Download - Aggegate planning 2016.pdf
-
7/25/2019 Aggegate planning 2016.pdf
1/78
Aggregate Planning
Prof Indrajit Mukherjee
-
7/25/2019 Aggegate planning 2016.pdf
2/78
Dependent and Independent Demand
Independent Demand
100 tables
Dependent Demand
100 x 1 tabletops
100 x 4 table legs
-
7/25/2019 Aggegate planning 2016.pdf
3/78
Demand (Priority) Resources(Capacity)
Why Plan?
To satisfy customer demand, ensure the availability of resources
-
7/25/2019 Aggegate planning 2016.pdf
4/78
How do you cope with fluctuations indemand?
Absorb
Demand Adjust output tomatch demand
Change
demand
-
7/25/2019 Aggegate planning 2016.pdf
5/78
Absorb demand
Keep output level
Make tostock
Makecustomer
wait
Part finished,
Finished Goods, or
Customer Inventory
Queues
Backlogs
Haveexcess
capacity
-
7/25/2019 Aggegate planning 2016.pdf
6/78
6
Adjust output tomatch demand
Hire Fire
TemporaryLabour
Lay-off
Overtime
Subcontract
Short time
3rd party work
-
7/25/2019 Aggegate planning 2016.pdf
7/78
Change demand
Change pattern of demand
Develop alternative products and/orservices
-
7/25/2019 Aggegate planning 2016.pdf
8/78
Advertising, pricing & product promotion(shifting demand into other time periods)
Counter cyclicbut similar type of products.Similar products with negatively correlated
demands, e.g. Snow blowers and Lawn
Mowers. AC pumps and Heater pumps
Partnering with suppliers to reduce informationdistortion along the supply chain
Prior Reservation systems
Demand Influencing Tactics
-
7/25/2019 Aggegate planning 2016.pdf
9/78
Ways of reconci ling capacity and demand
Capacity
Demand Demand
Capacity
Demand
Capacity
-
7/25/2019 Aggegate planning 2016.pdf
10/78
Masterproduction
schedule andMRP
systems
Detailedwork
schedules
Processplanning and
capacitydecisions
Aggregateplan for
production
A Holistic View of Planning
Productdecisions
Demandforecasts,
orders
Marketplaceand
demand
Research
and
technology
Rawmaterialsavailable
Externalcapacity
(subcontractors)
Workforce
Inventoryon
hand
Capacity
Demand
Lecture Slides by Indrajit SJMSOM
-
7/25/2019 Aggegate planning 2016.pdf
11/78
Pull and push philosophies of planning andcontrol
CENTRAL OPS. PLANNING AND CONTROL SYSTEM
Workcentre
DEMANDWorkcentre
Workcentre
Workcentre
Instruction onwhat to makeand where to
send it
FORECASTOR
PULL CONTROL
Workcentre
Workcentre
Workcentre
Workcentre DEMAND
Request Request Request Request
Delivery Delivery Delivery Delivery
PUSH CONTROL
-
7/25/2019 Aggegate planning 2016.pdf
12/78
MRPII
Manufacturing Resource Planning
An expanded system for determining manufacturing
resource requirements and for scheduling production.
Make sure you have enough parts when you needthem
Take future demands, factor in lead times (time
phase), compare to on-hand inventory, place
order
Determine order size and timing
-
7/25/2019 Aggegate planning 2016.pdf
13/78
Core MRP II (Manufacturing Resource Planning)
Aggregate ForecastAggregate Production
Plan
Detailed Forecast Master ProductionSchedule
Rough-Cut CapacityPlanning
Material
RequirementsPlanning
Capacity
RequirementsPlanning
Production Activity
Control
PASSENGER CAR FAMILY
(Independent Demand)
PASSENGER CAR MODE
PASSENGER CAR PARTS
(Dependent Demand)
Resource requirement
Planning
-
7/25/2019 Aggegate planning 2016.pdf
14/78
THE BIG PICTURE
A production and/or purchasing
schedule for Dependent DemandItems
(parts / components)
Material Requirement Plan(MRP)
Master Production Schedule(MPS)
A production schedule for IndependentDemand Items(end items / parent items)
The demand for this stuffis dependent on
the independent demand for this.
