AGGREGATE PLANNING

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AGGREGATE PLANNING

• Production Planning and Control

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Haeryip Sihombing Fakulti Kejuruteraan
PembuatanUniversiti Teknologi Malaysia Melaka
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Chapter Outline

• I. Introduction
• II. The Concept of Aggregation
• III. An Overview of Production-Planning
  Activities
• IV. Framework for Aggregate Production Planning
• V. Techniques for Aggregate Production Planning
• VI. Aggregate Planning in Service Companies
• VII. Implementing Aggregate Production Plans -
  Managerial Issues
• VIII. Hierarchical Production Planning

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Aggregate production planning is medium-term
capacity planning over a two to eighteen month
planning horizon. It involves determining the
lowest-cost method of providing the adjustable
capacity for meeting production requirements.
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Capacity Decisions Hierarchy
Linkages
Facilities Planning
Aggregate Planning
Scheduling
Time Frame
Facilities Planning
Aggregate Planning
Scheduling
Time
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Aggregation refers to the idea of focusing on
overall capacity, rather than individual products
or services.

• Aggregation is done according to
• Products
• Labor
• Time

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Production Planning

• Long Range Planning
• Strategic planning (1-5 years)
• Medium Range Planning
• Employment, output, and inventory levels (2-18
  months)
• Short Range Planning
• Job scheduling, machine loading, and job
  sequencing (0-2 months)

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Aggregate production planning involves managing...

• Work force levels - the number of workers
  required for production.
• Production rates - the number of units produced
  per time period.
• Inventory levels - the balance of unused units
  carried forward from the previous period.

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Common objectives of production planning...
MINIMIZEcost, inventory levels, changes in work
force levels, use of overtime, use of
subcontracting, changes in production rates,
changes in production rates, plant/personnel idle
time MAXIMIZEprofits, customer service
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Methods of Influencing Demand

• Price Incentives
• Reservations
• Backlogs
• Complementary Products or Services
• Advertising/promotion

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Methods of Influencing Supply

• Hiring/firing workers
• Overtime/slack time
• Part time/temporary labor
• Subcontracting
• Cooperative arrangements
• Inventories

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Aggregate Production Planning Variable Costs

• Hiring/firing costs
• Overtime/slack time costs
• Part time/temporary labor costs
• Subcontracting costs
• Cooperative arrangements costs
• Inventory carrying costs
• Backorder or stock out costs

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Aggregate Production Planning Strategies

• Chase strategy
• production rates or work force levels are
  adjusted to match demand requirements over
  planning horizon
• Level strategy
• constant production rate or work force level is
  maintained over planning horizon
• Mixed strategy
• both inventory level changes and work force level
  changes occur

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Aggregate Production Planning Techniques

• Trial-and-error method
• Mathematical techniques

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Trial-and-Error Method

• Examples of alternative strategies
• Vary work force levels
• Level work force, vary inventories and backorders
• Level work force, use subcontracting
• Level work force, use overtime and subcontracting

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Mathematical Techniques

• - Linear Decision Rule
• - Mgmt. Coefficient Models
• - Parametric Prod. Planning
• - Search Decision Rule
• - Production-Switching Heuristic

• - Linear Programming
• - Transportation Method
• - Goal Programming
• - Mixed Integer Programming
• - Simulation Models

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Managerial Issues in Aggregate Production
Planning
1. APP should be tailored to the particular
company and situation. 2. APP may be
constrained by union contracts or company
policies. 3. Mathematical techniques will likely
have to be balanced with managerial
judgment and experience. 4. A tendency to
blur the distinction between production
planning and production scheduling.
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Aggregate Planning in Services

• For service companies, aggregate planning
  results in staffing plans that call for changing
  the number of employees or subcontracting.

