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Aggregate Planning

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Title: Aggregate Planning


1
Aggregate Planning
  • Aggregate Planning is translating annual or
    quarterly business plans into broad labor and
    output plans for the intermediate term (6 to 18
    months). Its objective is to minimize the cost of
    resources required to meet demand over that
    period.
  • Optimal combination of production rate, workforce
    level, and inventory on hand is sought throughout
    the intermediate term.

2
Hierarchy of Planning Problems
Process Planning
Long- range
Strategic Capacity Planning
Intermediate- range
Aggregate Planning
Master Production Scheduling
Material Requirements Planning
Order Scheduling
Short- range
3
Aggregate Unit
  • Aggregate Planning requires aggregation of
    different goods or services.
  • If the items are similar, we can use units.
  • Otherwise it may be more appropriate to use
  • Weight (tons of steel)
  • Volume (barrels of gasoline)
  • Dollar value
  • …

4
Aggregation Unit - Example
  • Six different models of washing machines

Can we use dollars as an aggregation unit? Use a
fictitious washing machine as an aggregation unit
which requires (.32)(4.2)(.21)(4.9)(.17)(5.1)(.
14)(5.2)(.10)(5.4)(.06)(5.8)4.856 hours of
labor time
5
Different levels of aggregation
  • Items Final products to be delivered to the
    customer
  • Families A group of items that a share common
    manufacturing setup cost
  • Types Groups of families with production
    quantities that are determined by a single
    aggregate production plan
  • Above may not always work, the aggregation method
    should be consistent with the firms
    organizational structure, product line, planning
    needs and availability of forecast and other data.

6
Objective of Aggregate Planning
  • The goal of aggregate planning is to determine
    aggregate production targets and the levels of
    resources required to achieve these production
    goals given the demand projection for the
    intermediate term.
  • Namely given demand as D1, D2,…,DT
  • Find the number of workers employed in each
    period
  • Find the number of aggregate units to be produced
    in each period

7
Production Rate
What is the best production curve in seasonal
business
Demand
Time
8
Costs in Aggregate Planning
  • Smoothing costs
  • Holding costs
  • Shortage costs
  • Regular time costs
  • Overtime or subcontracting costs
  • Idle time costs

9
Smoothing costs
10
Holding and Back-Order Costs
11
Prototype Problem
  • Forecast demands over the next six months for
    disk drives.

Inventory at the end of December expected to be
500 Company requires ending inventory of 600 at
the end of June Initial workforce is 300
Cost of hiring one worker CH 500 Cost of
firing one worker CF 1000 Inventory holding
cost per unit, per month CI 80
12
Prototype Problem
  • Plant manager observed that over 22 working days,
    with the workforce level of 76, the firm produced
    245 disk drives.
  • K of aggregate units produced per worker per
    day 245 / (22 76) 0.14653

13
Zero Inventory Plan (Chase)
14
Zero Inventory Plan (Chase)
Hiring costs 755 500 377,500 Firing costs
145 1000 145,000 Holding costs (60030)
80 50,400 Total cost 572,900
15
Constant workforce plan (Level)
16
Constant workforce plan (Level)
  • Hiring costs (411-300) 500 55,500
  • Holding costs (5962600) 80 524,960
  • Total cost 580,460

17
Mixed Strategies
18
Other scenarios
Section 1
19
Comparing strategies
20
Mathematical programming formulations
  • Cost definitions and input data
  • CH Cost of hiring one worker
  • CF Cost of firing one worker
  • CI Cost of holding one unit of stock
  • CR Cost of producing one unit in regular time
  • CO Cost of producing one unit in over time
  • CU Idle cost per unit of production
  • CS Cost to subcontract one unit of production
  • nt Number of production days in period t
  • K Number of aggregate units produced by one
    worker in one day
  • I0 Initial inventory
  • W0 Initial workforce
  • Dt Forecast of demand in period t

21
Problem variables
  • Wt Workforce level in period t
  • Pt Number of units produced in period t
  • It Inventory level at the end of period t
  • Ht Number of workers hired in period t
  • Ft Number of workers fired in period t
  • Ot Overtime production in units
  • Ut Worker idle time in units
  • St Number of units subcontracted from outside