-
7/25/2019 Aggegate planning 2016.pdf
15/78
Production Plan
Months January February
Aggregate Production Plan 1,500 1,200(Shows the totalquantity of amplifiers)
Weeks 1 2 3 4 5 6 7 8Master Production Schedule(Shows the specific type andquantity of amplifier to beproduced
240-watt amplifier 100 100 100 100
150-watt amplifier 500 500 450 45075-watt amplifier 300 100
-
7/25/2019 Aggegate planning 2016.pdf
16/78
Demand Patterns
A pattern plus random influences Lumpy because of production lots
-
7/25/2019 Aggegate planning 2016.pdf
17/78
Dickson Chiu 2006
Why demand becomes lumpy
Time
Time
Orderpoint
Orderpoint
Le
vel
Level
0
0
Production orderrelease
Orderplacement
(a) Field inventory (Finished product in warehouse)
(b) Factory inventory (Finished product at plant)
Time
Order
point
Level
0
Purchase orderrelease
(c) Component inventory (Supply stocks at plant)
-
7/25/2019 Aggegate planning 2016.pdf
18/78
Dependent vs Independent Demand
Time
Time Time
Time
Demand
Demand
Stable demand Lumpy demand
Amountonhand
Amounto
nhand
Safety stock
-
7/25/2019 Aggegate planning 2016.pdf
19/78
Historical Perspective
mrp material
requirements
planning
MRP II Manufacturing
Resource Planning
ERP- Enterprise
Resource Planning
-
7/25/2019 Aggegate planning 2016.pdf
20/78
ERP
Sales and
distribution
Financial
accounting
Financial
controlling
Fixed assets
management
Human resources Work flow Industry solutions Materials
management
Production
planning (including
MRP and CRP)
Quality
management Plant maintenance Project systems
-
7/25/2019 Aggegate planning 2016.pdf
21/78
Aggregate Plan
*Aggregate Plan: A statement of a companys production rates,
workforce levels, and inventory holding based on estimates
of customer requirements and capacity limitations
Service Industry Staffing Plan
Manufacturing Industry Production Plan
*Decisions made at a product family (not SKU) level
-
7/25/2019 Aggegate planning 2016.pdf
22/78
Product Classification Hierarchy
Product typePump
Family ASmall pumps
Family BMedium pumps
Family CLarge pumps
S1 S2 S3 M1 M2 M3 L1 L2 L3
-
7/25/2019 Aggegate planning 2016.pdf
23/78
Aggregate
Farm tools:
Shovels
Spades
Forks
Aggregate by similar characteristics
Generic tool, call itShovel
Same characteristics?
-
7/25/2019 Aggegate planning 2016.pdf
24/78
13-5
Aggregate Planning: Objectives and
Approaches
Objectives:
Match aggregate Capacity with aggregate Demand(Effectiveness)
Minimize Costs (Efficiency)
Demand(Priority)
Supply (Capacity:(Machine, Manpower;Material and Products in
Inventory)
-
7/25/2019 Aggegate planning 2016.pdf
25/78
Objectives of capacity planning and control
Step 1 - Measure aggregatecapacity and demand.
Step 2 - Identify the alternativecapacity plans.
Step 3 - Choose the most
appropr iate capacityplan.
Time
Estimate of current capacity
Forecast demand
Aggregatedoutput
-
7/25/2019 Aggegate planning 2016.pdf
26/78
Strategies for Adjusting Capacity
Level production Producing at a constant rate and
using inventory to absorb
fluctuations in demand Chase demand
Hiring and firing workers tomatch demand
Subcontracting demand
Maintaining resources forlowest-demand levels and subcontact in case of higherdemand
Overtime and under-time Increasing or decreasing
working hours
Part-time workers Hiring part time workers to
complete the work
Backordering Providing the service or product
at a later time period
-
7/25/2019 Aggegate planning 2016.pdf
27/78
Ways of reconci ling capacity and demand
Level capacity
Capacity
Demand
Chase demand
Demand
Capacity
Demandmanagement
Demand
Capacity
-
7/25/2019 Aggegate planning 2016.pdf
28/78
Pure Aggregate Strategies
-
7/25/2019 Aggegate planning 2016.pdf
29/78
Example
Youve started a newcompany. Youve developed 2
production plans:
Month Forecast Plan 1 Plan 2
Jan 900 900 800Feb 700 700 800
Mar 800 800 800
You estimate 1 worker can make 100 units per month.