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• The END

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Production Planning Environment
Competitors Behavior
Raw Material Availability
Market Demand
Planning for Production
External Capacity (outsourcing)
Economic Conditions
Current Physical Capacity
Current Inventory
Current Work Force
Required Production Activities
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Planning Production

• Long-range plan (3-10 years) updated yearly
• Inputs aggregate forecasts (units) and current
  plant capacity (hours)
• Decision build new plant, expand an existing
  plant, create new product line, expand, contract,
  or delete existing product lines
• Level of detail Very Aggregated
• Degree of uncertainty High

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Planning Production

• Intermediate-range plan (6 month 2 years)
  updated quarterly
• Inputs aggregate capacity and product decisions
  from the long-term plan, units are aggregated by
  product line or family and plant department
• Decision changes in work force, additional
  machines, subcontracting, overtime
• Level of detail Aggregated
• Degree of uncertainty Medium

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Planning Production

• Short-range plan (1 week 6 month) updated
  daily or weekly
• Inputs decisions from the intermediate-term
  plan, units are aggregated by particular product
  and capacity available hours on a particular
  machine, short range forecast, inventory levels,
  work force levels, processes
• Decision overtime and undertime, possibility of
  not fulfilling all demand, subcontracting,
  delivery dates for suppliers, product quality
• Level of detail Very Detailed
• Degree of uncertainty Low

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Production Planning Example

• Small company makes one product plastic cases
  to hold CDs.
• Two different types of mold are used to produce
  top bottom.
• Two halves are manually put together, placed in
  the boxes shipped.
• The injection molding machines can make 550
  pieces per hour.
• A worker can finish 55 cases in 1 hour (10
  workers / machine)
• Forecasts of demand 80,000 cases per month for
  next year ? at 4 weeks/month the demand should be
  20,000 cases per week.
• Company runs 5 out of 7 days per week 4,000
  cases per day needed.
• Each worker can not work more than 8 hours per
  day
• 4,000/8 500 pieces per hour have to be
  produced.
• Plan 1 machine, 10 workers, 8 hours/day, 5
  days/week

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The Hierarchy of Production Planning Decisions
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Introduction to Aggregate Planning

• Goal To plan gross work force levels and set
  firm-wide production plans.
• Concept is predicated on the idea of an
  aggregate unit of production. May be actual
  units, or may be measured in weight (tons of
  steel), volume (gallons of gasoline), time
  (worker-hours), or dollars of sales. Can even be
  a fictitious quantity. (Refer to example in text
  and in slide below.)

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Introduction to Aggregate Planning

• Constant production rate can be satisfied with
  constant capacity.
• Work force is constant, production rate slightly
  less that capacity of people machines good
  utilization without overloading the facilities.
• Raw material usage is also constant.
• If supplier and customers are also close,
  frequent deliveries of raw material and finished
  goods will keep inventory low.
• How realistic is this example?
• Strategies to cope with fluctuating demand?

-- change the demand -- produce at constant rate
anyway -- vary the production rate -- use
combination of above strategies
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Introduction to Aggregate PlanningInfluencing
Demand

• Do not satisfy demand during peak periods
• Capacity lt Peak demand , constant production rate
• Loss of some sales
• Japanese car manufacturers often take this
  stance
• Determine percentage of the market share
• Constant production is set at this level
• Sales personal expected to sell produced amount
• Ease of planning must be compared to lost revenue

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Introduction to Aggregate PlanningInfluencing
Demand

• Shift demand from peak periods to non-peak
  periods / create new demand for non-peak periods
• Creating new demand can be done through
  advertising or incentive programs (automobile
  industry rebates telephone companys
  differential pricing system)
• Smoothing demand

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Introduction to Aggregate PlanningInfluencing
Demand

• Produce several products with peak demand in
  different periods
• Products should be similar, so that manufacturing
  them is not too different
• Snowmobiles and jetskis same engines, similar
  body work
• Lawn-mowers snowblowers baseball football
  equipment

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Medium Range Planning Aggregate Production
Planning

• Establish production rates by major product
  groups
• by labor hours required or units of production
• Attempt to determine monthly work force size and
  inventory levels that minimizes production
  related costs over the planning period (for 6-24
  month)

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Relevant Costs Involved

• Regular time costs
• Costs of producing a unit of output during
  regular working hours, including direct and
  indirect labor, material, manufacturing expenses
• Overtime costs
• Costs associated with using manpower beyond
  normal working hours
• Production-rate change costs
• Costs incurred in substantially altering the
  production rate
• Inventory associated costs
• Out of pocket costs associated with carrying
  inventory
• Costs of insufficient capacity in the short run
• Costs incurred as a result of backordering, lost
  sales revenue, loss of goodwill costs of
  actions initiated to prevent shortages
• Control system costs
• Costs of acquiring the data for analytical
  decision, computational effort and implementation
  costs

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Overview of the Problem

• Suppose that D1, D2, . . . , DT are the
  forecasts of demand for aggregate units over the
  planning horizon (T periods.) The problem is to
  determine both work force levels (Wt) and
  production levels (Pt ) to minimize total costs
  over the T period planning horizon.

