models of Operations Research Fall 2022
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Case Study:
MEsLIER SERvIcEs
MATP4700 Math Models of Operations Research
Fall 2022
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
MEsLIER SERvIcEs operates a telephone call center. Its employees answer incoming service calls for customers of two financial institutions, ApAMs FINANcIAL and BAMFoRp BANK. It aims to limit waiting times for customers while minimizing its employment costs. It wants to set up a linear programming problem to model its operations.
(The solution to your linear program may be fractional. This is OK, you don’t need to require the number of workers to be integer. We will return to this issue in the third part of the project.)
2 Shifts
Calls can come in at any time. MEsLIER SERvIcEs hires workers in shifts. There are 24 possible shifts, starting at each hour of the day. Each shift is seven hours long, consisting of working for three hours, a break for one hour, and then working for another three hours. Workers are paid for seven hours of employment for each shift, at a rate of $18 per hour for each hour worked between 9am and 5pm and at a rate of $27 per hour for each hour worked between 5pm and 9am. Thus, workers who start work at 3pm would earn $171 (= 2 X 18 + 5 X 27), they would have a break between 6pm and 7pm, and they would finish work at 10pm. Up to 100 workers can work any shift.
The workers in each shift are assigned to either ApAMs FINANcIAL or BAMFoRp BANK. They then work for that same bank for the duration of their shift.
Due to capacity constraints, MEsLIER SERvIcEs can have at most 420 workers working at the same time. Any worker who is scheduled to be working must be working at that time; they cannot be idle just due to the capacity constraint.
3 Demand
The two financial institutions’ demand for service varies through the day. In addition to varying by time of day, the demand also varies stochastically within each hour. For example, one day may see 80 calls from customers of BAMFoRp BANK between 9am and 10am, while another day may see 120 calls. In order to simplify the modeling process, MEsLIER SERvIcEs has assumed the demand is deterministic. It has estimated the number of employees it needs working at each hour of the day in order to adequately service ApAMs FINANcIAL and BAMFoRp BANK. These numbers are given in Table 1.FINANcIAL
120
95
90
50
30
30
60
110
140
200
180
180
240
220
220
230
250
260
230
250
260
250
220
190
Table 1: Demand for call center employees
4 Short-staffing and penalties
MEsLIER SERvIcEs is prepared to have fewer employees working at a particular time than the number given in the table. For example, it doesn’t require it has 100 people available to answer phones between 6am and 7am. However, in order to try to control the quality of service (QoS), it has imposed a limit on how short-staffed it can be. In particular, it requires that it is no more than ten workers short for either client for any hour, and it ensures that it is no more than 40 (respectively, 30) employees short over the course of the whole day for servicing the calls from ApAMs FINANcIAL (respectively, BAMFoRp BANK ). Thus, it may have only 80 employees working between 6am and 7am, with 50 of those assigned to calls from customers of ApAMs FINANcIAL and 30 assigned to calls from customers of BAMFoRp BANK.
BAMFoRp BANK is more worried about missed calls between 7am and 6pm. Therefore, for those eleven hours, MEsLIER SERvIcEs has decided to model a cost of $60 per worker for a staffing level that falls below the desired level of service to customers of BAMFoRp BANK. Between 6pm and 7am the corresponding cost is $30 per worker.
ApAMs FINANcIAL is not so sensitive to the time of day, with all calls being equally important. MEsLIER SERvIcEs models this as a cost of $45 per worker for a staffing level that falls below the desired level to service customers of ApAMs FINANcIAL.
MEsLIER SERvIcEs has imposed an additional restriction to try to control the QoS. Namely, it requires that there are at least 80 employees available to answer phones at any one time.
(Technical aside: MEsLIER SERvIcEs expects that the required service level for any given hour will be no larger than the number given in the table with 95% confidence. For the times with low demand, a sudden rise in demand could swamp the employees, so to guard against this MEsLIER SERvIcEs requires a minimum total employee level. MEsLIER SERvIcEs has contracts with each bank that stipulate a desired QoS with penalties for failure to meet those quotas; the short-staffing costs are estimates of the effect of short-staffing on the QoS penalties.)
5 Variables
You will probably need several sets of variables. The principal decision variables are the number of workers starting each shift who are working for each bank. 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.
6 Constraints
You will need several sets of constraints in your model.
7 The data
All the data can be found on LMS, and also on the course webpage at
http://www.rpi.edu/~mitchj/matp4700/callcenter2022/shiftsAB.dat The parameter definitions in the model file can be found on LMS and at
http://www.rpi.edu/~mitchj/matp4700/callcenter2022/shiftsABparam.mod
8 Questions
1. Due: Friday, October 21, 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 $150,000 and $200,000.
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 11.
3. Extra conditions (30 points): To be distributed later. Due on Tuesday December 6.
9 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.
Amos Eaton 325
x6915.
Tuesday 2.30–4pm in AE 325;
Thursday 1–3pm webex: https://rensselaer.webex.com/meet/mitchj
2022-10-19