Which plan do you use? How many workers do you hire?How do you meet demand?
-
7/25/2019 Aggegate planning 2016.pdf
30/78
Level Planning
Month Expected Demand Production DaysDemand Per Day
(computed)
Jan 900 22 41
Feb 700 18 39
Mar 800 21 38
Apr 1,200 21 57
May 1,500 22 68
June 1,100 20 55
6,200 124
= = 50 units per day6,200
124
Averagerequirement =
Total expected demand
Number of production days
-
7/25/2019 Aggegate planning 2016.pdf
31/78
Level Production Planning Strategy
70
60
50
40
30
0 Jan Feb Mar Apr May June = Month
22 18 21 21 22 20 = Number of working days
Productionratep
erworkingday
Level product ion using averagemonthly forecast demand
Forecast demand
Lecture Slides by Indrajit SJMSOM
-
7/25/2019 Aggegate planning 2016.pdf
32/78
Aggregate Production Planning
Production rate: quantity of product produced perunit of time (autos/day).
Workforce level: number of workers required to
meet a specific level of output.
Ending Inventory Position: unsold units carried
over from one period to the next.
Ending inventory for any period = opening inventory + product ion demand= _____ + _____ ____= _____ units
-
7/25/2019 Aggegate planning 2016.pdf
33/78
Aggregate Planning
Aggregate Plans(1) Production Planning
(2) Resource Requirement planning [e.g., Labor, material] Plan
(Typically these two are generated simultaneously)
Purpose is to specify the optimal combination of production rate,
the workforce level, and inventory-on-hand
Given the demand forecast Ft for each period t in the planninghorizon over T periods, determine the production rate Pt,
Inventory level It, and workforce level Wt for periods t = 1, 2, , T,
that minimizes the relevant costs over the planning horizon (T).
-
7/25/2019 Aggegate planning 2016.pdf
34/78
Aggregate Planning Approach
1. Trial and Error Approach
2. Mathematical Approaches
a) Linear Programming
b) Search Heuristics
Lecture Slides by Indrajit SJMSOM
-
7/25/2019 Aggegate planning 2016.pdf
35/78
The Aggregate Production Plan
Consider the following aggregate demand forecast:
The plan is stated in the common unit of capacity direct labor
hours
Period Q1 Q2 Q3 Q4
Forecasted Demand (D) 5,000 0 20,000 5,000
Aggregate Production (AP)
-
7/25/2019 Aggegate planning 2016.pdf
36/78
Suppose we want to try to match demand:
Period Q1 Q2 Q3 Q4Forecasted Demand (D) 5,000 0 20,000 5,000
Aggregate Production (AP) 5,000 0 20,000 5,000
The Aggregate Chase ProductionPlan
Lecture Slides by Indrajit SJMSOM
-
7/25/2019 Aggegate planning 2016.pdf
37/78
Costs of Maintaining and Changing Production
Levels
Costs of keeping production steady:
Cc -- Inventory Holding cost ($ / unit / quarter)
Cb -- Backorder cost ($ / unit / quarter)
Costs of changing production
CH -- Hiring cost ($ / Worker) CF -- Firing cost ($ / Worker)
Also need to worry about:
Cw -- Wage rate ($/unit)
-
7/25/2019 Aggegate planning 2016.pdf
38/78
Projected Available Balance is simply the estimated ending
inventory for each period
Period Q1 Q2 Q3 Q4Forecasted Demand (D) 5,000 0 20,000 5,000
Aggregate Production (AP) 5,000 0 20,000 5,000
Projected Available Balance (EI) 0 0 0 0 0
Workers 10
The Aggregate Production Plan
Lecture Slides by Indrajit SJMSOM
-
7/25/2019 Aggegate planning 2016.pdf
39/78
The # workers needed is based on the planned production and
average worker productivity here, say 500 Item
Produced/Worker/Quarter
Period Q1 Q2 Q3 Q4Forecasted Demand (D) 5,000 0 20,000 5,000
Aggregate Production (AP) 5,000 0 20,000 5,000