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Important Issues

• Smoothing. Refers to the costs and disruptions
  that result from making changes from one period
  to the next.
• Bottleneck Planning. Problem of meeting peak
  demand because of capacity restrictions.
• Planning Horizon. Assumed given (T), but what is
  right value? Rolling horizons and end of
  horizon effect are both important issues.
• Treatment of Demand. Assume demand is known.
  Ignores uncertainty to focus on the
  predictable/systematic variations in demand, such
  as seasonality.

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Relevant Costs

• Smoothing Costs
• changing size of the work force
• changing number of units produced
• Holding Costs
• primary component opportunity cost of investment
• Shortage Costs
• Cost of demand exceeding stock on hand. Why
  should shortages be an issue if demand is known?
• Other Costs payroll, overtime, subcontracting.

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Cost of Changing the Size of the Workforce
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Holding and Back-Order Costs
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Aggregate Units

• The method is based on notion of aggregate units.
  They may be
• Actual units of production
• Weight (tons of steel)
• Volume (gallons of gasoline)
• Dollars (Value of sales)
• Fictitious aggregate units

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Example of fictitious aggregate units.(Example.1)

• One plant produced 6 models of washing machines
• Model hrs. Price sales
• A 5532 4.2 285 32
• K 4242 4.9 345 21
• L 9898 5.1 395 17
• L 3800 5.2 425 14
• M 2624 5.4 525 10
• M 3880 5.8 725 06
• Question How do we define an aggregate unit here?

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Example continued

• Notice Price is not necessarily proportional to
  worker hours (i.e., cost) why?
• One method for defining an aggregate unit
  requires .32(4.2) .21(4.9) . . . .06(5.8)
  4.8644 worker hours. Forecasts for demand for
  aggregate units can be obtained by taking a
  weighted average (using the same weights) of
  individual item forecasts.

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Prototype Aggregate Planning Example(this
example is not in the text)

• The washing machine plant is interested in
  determining work force and production levels for
  the next 8 months. Forecasted demands for
  Jan-Aug. are 420, 280, 460, 190, 310, 145, 110,
  125. Starting inventory at the end of December is
  200 and the firm would like to have 100 units on
  hand at the end of August. Find monthly
  production levels.

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Step 1 Determine net demand.(subtract
starting inv. from per. 1 forecast and add ending
inv. to per. 8 forecast.)

• Month Net Predicted Cum. Net Days
• Demand
  Demand
• 1(Jan) 220 220 22
• 2(Feb) 280 500 16
• 3(Mar) 460 960 23
• 4(Apr) 190 1150 20
• 5(May) 310 1460 21
• 6(June) 145 1605 17
• 7(July) 110 1715 18
• 8(Aug) 225 1940 10

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(No Transcript)
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Step 2. Graph Cumulative Net Demand to Find Plans
Graphically
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Constant Work Force Plan

• Suppose that we are interested in determining
  a production plan that doesnt change the size of
  the workforce over the planning horizon. How
  would we do that?
• One method In previous picture, draw a
  straight line from origin to 1940 units in month
  8 The slope of the line is the number of units
  to produce each month.

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Monthly Production 1940/8 242.2 or rounded
to 243/month. But there are stockouts.
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How can we have a constant work force plan with
no stockouts?

