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Contents College 9. Chapter 9 additional (sheets): workforce planning resource loading. Hierarchical capacity planning. Aggregate/ strategic planning. Strategic. Rough-cut process planning. Resource loading. Tactical. Engineering & process planning. Scheduling. Operational.

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contents college 9
Contents College 9
  • Chapter 9
  • additional (sheets):
    • workforce planning
    • resource loading
hierarchical capacity planning
Hierarchical capacity planning

Aggregate/ strategic planning


Rough-cut process planning

Resource loading


Engineering & process planning



workforce planning
Workforce planning

Long term


  • Strategic level: workforce planning
    • hiring/firing staff
    • training staff
  • Tactical level: resource loading
    • planning overtime work
    • hiring temporary staff (e.g. Randstad)
    • subcontracting
  • Operational level: workforce scheduling
    • staffing
    • shift scheduling
    • days-off scheduling




Short term


capacity planning solution techniques
Capacity planningsolution techniques
  • (Strategic) workforce planning:
    • linear programming
  • (Tactical) resource loading:
    • integer linear programming
    • stochastic techniques
  • (Operational) workforce scheduling:
    • integer linear programming
    • cyclic staffing algorithms
workforce planning lp model 1 product various machines
Workforce planning LP model(1 product, various machines)
  • Demand forecast:

 sales (St) restriction:

  • Machine capacity restriction (machine j):

aj = production time on machine j

cjt = capacity machine j in period t

Xt = amount produced in period t

workforce planning lp model cont
Workforce planning LP model (cont.)
  • Inventory balance restrictions:

It = inventory in period t 

  • Workforce balance restrictions (operators):

Wt = workforce in period t (initial workforce=w)

Ht = hired workforce in period t

Ft = fired workforce in period t

workforce planning lp model cont1
Workforce planning LP model (cont.)
  • Operator capacity restriction (machine j):

b = man-hours required to produce one item

Wt = Workforce in period t

Ot = Overtime in period t

Xt = amount produced in period t

  • objective: maximize net profit, including labor, overtime, inventory, hiring/firing costs
resource loading example furniture manufacturer
Resource loading example:furniture manufacturer

Furniture manufacturer:

  • Produces large quantities of furniture to order
  • Mostly standard products (manufacture-to-order)
  • 6 production activities:

1. Sawmilling (S)

2. Assembly (A)

3. Cleaning (C)

4. Painting (P)

5. Decoration (D)

6. Quality control (Q)


Resource loading example (cont.)

Order acceptance:

  • Sales department negotiates prices & due dates (i.e. delivery dates)with customers
  • Given are for each department:
    • available machine capacity (in hours per week)
    • operator capacity per week (regular & overtime)
resource loading questions
Resource loading questions

During order acceptance, for any given set of orders, these questions need to be answered:

  • When should orders be released for production?
  • Can the delivery dates be met?
  • How much operator & machine capacity is required per department per week?
  • Is irregular capacity (e.g. overtime work, subcontracting) required?

This problem is the so-called resource loading problem

resource loading example cont loading method of planner
Resource loading example (cont.)loading method of planner
  • customer order consists of x jobs, each corresponding to a production activity
  • lead time = 1 week per job
  • operator and machine capacity check per department
  • repair plan, if infeasible
  • let each department solve their scheduling problem

Machine capacity check for D-department:

Machine capacity check for Q-department:


Machine capacity check for D-department:

Optimal loading of D-department:


Machine capacity check for Q-department:

Optimal loading of Q-department:

repairing an infeasible plan
Repairing an infeasible plan

Tardy orders may induce penalty costs

Options to come to a feasible plan:

  • Shift jobs in time; split jobs over 2 or more weeks
  • Increase order lead time
  • Expand operator capacity (overtime, hiring staff)
  • Subcontracting jobs or entire orders
resource loading problems formal description
Resource loading problemsformal description
  • Production system: job shop with operators and machine groups
  • Demand: a predefined set of orders that consist of jobs
  • Time horizon/time unit: T periods (=weeks) / hours
  • Order constraints: release and due dates (weeks)
  • Job constraints:
    • Pre-emption is allowed
    • Precedence relations (generic)
    • Machine and operator requirement (hours)
    • Optional: one-job-per-order-per-week production policy
    • Minimal duration
resource loading problems cont formal description
Resource loading problems (cont.)formal description
  • Resource capacity constraints:
    • Operator capacity: regular, overtime, hiring, subcontracting
    • Machine group capacity: regular
  • Objective:

assign jobs to machines and operators, and:

minimize the cost of the use of non-regular capacity


Resource loadingModel formulation

Order plan:

indicates per job per order per week whether this job may be processed in that week

Input for the model as binary columns

Order schedule:

indicates per job per order per week the fraction that is assigned to that week

 Output of the model; must match with order plan

resource loading model formulation cont
Resource loadingModel formulation (cont.)

Order plan represented by a binary column:





Resource loadingModel objective

Select one order plan per order, and determine the corresponding order schedule. Minimize the use of non-regular capacity.

PROBLEM: there are exponentially many feasible order plans

SOLUTION: column generation approach

column generation algorithm
Column generation algorithm


Restricted Linear Program (RLP)

Add columns

to RLP

Solve pricing algorithm

Columns exist with negative reduced costs?