Projected Available Balance (EI) 0 0 0 0 0
Workers 10 10 0 40 10
The Aggregate Production Plan
Lecture Slides by Indrajit SJMSOM
-
7/25/2019 Aggegate planning 2016.pdf
40/78
The Aggregate Production Plan
For our example, the wage rate is $ 10/Item
Period Q1 Q2 Q3 Q4
Forecasted Demand (D) 5,000 0 20,000 5,000
Aggregate Production (AP) 5,000 0 20,000 5,000
Projected Available Balance (EI) 0 0 0 0 0
Workers 10 10 0 40 10
Wages Cost 50,000 0 200,000 50,000
Hiring Cost
Firing CostHolding Cost
Backorder Cost
Total Cost
Lecture Slides by Indrajit SJMSOM
-
7/25/2019 Aggegate planning 2016.pdf
41/78
The hiring and firing costs are $ 2000 and $ 500, respectively
Period Q1 Q2 Q3 Q4
Forecasted Demand (D) 5,000 0 20,000 5,000
Aggregate Production (AP) 5,000 0 20,000 5,000
Projected Available Balance (EI) 0 0 0 0 0
Workers 10 10 0 40 10
Wages Cost 50,000 0 200,000 50,000
Hiring Cost 0 0 80,000 0
Firing Cost 0 5,000 0 15,000
Holding Cost
Backorder Cost
Total Cost
The Aggregate Production Plan
Lecture Slides by Indrajit SJMSOM
-
7/25/2019 Aggegate planning 2016.pdf
42/78
Holding and backorder costs are $ 1 and $ 3 /per unit/Quarter,
respectively
Period Q1 Q2 Q3 Q4Forecasted Demand (D) 5,000 0 20,000 5,000
Aggregate Production (AP) 5,000 0 20,000 5,000
Projected Available Balance (EI) 0 0 0 0 0
Workers 10 10 0 40 10
Wages Cost 50,000 0 200,000 50,000
Hiring Cost 0 0 80,000 0
Firing Cost 0 5,000 0 15,000
Holding Cost 0 0 0 0
Backorder Cost 0 0 0 0
Total Cost
The Aggregate Production Plan
Lecture Slides by Indrajit SJMSOM
-
7/25/2019 Aggegate planning 2016.pdf
43/78
Which gives us a projected cost of $ 400,000 for the year
Period Q1 Q2 Q3 Q4
Forecasted Demand (D) 5,000 0 20,000 5,000
Aggregate Production (AP) 5,000 0 20,000 5,000Projected Available Balance (EI) 0 0 0 0 0
Workers 10 10 0 40 10
Wages Cost 50,000 0 200,000 50,000
Hiring Cost 0 0 80,000 0
Firing Cost 0 5,000 0 15,000
Holding Cost 0 0 0 0
Backorder Cost 0 0 0 0
Total Cost 50,000 5,000 280,000 65,000
400,000
The Aggregate Production Plan
Lecture Slides by Indrajit SJMSOM
-
7/25/2019 Aggegate planning 2016.pdf
44/78
Chase, Level, & Mixed Strategies
This example is the first of the two "pure" strategies
Chase strategy: make only as much as you can sell
The other pure strategy has a different cost structure Level strategy: maintain a steady production rate
In reality, a firm's strategy will be a combination or mixed
strategyof chase and level
-
7/25/2019 Aggegate planning 2016.pdf
45/78
A level strategy will absorb fluctuations in demand through
fluctuations in inventory
Period Q1 Q2 Q3 Q4
Forecasted Demand (D) 5,000 0 20,000 5,000Aggregate Production (AP) 7,500 7,500 7,500 7,500
Projected Available Balance (IP) 0
Workers 10
Wages Cost
Hiring Cost
Firing Cost
Holding Cost
Backorder Cost
Total Cost
The Aggregate Production Plan
Lecture Slides by Indrajit SJMSOM
-
7/25/2019 Aggegate planning 2016.pdf
46/78
A level strategy will absorb fluctuations in demand through
fluctuations in inventory
Period Q1 Q2 Q3 Q4Forecasted Demand (D) 5,000 0 20,000 5,000
Aggregate Production (AP) 7,500 7,500 7,500 7,500
Projected Available Balance (IP) 0
Workers 10 15 15 15 15
Wages Cost
Hiring CostFiring Cost
Holding Cost
Backorder Cost
Total Cost
The Aggregate Production Plan
Lecture Slides by Indrajit SJMSOM
-
7/25/2019 Aggegate planning 2016.pdf
47/78
A level strategy will absorb fluctuations in demand through
fluctuations in inventory
Period Q1 Q2 Q3 Q4Forecasted Demand (D) 5,000 0 20,000 5,000
