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Aggregate Planning Planning the overall use of the organizations assets Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall,

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Presentation on theme: "Aggregate Planning Planning the overall use of the organizations assets Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall,"— Presentation transcript:

1 Aggregate Planning Planning the overall use of the organizations assets Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall, Inc. Philip A. Vaccaro, PhD

2 OVERVIEW OVERVIEW Objectives Objectives Definition of the Quasi-Unit Definition of the Quasi-Unit Strategies Strategies Computer Applications Computer Applications Master Production Schedule Master Production Schedule

3 Aggregate Planning Definition The overall employment of the firms facilities and other resources over the next 3 to 18 months so as to satisfy customer demand for all goods and services at minimum cost, as well as certain corporate policies such as no layoffs and no overtime pay.

4 Corporate Policy Requirements EXAMPLES A stable work force 100% in-house production Sufficient inventories to reduce the likelihood of stockouts to =< 5% Overtime labor hours not to exceed 10% of all regular labor hours

5 The Quasi-Unit A unit of product that is representative of all the firms goods and services. Examples are appliances and vehicles in manufacturing. Examples are airline passenger miles flown and restaurant meals in the service sector.

6 The Meal Quasi-Unit x ounces of meat x ounces of fish x ounces of poultry x gallons of water for cooking / cleaning x amount of gas and electricity for cooking x amount of labor time and cost for cooking x amount of labor time and cost for serving x ounces of vegetables of various types And much more! RESTAURANT

7 The Quasi - Unit The Quasi - Unit Gallons of Beer Not individual brands, bottles, kegs, cans, varieties Tons of Steel Not ingots or beams of varying tensile strength MANUFACTURING

8 The Quasi - Unit The Quasi - Unit Faculty-to-Student Contact Hours Transcends the number of hours of classroom instruction, student advising, directed studies, internships, and so on. Airline Passenger Miles Flown Transcends the number of flights by specific aircraft over specific routes. SERVICE SECTOR

9 The Pseudo / Composite Unit ALTERNATE NAMES FOR THE QUASI - UNIT Sometimes, the unit does not exist at all! It is a collection of square feet of sheet metal, several dozen screws or bolts, a variety of components, assemblies, ounces of glue, paint, fabric, and so on. STRICTLY USED FOR PLANNING PURPOSES

10 The New Quasi - Unit Definition SINGLE PRODUCT A SINGLE PRODUCT representing a family of individual products that: are processed on the same machines are processed by the same workers share the same general machine setup have similar cost structures, carry costs, and output rates share more / less the same parts and assemblies as well as physical charac- teristics

11 Quasi-Unit Interpretations Mid-sized Product Mid-sized Product in a line of similar products ( autos, refrigerators ) Only Product Only Product made in a particular manufacturing plant Basic Product Basic Product with minor differences between models Hybrid Product Hybrid Product or cross between several types of products produced in one or more plants Other

12 Basic Product with Minor Differences UNITED STATES MEXICO

13 Basic Product with Minor Differences 2013 Ford Fusion 2013 Lincoln MKZ

14 Basic Product with Minor Differences 2013 Ford Fusion 2013 Lincoln MKZ

15 Overview of the Process A quasi-unit demand forecast is developed for 3 to 18 months into the future. The firm then manipulates production rates, inventory levels, and work force levels to generate a series of aggregate plans that meet the forecasted demand. AGGREGATE PLANNING

16 Overview of the Process AGGREGATE PLANNING Cost estimates are developed and the lowest cost aggregate plan is adopted. Plan is then decomposed or disaggregated into a series of master production schedules that specify the exact number of products and services to be generated on a daily, weekly, or monthly basis. ( by make, model, color, and options )

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18 The Aggregate Plan I. Specified production rate for each time period. II. Specified level of inventory for each time period. III. Specified labor force size for each time period. * ( USUALLY 1 MONTH OR 1 QUARTER ) * MINIMUM DATA REQUIREMENTS

19 The Three Principal Strategies Inventory Cushion or Level Work Force Skeleton Force or Cadre Chase or Matching AGGREGATE PLANNING

20 Inventory Cushion Strategy Labor force size is fixed. Production rate is fixed. riseslow Inventories rise during slow demand periods. shrinkhigh Inventories shrink during high demand periods. Inventory stockout costs are reduced. Overtime pay, hiring, firing, subcontracting, and production rate change costs are eliminated.