• Answer using the graph, find the straight
  line that goes through the origin and lies
  completely above the cumulative net demand curve

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From the previous graph, we see that cum. net
demand curve is crossed at period 3, so that
monthly production is 960/3 320. Ending
inventory each month is found from

• Month Cum. Net. Dem. Cum. Prod.
  Invent.
• 1(Jan) 220 320 100
• 2(Feb) 500 640
  140
• 3(Mar) 960 960
  0
• 4(Apr.) 1150 1280
  130
• 5(May) 1460 1600
  140
• 6(June) 1605 1920
  315
• 7(July) 1715 2240
  525
• 8(Aug) 1940 2560
  620

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But - may not be realistic for several reasons

• It may not be possible to achieve the production
  level of 320 unit/month with an integer number of
  workers
• Since all months do not have the same number of
  workdays, a constant production level may not
  translate to the same number of workers each
  month.

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To overcome these shortcomings

• Assume number of workdays per month is given
• K factor given (or computed) where
• K of aggregate units produced by one worker
  in one day

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Finding K

• Suppose that we are told that over a period of 40
  days, the plant had 38 workers who produced 520
  units. It follows that
• K 520/(3840) 0.3421
• average number of units produced by one
  worker in one day.

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Computing Constant Work Force

• Assume we are given the following of working
  days per month 22, 16, 23, 20, 21, 17, 18, 10.
  March is still critical month. Cum. net demand
  thru March 960. Cum of working days
  221623 61. Find 960/61 15.7377 units/day
  implies 15.7377/.3421 46 workers required.

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Constant Work Force Production Plan

• Mo wk days Prod. Cum Cum Net
  End Inv
• Level
  Prod Dem
• Jan 22 346 346
  220 126
• Feb 16 252 598
  500 98
• Mar 23 362 960
  960 0
• Apr 20 315 1275
  1150 125
• May 21 330 1605
  1460 145
• Jun 22 346 1951
  1605 346
• Jul 21 330 2281
  1715 566
• Aug 22 346 2627
  1940 687

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Addition of Costs

• Holding Cost (per unit per month) 8.50
• Hiring Cost per worker 800
• Firing Cost per worker 1,250
• Payroll Cost 75/worker/day
• Shortage Cost 50 unit short/month

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Cost Evaluation of Constant Work Force Plan

• Assume that the work force at end of Dec was 40.
• Cost to hire 6 workers 6800 4800
• Inventory Cost accumulate ending inventory
  (126980. . .687) 2093. Add in 100 units
  netted out in Aug 2193. Hence Inv. Cost
  21938.518,640.50
• Payroll cost
• (75/worker/day)(46 workers )(167days) 576,150
• Cost of plan 576,150 18,640.50 4800
  599,590.50 600K

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Cost Reduction in Constant Work Force Plan

• In the original cum net demand curve,
  consider making reductions in the work force one
  or more times over the planning horizon to
  decrease inventory investment.

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Cost Evaluation of Modified Plan

• I will not present all the details here. The
  modified plan calls for reducing the workforce to
  36 at start of April and making another reduction
  to 22 at start of June. The additional cost of
  layoffs is 30,000, but holding costs are reduced
  to only 4,250. The total cost of the modified
  plan is 467,450.

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Zero Inventory Plan (Chase Strategy)

• Here the idea is to change the workforce each
  month in order to reduce ending inventory to
  nearly zero by matching the workforce with
  monthly demand as closely as possible. This is
  accomplished by computing the units produced by
  one worker each month (by multiplying K by days
  per month) and then taking net demand each month
  and dividing by this quantity. The resulting
  ratio is rounded up and possibly adjusted
  downward.

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• I got the following for this problem
• Period hired fired
• 1 10 Cost of
  this
• 2 20
  plan
• 3 9
  555,704.50
• 4 31
• 5 15
• 6 24
• 7 4
• 8 15

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Optimal Solutions to Aggregate Planning Problems
Via Linear Programming

• Linear Programming provides a means of solving
  aggregate planning problems optimally. The LP
  formulation is fairly complex requiring 8T
  variables and 3T constraints, where T is the
  length of the planning horizon. Clearly, this can
  be a formidable linear program. The LP
  formulation shows that the modified plan we
  considered with two months of layoffs is in fact
  optimal for the prototype problem.