Aggregate Production (AP) 7,500 7,500 7,500 7,500
Projected Available Balance (EI) 0 2,500
Workers 10 15 15 15 15
Wages Cost
Hiring CostFiring Cost
Holding Cost
Backorder Cost
Total Cost
The Aggregate Production Plan
Lecture Slides by Indrajit SJMSOM
-
7/25/2019 Aggegate planning 2016.pdf
48/78
A level strategy will absorb fluctuations in demand through
fluctuations in inventory
Period Q1 Q2 Q3 Q4Forecasted Demand (D) 5,000 0 20,000 5,000
Aggregate Production (AP) 7,500 7,500 7,500 7,500
Projec ted Available Balance (IP) 0 2,500 10,000
Workers 10 15 15 15 15
Wages Cost
Hiring CostFiring Cost
Holding Cost
Backorder Cost
Total Cost
The Aggregate Production Plan
Lecture Slides by Indrajit SJMSOM
-
7/25/2019 Aggegate planning 2016.pdf
49/78
A level strategy will absorb fluctuations in demand through
fluctuations in inventory
Period Q1 Q2 Q3 Q4Forecasted Demand (D) 5,000 0 20,000 5,000
Aggregate Production (AP) 7,500 7,500 7,500 7,500
Projected Available Balance (IP) 0 2,500 10,000 -2,500 0
Workers 10 15 15 15 15
Wages Cost
Hiring CostFiring Cost
Holding Cost
Backorder Cost
Total Cost
The Aggregate Production Plan
Lecture Slides by Indrajit SJMSOM
-
7/25/2019 Aggegate planning 2016.pdf
50/78
A level strategy will absorb fluctuations in demand through
fluctuations in inventory
Period Q1 Q2 Q3 Q4Forecasted Demand (D) 5,000 0 20,000 5,000
Aggregate Production (AP) 7,500 7,500 7,500 7,500
Projected Av ailable Balance (IP) 0 2,500 10,000 -2,500 0
Workers 10 15 15 15 15
Wages Cost 75,000 75,000 75,000 75,000
Hiring Cost 10,000 0 0 0
Firing Cost 0 0 0 0
Holding Cost 2,500 10,000 0 0
Backorder Cost 0 0 7,500 0
Total Cost 87,500 85,000 82,500 75,000
330,000
The Aggregate Production Plan
Holding and backorder costs are Rs 1 and Rs 3/AP/Quarter, respectively
Holding costs - charged on inventory at the end
-
7/25/2019 Aggegate planning 2016.pdf
51/78
Subcontracting
Advantages:
No excess capacity
Level production
Disadvantage:
Costs of subcontracting
Meet demand bysubcontracting
Lecture Slides by Indrajit SJMSOM
-
7/25/2019 Aggegate planning 2016.pdf
52/78
Subcontracting Planning
Month Expected DemandProduction
DaysDemand Per Day
(computed)
Jan 900 22 41
Feb 700 18 39
Mar 800 21 38
Apr 1,200 21 57May 1,500 22 68
June 1,100 20 55
6,200 124
Minimum requirement = 38 units per day
Lecture Slides by Indrajit SJMSOM
-
7/25/2019 Aggegate planning 2016.pdf
53/78
Subcontracting Planning
70
60
50
40
30
0 Jan Feb Mar Apr May June = Month
22 18 21 21 22 20 = Number of working days
Productionrateperworkingday
Level productionusing lowest
monthly forecast
demand
Forecast demand
Combination or Mixed Strategy
-
7/25/2019 Aggegate planning 2016.pdf
54/78
Combination or Mixed Strategy
Demand is matched to some extent , production is partially smoothenedIn peak period some subcontracting take place
Mi d O i l S U i LPP
-
7/25/2019 Aggegate planning 2016.pdf
55/78
Mixed Optimal Strategy Using LPP
Hiring cost = $100 per worker
Firing cost = $500 per worker
Inventory carrying cost = $0.50 pound per quarter/item
Production per employee = 1,000 pounds per quarter
Beginning work force = 100 workers
QUARTER SALES FORECAST
Spring 80,000
Summer 50,000
Fall 120,000
Winter 150,000
A candy company makes variety of candies in three factoriesworldwide. Its line of chocolate candies exhibit highly seasonal
demand pattern with peak during winter, and valley during summer.Given the cost and quarterly sales forecast, determine a suitablecost effective production Plan.
