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Contents College 9 - PowerPoint PPT Presentation

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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• Chapter 9

• workforce planning

Hierarchical capacity planning

Aggregate/ strategic planning

Strategic

Rough-cut process planning

Tactical

Engineering & process planning

Scheduling

Operational

Long term

decisions

• Strategic level: workforce planning

• hiring/firing staff

• training staff

• planning overtime work

• hiring temporary staff (e.g. Randstad)

• subcontracting

• Operational level: workforce scheduling

• staffing

• shift scheduling

• days-off scheduling

Intermediate

term

decisions

Short term

decisions

Capacity planningsolution techniques

• (Strategic) workforce planning:

• linear programming

• integer linear programming

• stochastic techniques

• (Operational) workforce scheduling:

• integer linear programming

• cyclic staffing algorithms

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

• 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

• 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

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)

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)

Example customer order:

S

A

C

P

D

Q

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?

• 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 Q-department:

Tardy orders may induce penalty costs

Options to come to a feasible plan:

• Shift jobs in time; split jobs over 2 or more weeks

• Expand operator capacity (overtime, hiring staff)

• Subcontracting jobs or entire orders

• 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 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

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

Order plan represented by a binary column:

2

1

3

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

Solve

Restricted Linear Program (RLP)

to RLP

Solve pricing algorithm

Columns exist with negative reduced costs?

yes

no

LP relaxation solved

objective function

minimize

subject to:

select order plan

match order schedule with order plan

all work must be done

machine capacity restriction

operator capacity restriction

variable restrictions

• 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)

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

W = required workforce

3 lower bounds on W:

1. weekend constraint:

2. total demand constraint:

3. maximum daily demand constraint:

Define:

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

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

Assumption:

the first day to be scheduled is a Saturday

• 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

required

during

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

• 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.

Demand requirement:

n = max(d1,d7)=2

surplus:

day-pairs: Sun-Mon & Mon-Mon

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

• 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

• 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

N patterns

m periods

bi = requirement

• 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

• 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

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’’

• 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)

• 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’

model:

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

• 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

N round trips

m jobs