22
Constraints
  • Conservation of workforce
  • Wt Wt-1 Ht Ft, for all t1,..,T
  • Inventory balance
  • It It-1PtSt-Dt, for all t1,..,T
  • Relationship between production and workforce
  • Pt K nt Wt Ot - Ut, for all t1,..,T

23
Constraints
  • Initial levels
  • I0I0, W0 W0
  • Ending levels
  • ITIT, WT WT
  • Non-negativity constraints
  • Wt, Pt, It, Ht, Ft, Ot, Ut, St gt0 for all
    t1,..,T

24
Linear program
25
Properties of the optimal solution
  • If cFgt0 and cHgt0, there could be either hiring
    or firing in one period (if at all), not both.
    i.e.,
  • HtFt0, for all t1,..,T
  • If cOgt0 and cIgt0, there could be either
    overtime or idle time in one period (if at all),
    not both. i.e.,
  • OtUt0, for all t1,..,T

26
Extensions
  • Adding buffers for uncertainty
  • It gt Bt, for all t1,..,T
  • Bt Buffer stock for period t defined in advance
  • Storage constraints for inventory
  • It lt Vt, for all t1,..,T
  • VtStorage capacity in period t

27
Extensions
  • Limits on overtime
  • OtltMt, for all t1,..,T
  • Mt maximum overtime in period t
  • Capacity constraints on production
  • PtltCt, for all t1,..,T
  • Ct capacity in period t

28
Allowing for backorders
29
Backorders and inventory
  • In any period, there are either backorders or
    positive inventory (if at all), but not both.
    i.e.,

30
Convex piecewise-linear costs
31
Linearization
32
Example (pp 153)
  • The Paris Paint Company is in the process of
    planning labor force requirements and production
    levels for the next 4 quarters. The marketing
    department has provided production with the
    following forecasts of demand for Paris Paint
    over the next year.

Assume that there are currently 280 employees
with the company. Employees are hired for at
least one full quarter. Hiring costs amount to
1,200 per employee and firing costs are 2,500
per employee. Inventory costs are 1 per gallon
per quarter. It is estimated that one worker
produced 1,000 gallons of paint each quarter.
Assume that Paris currently has 80,000 gallons of
paint in inventory and would like to end the year
with an inventory of at least 20,000
gallons. Formulate the problem as an LP.
33
Linear Program
34
Example 2
  • Sun Microsystems is the producer of computer
    workstations. For the year 2003, they estimate
    their quarterly demand for high-end workstations
    to be the following

Sun Microsystems focuses on innovation and design
rather than manufacturing. Therefore, Sun
developed strategic relationships with contract
manufacturers Solectron and Celestica Solectron
promising 6000, Celestica promising 2000 units of
maximum manufacturing capacity per quarter for
Sun. Solectron is the preferred vendor for Sun
and charges 1000 for each high end workstation
that it manufactures. Sun may chose to use any
amount of the capacity that Solectron promised
without paying any penalties. Celestica, on the
other hand, is the secondary vendor for Sun. It
charges 1100 for each high-end workstation.
Celestica requires that Sun declares the Q1
subcontracted amount in advance. Also, Sun can
increase the subcontracted amount each quarter
only if it pays 100 per unit of increase. In
addition, Sun can never reduce the amount it
subcontracted from Celestica from quarter to
quarter. Inventory holding costs are 50 per
quarter per unit. How much should Sun subcontract
from Solectron and Celestica each quarter?
35
Linear Program
36
Example 3 (pp 128 modified)
  • The personal department of the AM Corporation
    wants to know how many workers will be needed
    each month for the next 4 month production
    period. The demands would be 1250, 1100, 950, 900
    in months July, August, September, October.
  • The inventory on hand at the end of June was 500
    units. The company wants to maintain a minimum
    inventory of 300 units each month. Each unit
    requires 5 employee hours. There are 20 working
    days each month, and each employee works an
    8-hour day. The workforce at the end of June was
    35 workers.
  • The workers can also work overtime, but overtime
    cannot exceed 30 of the regular time in each
    month. Overtime costs an additional 20 per unit
    produced.
  • Hiring costs 2500 per employee, firing costs
    4000 per employee, payroll costs 3000 per
    employee per month, and inventory holding cost is
    100.
  • Formulate the problem as an LP and solve