-
7/25/2019 Aggegate planning 2016.pdf
56/78
General Linear Programming (LP) Model
LP gives an optimal solution, but demand andcosts must be linear
Let
Wt= workforce size for period t Pt =units produced in period t
It =units in inventory at the end of period t
Ft =number of workers fired for period t
Ht= number of workers hired for period t
-
7/25/2019 Aggegate planning 2016.pdf
57/78
whereH
t= # hired for period t
Ft
= # fired for period tIt
= inventory at end
of period tP
t= units produced
in period tW
t= workforce size
for period t
Minimize Z = $100 (H1 + H2 + H3 + H4)
+ $500 (F1 + F2 + F3 + F4)+ $0.50 (I1 + I2 + I3 + I4)
Subject to
P1 - I1 = 80,000 (1)
Demand I1 + P2 - I2 = 50,000 (2)
constraints I2 + P3 - I3 = 120,000 (3)I3 + P4 - I4 = 150,000 (4)
Production 1000 W1 = P1 (5)
constraints 1000 W2 = P2 (6)
1000 W3 = P3 (7)
1000 W4 = P4 (8)100 + H1 - F1 = W1 (9)
labour/Work force W1 + H2 - F2 = W2 (10)
constraints W2 + H3 - F3 = W3 (11)
W3 + H4 - F4 = W4 (12)
APP Mixed Optimal Strategy Using LPP
G li d f l ti f A t
-
7/25/2019 Aggegate planning 2016.pdf
58/78
Generalized formulation for Aggregate
Planning Problems using Linear Programming
Dt the forecasts of demand for aggregate units for period t, t = 1 T
nt number of units that can be made by one worker in period t
CtP cost to produce one unit in period t
CtW cost of one worker in period t
Ct
H cost to hire one worker in period t
CtL cost to layoff one worker in period t
CtI cost to hold one unit in inventory in period t
CtB cost to backorder one unit in period t
Wt number of workers available in period tPt number of units produced in period t
It number of units held in the inventory at the end of period t
Ht number of workers hired in period t
Ft number of workers fired in period t
known info
decision variables
-
7/25/2019 Aggegate planning 2016.pdf
59/78
Solver
Example LP Model Cost$32,00
0
Beg Wkforce 100 Beg Inv. 0 Firing cost $500Units/wker 1000 Inv. Cost $0.50 Hiring cost $100
Qtr Demand Production Inventory Wkers Needed Wkers Hired
Wkers
Fired
Demand
Constraint
Production
Constraint
Wkforce
Constrai
nt
1 80,000 80,000 0 80 0 20 80,000 80,000 802 50,000 80,000 30,000 80 0 0 50,000 80,000 803 120,000 90,000 0 90 10 0 120,000 90,000 904 150,000 150,000 0 150 60 0 150,000 150,000 150
Total 400,000 400,000 30,000 70 20
Note: LP approach ignores Economic Lot Sizing policy for production run
Solve
APP (Capacity Constraint) by the Transportation
-
7/25/2019 Aggegate planning 2016.pdf
60/78
APP (Capacity Constraint) by the TransportationMethod (A Special form of LPP)
1 900 1000 100 500
2 1500 1200 150 5003 1600 1300 200 500
4 3000 1300 200 500
Regular production cost per unit $20
Overtime production cost per unit $25Subcontracting cost per unit $28
Inventory holding cost per unit per period $3
Beginning inventory 300 units
Unused capacity cost $0
No Backorder Allowed (in this example)
EXPECTED REGULAR OVERTIME SUBCONTRACT
QUARTER DEMAND CAPACITY CAPACITY CAPACITY
(Assumed Cost & Constraint fuct ion are Linearand nobackorderor change in level of worker)
-
7/25/2019 Aggegate planning 2016.pdf
61/78
Network Diagram
Source 1
Source 2
Source 3
Source n
.
.
.
Destination 1
Destination 2
Destination 3
Destination m
.
.
.
