HAROSA 2013 Barcelona July WORKFORCE PLANNING AND SCHEDULING WITH CROSS-TRAINING AND LEARNING Scott E. Grasman Industrial & Systems Engineering Rochester Institute of Technology

 Optimized worker scheduling in a production process or service setting.  Worker allocation models 1.Single/Multi Period Worker-Task Allocation 2.Short-Term Cross-Training 3.Mid-Term Workforce Planning 4.Long-Term Workforce Development OUTLINE

 Other considerations for short-term worker allocation models:  Overtime assignments  Individual worker preferences  Factory rules and limitations  Reduction of operational gaps  Shift structures  Learning?  Approach/Solution: Two-Phase Goal (MIP) Programs  see Jarugumilli et al (2011) SINGLE/MULTI PERIOD WORKER-TASK ALLOCATION

CROSS-TRAINING MODELS  Workforce allocation may contain costly operational gaps: Approaches/Solution:  Overtime allocations are temporary solutions.  Effective cross-training plans are an essential component of worker allocation models.  Identification of operations and workers for cross-training and implementation of the plans on a shift-by-shift basis.  Learning is important. Workers of a particular skill level required at an operation Workers of that skill level assigned to the operation Operational Gap at that skill level

PROBLEM STATEMENT

FORMULATION – SETS

MATHEMATICAL FORMULATION Assignment Cost Overtime Allocation Cost Cost of Workforce Surplus Cost of Cross-training Cost of Slack in Cross-training Cost of surplus in Cross-training Requirements for training and for normal worker allocation

MATHEMATICAL FORMULATION Constraints which determine the binary variable that depends on the normal work assignment Limitation on the minimum shift duration that the worker can be assigned for Worker shift duration constraint Overtime Limit A worker is either allocated normally or trained in a shift

MATHEMATICAL FORMULATION Limit on cross-training for both individual shifts and the complete factory Training duration should be continuous Constraints that control the condition that training for L2 and L3 cannot take place if there is normal work of any skill level taking place at the same operation.

MATHEMATICAL FORMULATION The start of training is recorded and the workers are assigned to training if the training takes place. A worker cannot be trained multiple times on the same OP-Skill combo Training variables will be carried forward to the shifts of the next time horizon

MATHEMATICAL FORMULATION Connecting the end of training and the start of training Training can only occur on one skill level at any operation in any shift. Calculating the shifts where the worker can work or get trained in higher skill level after his training takes place.

MATHEMATICAL FORMULATION Constraints on the minimum and maximum possible qualifications of a worker. Workers can only be trained on L3, if they are qualified on L2; and they can only be trained on L2 if they are qualified on L1 for the same operation.

MATHEMATICAL FORMULATION Pre-allocation for training Partial allocation for work

RESULTS  Industrial Test Cases ~ workers and ~ operations  Solve in seconds (when linear) or minutes (when non-linear)  Reduction in overtime and operational gaps (see Gandhi et al 2013)

CONTINUED WORK  Extension  Worker Qualification is Non-Linear  Parallel Work: Mid-Term Workforce Planning  Stochastic Models  Future Work: Long-Term Workforce Development  Non-Linear Learning Models

Aim: determine the size of the workforce at the start of the planning horizon in order to increase flexibility within the workforce (without double counting workers):  shift swaps, cross-training, and overtime  hiring and firing  overtime once the actual demand is revealed Approach/Solution: Two-Stage Stochastic Programming 1.In the first stage, size of the workforce will be built using workers from previous planning horizon, shift-swaps, hiring and cross training 2.In the second stage, actual value of the demand will be revealed that will decide the recourse, i.e. required overtime WORKFORCE PLANNING

MODEL :OBJECTIVE FUNCTION

MODEL : CONSTRAINTS

SOLUTION APPROACH Scenario based Approach  see Kulkarni et al (2013) 1.Generate demand scenarios, e.g., low, average, high 2.Determine the probability of occurrence of each demand scenario  Converts the problem into an extended deterministic version of the “original” model!

SOLUTION APPROACH: IMPLEMENTATION

COMPLEXITY OF THE PROBLEM For: j - operations, k - shifts l - skill levels r - possible demands for each operation Scenarios3 demand cases for j operations rjrj Constraints15 linear constraints2*j*k*l*(n r ) Variables8 continuous variables3*j*k*l*(n r )

Solved using Cplex 12.5 using C# Larger problems solved by limiting the scenarios Good opportunity for an analytical method or simulation optimization. TEST CASES OperationsSkillsShifts Demand CasesScenarios Time* (Seconds) First Stage Variables Second stage VariablesTotal Constraints

The learning organization is “continually expanding its capacity to create its future” - Peter Senge, The Fifth Discipline: The Art & Practice of the Learning Organization LONG-TERM WORKFORCE DEVELOPMENT

 Quantitative, descriptive models of how humans learn are non-linear: WHAT’S THE ISSUE?

LEARNING MODEL Production rate of worker i on task j in period  Did i do task j in period t? Learning term Forgetting term Model proposed in:

LEARNING AND FORGETTING Forgetting leads to reduction in rate Rate in period 10 same under all plans

 5 workers, 10 tasks, and 10 periods  Knitro 7.0 unable to find a feasible solution for any instance within 2 hours HOW BIG OF AN INSTANCE CAN YOU SOLVE?

A VERY SIMPLE IDEA..... Suppose a model includes the following relation between variables y and x Because x has a finite and countable domain we can model it and the non-linear function with binary variables

RETURNING TO OUR PROBLEM.... Learning function has a finite and countable domain (number of times i performed task j between periods 0 and  ) 1. Introduce a new binary variable (and linking constraints) to the model: Did i do j k times between periods 0 to t? 2. Enumerate production rates in a period.

 Used Gurobi 5.1  Solver parameters:  one hour time limit  1% relative gap  1 unit absolute gap (for these instances objective function values are typically less than 20 so relative gap not as meaningful) EXPERIMENTAL SETUP TO TEST REFORMULATION

 Quantitative, descriptive models of how humans learn are non-linear: RECALL LEARNING CURVES…

INITIAL RESULTS instances with a heterogeneous workforce are the hardest to solve even though more workers means more z variables, instances easier to solve

 The reformulated problems are significantly easier to solve and thus much bigger instances than what is seen in the literature can be solved (though solution time does depends on size and initial buffers/capacity).  see Chacosky et al (2012) Future  Develop an effective heuristic based on the reformulation?  Tighten the formulation?  Dynamic formulation?  Integrate uncertainty in learning rates? Simulation??  Interface with planning and cross-training models! CONCLUSIONS & FUTURE WORK

Scott E. Grasman, Ph.D.  Professor and Department Head, Industrial and Systems Engineering  Kate Gleason College of Engineering Rochester Institute of Technology  81 Lomb Memorial Drive, Rochester, NY  (office) (fax)  (or  YOU ARE ALL WELCOME TO VISIT ROCHESTER!!! QUESTIONS?