37
Linear Program
38
Example 4- Component availability constraints
  • Seagate is a manufacturer of hard drives for
    personal and enterprise use. The enterprise
    division is trying to plan its production for the
    first six months in 2003. The forecast for these
    6 months are given below

Seagate does not have any constraints on
workforce or equipment for manufacturing hard
drives. However, the enterprise hard drive
requires two counts of a specific chip
(application specific integrated circuit ASIC)
which is sourced from Texas Instruments. TIs
capacity is fixed for the first 4 months and it
can allocate only a portion of its capacity to
Seagate. This amounts to 26000 such chips in each
of the first 4 months in 2003. Seagate may also
choose to use subcontractors for manufacturing
the hard drives which would cost an additional
(on top of its own manufacturing) 50 per
drive. If the inventory holding costs are 20 per
unit per month for the hard drives and 5 per
unit per month for the ASICs, find the optimal
production plan for the hard drives and optimal
purchase plan for the hard drives and ASICs.
39
Linear Program
40
Production Planning problems with concave costs
  • Why concave costs?
  • Economies of scale
  • Setup costs associated with production,
    subcontracting, overtime, alternate resources
  • Cannot be modeled as linear programs
  • May be more difficult to solve

41
Modeling fixed (setup) costs
  • Assume there is a fixed cost associated with
    subcontracting. No idle time or overtime allowed.

42
Example 5- Component availability constraints
  • Seagate is a manufacturer of hard drives for
    personal and enterprise use. The enterprise
    division is trying to plan its production for the
    first six months in 2003. The forecast for these
    6 months are given below

Seagate does not have any constraints on
workforce or equipment for manufacturing hard
drives. However, the enterprise hard drive
requires two counts of a specific chip
(application specific integrated circuit ASIC)
which is sourced from Texas Instruments. TIs
capacity is fixed for the first 4 months and it
can allocate only a portion of its capacity to
Seagate. This amounts to 26000 such chips in each
of the first 4 months in 2003. Seagate may also
choose to use subcontractors for manufacturing
the hard drives which would cost an additional
(on top of its own manufacturing) 10 per drive.
In addition, there is a fixed cost of 20,000
working with a subcontractor in any month. If the
inventory holding costs are 20 per unit per
month for the hard drives and 5 per unit per
month for the ASICs, find the optimal production
plan for the hard drives and optimal purchase
plan for the hard drives and ASICs.
43
Mixed Integer Program
44
General concave cost models with no capacity
constraints and no backlogging
45
Properties of the optimal solution
  • There is either production in one period or
    inventory carried over from an earlier period,
    not both.

46
Forward Algorithm
47
Example
  • Production to be planned for 4 periods

48
Example continued
49
Example continued
50
Network flow model general costs
D1D2D3..DT
0
P1
P2
P3
PT
1
T
3
2
IT-1
I3
I2
I1
DT
D1
D2
D3
51
Shortest path problem concave costs
M04
M03
M02
1
4
3
2
0
M23
M01
M12
M34
M13
M14
M24
52
Shortest path problem concave costs
M04
M03
M02
1
4
3
2
0
M23
M0190
M12130
M34
M13
M14
M24340
53
Backlogging case concave costs
54
Example
55
Shortest path solution
M044830
M032630
M021370
1
4
3
2
0
M231280
M01570
M12840
M341560
M131960
M143960
M243280
56
Static concave cost model
  • Consider the case where
  • DtD, for all i1,..,T
  • hth, for all t1,..,T
  • Atk, for all t1,..,T
  • CtC, for all t1,..,T
  • There will be re-generation points every n
    periods.
  • Minimize total cost per cycle
  • Variable production costs can be ignored
  • What is optimal n?