Supply
S1
S2
S3
Sn
Demand
D1
D2
D3
Dm
c11
c12c13c1m
c21c22
c23c2mc31
c32c33
c3m
cn1cn2
cn3
cnm
-
7/25/2019 Aggegate planning 2016.pdf
62/78
In general, a transportation problem is
specified by the following information:
A set of m supply points from which a good is
shipped. Supply point ican supply at most si units.
A set of n demand points to which the good is
shipped. Demand pointjmust receive at least di
units of the shipped good.
Each unit produced at supply point iand shipped
to demand pointjincurs a variable cost of cij.
-
7/25/2019 Aggegate planning 2016.pdf
63/78
A transportation problem is specified by the
supply, the demand, and the shipping costs.Relevant data can be summarized in a
transportation table given below.
Fr o m To
Ci t y 1 Ci t y 2 Ci t y 3 Ci t y 4 Su p p l y ( M i l l i o n
k w h )
P la n t 1 $8 $6 $10 $9 35
P la n t 2 $9 $12 $13 $7 50
P la n t 3 $14 $9 $16 $5 40
D em a n d ( M i l l io n
k w h )
45 20 30 30
Solution
-
7/25/2019 Aggegate planning 2016.pdf
64/78
Solution
Decision Variables
Powerco must determine how much power is sntfrom each plant to each city soxij = Amount ofelectricity produced at plant iand sent to cityj
x14 = Amount of electricity produced at plant 1 and sentto city 4
Constraints A supply constraint ensures that the total quality
produced does not exceed plant capacity. Each plantis a supply point.
A demand constraint ensures that a locationreceives its demand. Each city is a demand point.
Since a negative amount of electricity can not beshipped allxijs must be non negative
Method for Handling Supply Not
-
7/25/2019 Aggegate planning 2016.pdf
65/78
Method for Handling Supply Not
Equal to Demand
When supply does not equal demand, you can use
the idea of a slack variable to handle the excess.
A slack variable is a variable that can be incorporated
into the model to allow inequality constraints tobecome equality constraints.
If supply is greater than demand, then you need a slack
variable known as a dummy destination.
If demand is greater than supply, then you need a slackvariable known as a dummy source.
-
7/25/2019 Aggegate planning 2016.pdf
66/78
Powerco Solution
LP Formulation of the ProblemMinZ= 8x11+6x12+10x13+9x14+9x21+12x22+13x23+7x24
+14x31+9x32+16x33+5x34
S.T.: x11+x12+x13+x14 = 30
x14+x24+x34 >= 30
xij >= 0 (i= 1,2,3; j= 1,2,3,4)
Farnsworth Tire Company
-
7/25/2019 Aggegate planning 2016.pdf
67/78
Farnsworth Tire Company developed data that relates to production,demand, capacity, and cost at its West Virginia plant. The data is providedbelow:
Sales Period
Mar. Apr. MayDemand 800 1000 750
Capacity
Regular 700 700 700
Overtime 50 50 50
Subcontract 150 150 130
Beg. Inv 100 tires
Cost
Regular time $40/tire
Overtime $50/tire
Subcontract $ 70/tire
Carrying cost $2/tire/month
Backorder cost $ 4/tire/month (Solve the problem without thisfirst, and then with backorder)
Backorder is not a viable alternative for this company. Give an initialsolution to the company and check if its optimal (use u-v method).
Solution (Assumed Cost & Constraints Linearity and no
-
7/25/2019 Aggegate planning 2016.pdf
68/78
( y
backorder or change in level of worker)
Supply From Period 1 Period 2 Period 3DummyCapacity
TotalCapacity
Avai lable
Beg Inv 100
Period 1
Reg Time 700
Overtime 50
Subcontract 150
Period 2
Reg Time X 700
Overtime X 50
Subcontract X 150
Period 3
Reg Time X X 700
Overtime X X 50
Subcontract X X 130
Total Demand 800 1000 750 230 2780
Initial Solution= $105900 and optimal is $105700 (without backorder)
PERIOD OF USEPERIOD OF Unused
-
7/25/2019 Aggegate planning 2016.pdf
69/78
PRODUCTION 1 2 3 4 Capac ity Capac ity
Beginning inventoryi i+h i+2h i+3h I
1 Regularr r+h r+2h r+3h R1
Overtime
o o+h o+2h o+3h O1
Subcontracts s+h s+2h s+3h S1
2 Regularr+b r r+h r+2h R2
Overtimeo+b o o+h o+2h O2
Subcontracts+b s s+h s+2h S2
3 Regularr+2b r+b r r+h R3
Overtimeo+2b o+b o o+h O3
Subcontracts+2b s+b s s+h S3
4 Regularr+3b r+2b r+b r R4
Overtimeo+3b o+2b o+b o O4
Subcontracts+3b s+2b s+b s S4
Unmet Demand
D1 D2 D3 D4
Demand of
period 1 is
produced/
satisfied in
period 2.Addition
cost of
back-
ordering
(loss of
goodwill,
cost ofexpediting
etc is
added.