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Aggregate Units

• The method is based on notion of aggregate
  units.
• They may be
• Actual units of production
• Weight (tons of steel)
• Volume (gallons of gasoline)
• Dollars (value of sales)
• Fictitious aggregate units

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Overview of the Problem

• D1, D2, . . . , DT - the forecasts of demand for
  aggregate units over the planning horizon
  (T periods)
• Determine Wt - work force levels
• Pt - production levels
• It inventory levels
• Ht number of workers hired in this period
• Ft number of workers fired in this period
• Ot overtime production in units
• Ut undertime, worker idle time in units
• St number of units subcontracted from
  outside
• to minimize total costs over the T period
  planning horizon

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Example of fictitious aggregate units

• One plant produced 6 models of washing machines
• Model hrs. Price sales
• A 5532 4.2 285 32
• K 4242 4.9 345 21
• L 9898 5.1 395 17
• L 3800 5.2 425 14
• M 2624 5.4 525 10
• M 3880 5.8 725 06
• Question How do we define an aggregate unit here?

Price/hours 67.86 70.41 77.45 81.73 97.22 125.0
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Example (continued)

• Notice Price is not necessarily proportional to
  worker hours (i.e., cost) why?
• One method for defining an aggregate unit
• 0.32(4.2) 0.21(4.9) 0.17(5.1) 0.14(5.2)
  0.10(5.4) 0.06(5.8) 4.856 worker hours
• Forecasts for demand for aggregate units can be
  obtained by taking a weighted average (using the
  same weights) of individual item forecasts.

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Example (continued)

• The washing machine plant is interested in
  determining work force and production levels for
  the next 8 months
• Forecasted demands for Jan-Aug. are
• 420, 280, 460, 190, 310, 145, 110, 125
• Starting inventory at the end of December is 200
  and the firm would like to have 100 units on hand
  at the end of August
• Find monthly production levels

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Step 1 Determine net demand.(subtract
starting inventory from period 1 forecast and add
ending inventory to period 8 forecast)

• Month Forecasted Net Predicted Cum. Net
• Demand Demand Demand
• 1(Jan) 420 420-200220 220
• 2(Feb) 280 280 500
• 3(Mar) 460 460 960
• 4(Apr) 190 190 1150
• 5(May) 310 310 1460
• 6(June) 145 145 1605
• 7(July) 110 110 1715
• 8(Aug) 125 125100225 1940
• Starting inventory - 200 and final inventory -
  100 units

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Step 2. Graph Cumulative Net Demand to Find
Plans Graphically
Draw a straight line from first point 220 to 1940
units in month 8 The slope of the line is the
number of units to produce each month.
Determine a production plan that doesnt change
the size of the workforce over the planning
horizon. What to do?
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Monthly Production 1940 / 8 242.5
(rounded to 243/month)
Any shortfalls in this solution?
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How can we have a constant work force plan with
no stockouts?

• Using the graph, find the straight line that
  goes through the origin and lies completely above
  the cumulative net demand curve

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From the previous graph, we see that cum. net
demand curve is crossed at period 3, so that
monthly production is 960/3 320. Ending
inventory each month is found from
70
However

• This solution may not be realistic for several
  reasons
• It may not be possible to achieve the production
  level of 320 unit/mo with an integer number of
  workers
• Since all months do not have the same number of
  workdays, a constant production level may not
  translate to the same number of workers each
  month
• Some thoughts
• Final inventory is 620 units, not 100 units
• Cost of carrying inventory in each period

71
Production Strategies

• Constant production rate with Zero inventory
• stockouts
• carrying inventory
• Constant production rate with no stockouts
• carrying inventory
• extra inventory at the period T
• Mixed strategy
• few changes in the workforce allowed
• more flexibility
• lower costs

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Example 2 (based on example 1)

• The plant has 38 workers who produced 630 units
  in a period of 40 days
• K 630/(3840) 0.414 ? average number of units
  produced by one worker in one day
• Assume we are given the following working days
  per month
• jan 22 apr 20 jul 18
• feb 16 may 21 aug 10
• mar 23 jun 17

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Constant Work Force Production Plan 38 workers,
K .414

• Month wk Prod. Cum Cum Nt
  End Inv
• days Dem Level Prod Dem
• Jan 22 220 346 346
  220 126
• Feb 16 280 252 598
  500 98
• Mar 23 460 362 960
  960 0
• Apr 20 190 315 1275
  1150 125
• May 21 310 330 1605
  1460 145
• Jun 22 145 346 1951
  1605 346
• Jul 21 110 330 2281
  1715 566
• Aug 22 125 346 2627
  1940 687
• 100