LP relaxation solved

model formulation cont milp model for resource loading
Model formulation (cont.)MILP model for resource loading

objective function


subject to:

select order plan

match order schedule with order plan

all work must be done

machine capacity restriction

operator capacity restriction

variable restrictions

workforce scheduling topics chapter 9
Workforce scheduling topics (chapter 9)
  • Days-off scheduling
    • assigning employees to work-patterns
    • various assignment patterns over the cycle
  • Shift scheduling
    • assigning employees to shift-patterns
    • each shift has its own cost
    • objective: minimize cost
  • Cyclic staffing problem & extensions
    • assigning people to m-period cyclic schedule, so that requirement in each period is met
  • Crew scheduling
    • transportation crew scheduling (airline industry)
days off scheduling
Days-off scheduling

Find the minimum number of employees to cover a 7-day-a-week operation, so that the following constraints are satisfied:

1. The demand per day (nj) is met (n1=Sunday)

2. Each employee is given each k1 out of k2 weekends off

3. Each employee works exactly 5 out of 7 days

4. Each employee works no more than 6 consecutive days

days off scheduling cont
Days-off scheduling (cont.)

W = required workforce

3 lower bounds on W:

1. weekend constraint:

2. total demand constraint:

3. maximum daily demand constraint:

days off scheduling cont1
Days-off scheduling (cont.)


n = max (n1,n7) = maximum weekend demand

uj = surplus of employees = W - nj (if j=2,…,6) n - nj (if j=1,7)


the first day to be scheduled is a Saturday

days off scheduling algorithm
Days-off scheduling algorithm
  • STEP 1:schedule the weekends off

maximum demand in weekend is n  W-n empl. Free

 Cyclically assign the weekends of to W-n employees

  • STEP 2:Determine the additional off-day pairs

Construct a list of n “off day”-pairs (k,l):

    • choose day k, such that uk =max (u1,…,u7)
    • choose day l (l  k) such that ul 0, if ul = 0 for all l, then set l = k.
    • decrease uk and ul with 1

Pairs (k,k) are non-distinctive pairs

days off scheduling algorithm cont




week i

Days-off scheduling algorithm (cont.)
  • STEP 3:Categorize employees in week i

type T1 weekend i off weekend i+1 off 0

type T2 weekend i off weekend i+1 on 1

type T3 weekend i on weekend i+1 off 1

type T4 weekend i on weekend i+1 on 2

note: |T3|+|T4| = n, and |T2|+|T4| = n |T2|=|T3|

 pair each employee of T2 with an employee of T3

days off scheduling algorithm cont1
Days-off scheduling algorithm (cont.)
  • STEP 4:Assign the n off-day pairs in week i

First assign off-day pairs to the employees of T4

Then assign off-day pairs to the T2-T3 pairs of employees. Assign the earliest day to the T3-empl.

Set i = i+1 and GO TO STEP 3.

days off scheduling example
Days-off scheduling example

Demand requirement:

n = max(d1,d7)=2


day-pairs: Sun-Mon & Mon-Mon

days off scheduling example cont2
Days-off scheduling example (cont.)

Minimum 4-day, maximum 6-day work stretch:

days off scheduling algorithm properties
Days-off scheduling algorithm properties
  • If all off-pairs are distinct  maximum work-stretch is 5 days
  • Schedule always satisfies the constraints
  • There exists an optimal cyclic schedule, that may be found by the algorithm
shift scheduling
Shift scheduling
  • Generalization of days-off scheduling problem
  • Cycle of periods is predetermined (e.g. 1 day cycle, with periods of hours)
  • Several shift patterns, with associated costs cj. Shift pattern j is binary vector a(i,j).
  • Personnel requirement bi per period

solution method: linear programming

shift scheduling ilp model
Shift scheduling ILP model

N patterns

m periods

bi = requirement

shift scheduling ilp model properties
Shift scheduling ILP model properties
  • Strongly NP-hard problem
  • When shift-columns contain a contiguous set of 1’s, the LP-relaxation solution is integer (note: LP optimization in polynomial time)
  • Other special cases of shift scheduling also solvable: cyclic staffing problem
cyclic staffing problem
Cyclic staffing problem
  • Special case of shift planning problem
  • m-period cyclic schedule: period m is followed by period 1
  • Personnel requirement is bi per period
  • Each person works k consecutive periods, and is free for the remaining m-k periods
  • Example: (3,5)-cyclic staffing problem:
  • Columns of A do not always have a

contiguous set of ones  LP solution

close to ILP solution

exact cyclic staffing algorithm
Exact cyclic staffing algorithm

STEP 1: solve LP-relaxation  solution vector x*

if x* integer  done, else: go to step 2

STEP 2: form two linear programs LP’ and LP’’, so that these constraints are added respectively:

LP’’ has an optimal solution that is integer

if LP’ is not feasible  LP’’ solution optimal

otherwise: LP’ has an optimal solution that is integer, and the optimal solution is the best of the solutions of LP’ and LP’’

extensions of cyclic staffing
Extensions of cyclic staffing
  • Days-off scheduling with a fixed cycle
    • each employee 2 days off a week
    • every other weekend off (e.g. 1st, 3rd, 5th, etc.)

 solved by column generation

  • Cyclic staffing with overtime
    • 3 work shifts: 8:00-16:00, 16:00-0:00, 0:00-8:00
    • matrix A:

 solved by algorithm 9.4.1 (cyclic staffing algorithm)

extensions of cyclic staffing cont
Extensions of cyclic staffing (cont.)
  • Cyclic staffing with linear penalties for under- and overstaffing
    • varying demand bi
    • penalty ci’ and ci’’ for under- and overstaffing
    • understaffing = x’

 overstaffing = bi- (ai1x1+ … + ai1x1) - xi’


if c-c’’A0 and c’-c’’ 0  solvable by alg. 9.4.1

crew scheduling
Crew scheduling
  • Applications in airline/transportation industry
  • Each crew performs a number of jobs (flight legs) in a so-called ‘round trip’
  • There are m jobs (flight legs), and n feasible round trips
  • A round trip has costs cj
  • objective: choose a set of round trips, so that each flight leg is covered exactly once, by one and only one round trip  set partitioning problem
  • solution method: column generation heuristic on ILP model