h i blDummy Demand
-
7/25/2019 Aggegate planning 2016.pdf
70/78
The Transportation Tableau
Unused
PERIOD OF PRODUCTION 1 2 3 4 Capacity Capacity
Beginning 0 3 6 9 0
Inventory 300 300
Regular 600 300 100 0 1000
Overtime 0 100 0 100
Subcontract 500 0 500
Regular 1200 1200
Overtime 150 150
Subcontract 350 150 0 500
Regular 1300 1300
Overtime 200 200
Subcontract 500 500
Regular 1300 1300
Overtime 200 200
Subcontract 500 500
Demand 900 1500 1600 3000 750 7750
1
2
3
4
PERIOD OF USE
20 23 26 29
25 28 31 34
28 31 34 37
20 23 26
25 28 31
28 31 34
20 23
25 28
28 31
20
25
28
yBalanced Problem
M
High Costsay 9999if usingsolver or
puttingextremelylargenumberinto thecost ofproducingthat willforce thevalue toequalzero
No backorder allowed
InitialBasic
Feasible
Solution(Vogels,LC, orNWC)andthen
Simplex,Steep-ing
Stone orU-V
method
Determine a BFS for the below Production Plan. Is the BFS also optimal?-Home
Exercise
-
7/25/2019 Aggegate planning 2016.pdf
71/78
2007 Pearson Education
Exercise
Transportation Tableau for Tru-Rainbow Company
-
7/25/2019 Aggegate planning 2016.pdf
72/78
There are three basic methods to find the
starting basic feasible solution (bfs) for a
balanced TP problem
Northwest Corner Method
Minimum Cost Method
Vogels Approximation Method
-
7/25/2019 Aggegate planning 2016.pdf
73/78
The minimum cost method uses shipping
costs in order come up with a BFS that hasa lower cost.
To find the BFS (basic feasible solution) by
the Minimum Cost method:
Find the decision variable with the smallest
shipping cost (xij). Then assignxij its largest
possible value, which is the minimum of si and
dj
-
7/25/2019 Aggegate planning 2016.pdf
74/78
Next, cross out row iand columnjand reduce
the supply or demand of the noncrossed-out
row or column by the value ofxij.
Choose the cell with the minimum cost of
shipping from the cells that do not lie in a
crossed-out row or column and repeat the
procedure.
Often the minimum cost method may yield a
costly BFS.
Vogels approximation method for finding a bfsusually avoids extremely high shipping costs.
Simplex in Solver: Integer Solutions
-
7/25/2019 Aggegate planning 2016.pdf
75/78
Simplex in Solver: Integer Solutions
Property
If all the supplies and demands have integer
values, then the transportation problem with
feasible solutions is guaranteed to have an
optimal solution with integer values for all itsdecision variables.
This implies that there is no need to add restrictionson the model to force integer solutions.
Home Exercise
-
7/25/2019 Aggegate planning 2016.pdf
76/78
-
7/25/2019 Aggegate planning 2016.pdf
77/78
Aggregate Planningfor Services
1. Most services cant be inventoried
2. Demand for services is difficult to predict
3. Capacity is also difficult to predict
4. Service capacity must be provided at theappropriate place and time
5. Laboris usually the most constrainingresource for services
Lecture Slides of Indrajit Mukherjee, SJMSOM
Daily demand profile
-
7/25/2019 Aggegate planning 2016.pdf
78/78
6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00 22.00
Number ofcustomers
Daily demand profile
Core shift team 1 Core shift team 2
PartTime
PartTime
Part-
time
Staff scheduling
Figure Chase capacity strategy for a fast-food restaurant
Time