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Addition of Costs

• Holding Cost (per unit per month) 8.50
• Hiring Cost per worker
  800.00
• Firing Cost per worker
  1,250.00
• Payroll Cost ( per worker/day)
  75.00
• Shortage Cost (unit short/month) 50.00

75
Cost Evaluation of Constant Work Force Plan

• Assume that the work force at end of Dec was 32
• Cost to hire 6 workers 6800 4,800
• Inventory Cost ? accumulate ending inventory
  (126980125145346567687) 2,095
• (100 units at the end of the august in
  included in 687 units inventory)
• Hence Inventory Cost 20958.517,809.37
• Payroll cost
• (75/worker/day)(38 workers )(167days)
  475,950
• Cost of plan
• 475,950 17,809.37 4800 498,559.37

76
Cost Reduction in Constant Work Force Plan

• In the original cum net demand curve, consider
  making reductions in the work force one or more
  times over the planning horizon to decrease
  inventory investment.

77
Cost Evaluation of Modified Plan with One
Workforce Adjustment

• The modified plan calls for
• hiring 6 workers in Jan (to 38)
• reducing the workforce to 23 (from 38) at start
  of April
• cost of hiring is 4,800.00
  4,800.00
• cost of layoffs is 18,750.00
  0.00
• payroll cost is 356,700.00
  475,950.00
• holding costs are 2,528.93
  17,809.37
• shortage costs are 7,770.40
  0.00
• The total cost of the modified plan is
  390,548.33
• Original plan had cost of 498,559.37

78
Cost Evaluation of Modified Plan with Two
Workforce Adjustment

• The modified plan calls for
• hiring 6 workers in January
• firing 8 workers at start of April
• firing 12 workers at start of June
• Two One None
• cost of hiring is 4,800.00
  4,800.00 4,800.00
• cost of layoffs is 25,000.00
  18,750.00 0.00
• payroll cost is 353,850.00
  356,700.00 475,950.00
• holding costs are 3,452.87
  2,528.93 17,809.37
• shortage costs are 0.00
  7,770.40 0.00
• The total cost 387,102.87
  390,548.33 498,559.37

79
Constant Work Force Production Plan 38 workers,
K .414

• Month wk Prod. Cum Cum Nt
  End Inv
• days Dem Level Prod Dem
• Jan 22 220 346 346
  220 126
• Feb 16 280 252 598
  500 98
• Mar 23 460 362 960
  960 0
• Apr 20 190 315 1275
  1150 125
• May 21 310 330 1605
  1460 145
• Jun 22 145 346 1951
  1605 346
• Jul 21 110 330 2281
  1715 566
• Aug 22 125 346 2627
  1940 687
• 100

80
Cost Reduction in Constant Work Force Plan
81
Zero Inventory Plan (Chase Strategy)

• Idea
• change the workforce each month in order to
  match the workforce with monthly demand as
  closely as possible
• This is accomplished by computing the units
  produced by one worker each month (by multiplying
  K by days per month)
• Then take net demand each month and dividing by
  this quantity. The resulting ratio is rounded up
  and possibly adjusted downward.

82

• At the end of December there are 32 workers
• Period hired fired
• 1 7 Cost of
  this
• 2 17
  plan
• 3 6
  461,732.08
• 4 25
• 5 13
• 6 20
• 7 4
• 8 13

83
Hybrid Strategies

• Use a combination of options
• Build-up inventory ahead of rising demand use
  backorders to level extreme peaks
• Finished goods inventories Anticipate demand
• Back orders lost sales Delay delivery or allow
  demand to go unfilled
• Shift demand to off-peak times Proactive
  marketing
• Overtime Short-term option
• Pay workers a premium to work longer hours

84
Hybrid Strategies

• Undertime Short-term option
• Slow the production rate or send workers home
  early (lowers labor productivity, but doesnt tie
  up capital in finished good inventories)
• Reassign workers to preventive maintenance during
  lulls
• Subcontracting Medium-term option
• Subcontract production or hire temporary workers
  to cover short-term peaks
• Hire fire workers Long-term option
• Change the size of the workforce
• Layoff or furlough workers during lulls