57
Static concave costs - continued
58
From Aggregate Plan to Master Production Schedule
  • The result of the aggregate plan is the
    production quantities, inventory levels and
    required resources at the aggregate level in the
    mid term.
  • The firms are expected to act (acquire workforce
    resources, contact suppliers) based on the
    aggregate plan.
  • Such actions will be input (constraints) to lower
    level (i.e. more detailed and shorter term)
    decisions in the company
  • Consistency between the aggregate plan and the
    master production schedule is desired, but not
    always possible
  • Initial master production schedule mandated by
    the demand plan (forecast) may not always be
    feasible.

59
Inputs to Master Production Schedule
  • Forecast by end item (sometimes also by customer
    class or distribution channel) usually weekly
    sometimes monthly
  • For each item, manufacturing/distribution process
    flow
  • For each stage of manufacturing/distribution
    process flow
  • Consumption rate of critical components
  • Consumption rate of critical resources
  • Availability for critical components and
    resources
  • Lead times (for time phasing)
  • Prioritization and allocation schemes for
    constrained situations

60
Feasible MPS
10
6
5
10
Assemble 1 week
Test 0 week
Procure 0 week
Deliver
Media B
Media B
Hard Drive 1
Hard Drive 1
30
16
15
20
20
20
Procure 0 week
ASIC
ASIC
7
8
8
7
Assemble 1 week
Test 0 week
Deliver
Procure 0 week
Media A
Hard Drive 2
Hard Drive 2
Media A
61
Unconstrained Plan
10
6
5
10
Assemble 1 week
Test 0 week
Procure 0 week
Deliver
Media B
Media B
Hard Drive 1
Hard Drive 1
30
16
15
20
20
20
Procure 0 week
ASIC
ASIC
7
18
15
7
Assemble 1 week
Test 0 week
Deliver
Procure 0 week
Media A
Hard Drive 2
Hard Drive 2
Media A
62
Material constrained plan
10
6
5
10
Assemble 1 week
Test 0 week
Procure 0 week
Deliver
Media B
Media B
Hard Drive 1
Hard Drive 1
30
16
15
20
20
20
Procure 0 week
ASIC
ASIC
7
18
15
7
Assemble 1 week
Test 0 week
Deliver
Procure 0 week
Media A
Hard Drive 2
Hard Drive 2
Media A
63
Capacity constrained plan
10
6
5
10
Assemble 1 week
Test 0 week
Procure 0 week
Deliver
Media B
Media B
Hard Drive 1
Hard Drive 1
30
16
15
20
20
20
Procure 0 week
ASIC
ASIC
7
18
15
7
Assemble 1 week
Test 0 week
Deliver
Procure 0 week
Media A
Hard Drive 2
Hard Drive 2
Media A
64
Material constrained plan-build ahead
10
6
5
10
Assemble 1 week
Test 0 week
Procure 0 week
Deliver
Media B
Media B
Hard Drive 1
Hard Drive 1
30
16
15
20
20
20
Procure 0 week
ASIC
ASIC
7
18
15
7
Assemble 1 week
Test 0 week
Deliver
Procure 0 week
Media A
Hard Drive 2
Hard Drive 2
Media A
65
Constrained plan Fair-share allocation
10
6
5
10
Assemble 1 week
Test 0 week
Procure 0 week
Deliver
Media B
Media B
Hard Drive 1
Hard Drive 1
30
16
15
20
20
20
Procure 0 week
ASIC
ASIC
7
18
15
7
Assemble 1 week
Test 0 week
Deliver
Procure 0 week
Media A
Hard Drive 2
Hard Drive 2
Media A
66
Constrained plan Priority allocation
10
6
5
10
Assemble 1 week
Test 0 week
Procure 0 week
Deliver
Media B
Media B
Hard Drive 1
Hard Drive 1
30
16
15
20
20
20
Procure 0 week
ASIC
ASIC
7
18
15
7
Assemble 1 week
Test 0 week
Deliver
Procure 0 week
Media A
Hard Drive 2
Hard Drive 2
Media A
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