85
Another APP Example
Quarter Sales Forecast (lb) Spring 80,000 Summer
50,000 Fall 120,000 Winter 150,000

• _________________________
• Hiring cost 100 per worker
• Firing cost 500 per worker
• Inventory carrying cost 0.50 per pound per
  quarter
• Production per employee 1,000 pounds per
  quarter
• Beginning work force 100 workers

86
Level Production Strategy

• Sales Production
• Quarter Forecast Plan Inventory
• Spring 80,000 100,000 20,000
• Summer 50,000 100,000 70,000
• Fall 120,000 100,000 50,000
• Winter 150,000 100,000 0
• 400,000 140,000
• Cost 140,000 pounds x 0.50 per pound 70,000

87
Chase Demand Strategy (Zero Inventory)
Hiring cost 100 per worker Firing cost
500 per worker Inventory carrying cost 0.50
per pound per quarter Production per employee
1,000 pounds per quarter Beginning work force
100 workers

• Sales Production Workers Workers Workers
• Quarter Forecast Plan Needed Hired Fired
• Spring 80,000 80,000 80 - 20
• Summer 50,000 50,000 50 - 30
• Fall 120,000 120,000 120 70 -
• Winter 150,000 150,000 150 30 -
• 100 50
• Cost (100 workers hired x 100) (50 workers
  fired x 500)
• 10,000 25,000 35,000

88
APP By Linear Programming

• Min 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)
• P1 - 1,000 W1 0 (5) Production
• P2 - 1,000 W2 0 (6) constraints
• P3 - 1,000 W3 0 (7)
• P4 - 1,000 W4 0 (8)
• W1 - H1 F1 100 (9) Work force
• W2 - W1 - H2 F2 0 (10) constraints
• W3 - W2 - H3 F3 0 (11)
• W4 - W3 - H4 F4 0 (12)

where Ht hired for period t Ft fired for
period t It inventory at end of period
t Wt workforce at period t Pt units
produced at period t
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Optimal Solutions to Aggregate Planning Problems
Via 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
• CtH 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 t
• Pt 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

90
Optimal Solutions to Aggregate Planning Problems
Via Linear Programming

• LP
• s.t constraints
• All variables are continuously divisible is it
  a problem?
• Solution Produce 214.5 of aggregated units
• Hire 56.38 workers
• IP
• s.t constraints
• Some variables are continuously divisible, some
  are real number only problem?

91
Linear Programming Objective Function and
Constraints

• Number of constraints is 3T, number of unknown
  is 5T
• W0, I0, B0 initial workforce, initial
  inventory/backlog

92
Linear Programming Product Mix Planning

• Multiple products processed on various
  workstation
• i an index of product, i 1, , m
• j an index of workstation, j 1, , n
• t an index of period, t 1, , T
• Dit the maximum demand for product i for period
  t
• dit the minimum sales allows of product i for
  period t
• aij time required on workstation j to produce
  one unit of product i
• cjt capacity of workstation j in period t in
  the same units as aij
• ri net profit from one unit of product i
• hi cost to hold one unit of product i for one
  period in the inventory
• Xit amount of product i produced in period t
• Sit amount of product i sold in period t
• Iit number of units of product i held in the
  inventory at the end of period t

93
Linear Programming Product Mixed Planning
Objective Function and Constraints
This model can be used to obtain information
on demand feasibility bottleneck
location product mix
94
Product Mix Planning

• Demand feasibility
• Determine if the set of demands is
  capacity-feasible
• If SitDit then demand is feasible, otherwise
  demand is infeasible
• If could not find a feasible solution, then
  lower bound dit is too high for a given capacity
• Bottleneck locations
• Constraints restrict production on each
  workstation in each period
• Observe binding constraints to determine which
  workstations limit capacity
• Consistently binding workstation is a
  bottleneck
• Require close management attention
• Product mix
• If capacity is an issue, then model will try to
  maximize revenue by utilizing products with high
  net profit

95
Homework Assignment

• Read Production Operations Analysis by
  S.Nahmias chapter 3, sections 1 4
• Problems
• 3.5
• 3.9 3.11
• 3.14 3.16

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The END