21 The Inventory Cushion The Inventory Cushion DURING HIGH DEMANDDECREASES DURING LOW DEMANDINCREASES THE INVENTORY CUSHION INSULATES THE FACTORY FROM DEMAND FLUCTUATIONS

22 Skeleton Force or Cadre Strategy The permanent labor force size is usually set for the lowest monthly or quarterly forecasted quasi- unit demand. Higher demand must then be met by scheduling overtime, hiring temporary workers, subcontract- ing, and so on. Lower demand, if any, is tolerated as paid idle time.

23 SKELETON FORCE STRATEGY U.S. ARMY ACTIVE DUTY 480,000 46% RESERVE / NATIONAL GUARD 563,000 54% $100,000. TO PAY AND EQUIP EACH SOLDIER PERMANENT WORK FORCE $25,000. TO PAY AND EQUIP EACH SOLDIER PART-TIME WORK FORCE * RETIRED AND STANDBY RESERVES ACTIVATED ONLY IF ABSOLUTELY NECESSARY - 12,000,000 TROOPS

24 SKELETON FORCE STRATEGY HIGHEREDUCATION In 1984, full-time faculty were 80% of all faculty In 1987, full-time faculty were 67% of all faculty In 2001, full-time faculty were 55% of all faculty In 2003, full-time faculty were 50% of all faculty BETWEEN 1995 – 1997, 67% OF ALL NEW PROFESSORS WERE HIRED AS ADJUNCTS, IN ORDER TO SAVE $$$

25 Chase or Matching Strategy Calls for monthly / quarterly adjustment of the labor force size as necessary to match produc- tion to demand. Eliminates or reduces inventory carry costs and stockout costs. It generates substantial hiring, training, and termination costs and risks the degradation of employee morale and loyalty.

26 Two Types of Costs Those that WILL change from one developed plan to the next Those that WILL NOT change from one developed plan to the next NON-RELEVANTRELEVANT

27 THEIR INCLUSION WOULD NOT DIFFERENTIATE ONE PLAN FROM ANOTHER NON-RELEVANT COSTS ARE OMITTED FROM THE AGGREGATE PLANNING ANALYSIS SINCE THEY DO NOT ASSIST THE PLANNER IN IDENTIFYING THE MOST COST- EFFECTIVE PLAN.

28 Possible Plan Costs Regular Hourly Labor Rate Overtime Hourly Labor Rate Direct Labor Unit Cost Direct Materials Unit Cost Overhead Unit Cost Labor Severance Cost per Unit Subcontracting Unit Cost Inventory Unit Carry Cost Inventory Unit Stockout Cost Production Rate Change Costs Labor Hire/Train Cost per Unit AGGREGATE PLANNING

29 Possible Data Input Productivity per worker per day Number of production days in the plan Accurate monthly or quarterly demand forecasts Existing plant capacity Required labor hours per unit Required direct materials per unit Corporate policy regarding overtime, subcontracting, backordering, etc. Machine processing hours per unit AGGREGATE PLANNING

30 Possible Decision Variables AGGREGATE PLANNING Production Rate Changes Production Subcontracting Overtime Labor Hours Equipment Rental Backordering Temporary Employees Part-time Employees USEFUL FOR DEVELOPING ALTERNATIVE PLANS UNDER VARIOUS STRATEGIES

31 The Selected Aggregate Plan MODERATE INVENTORY LEVELS MODERATE STOCKOUT COSTS A SMALL NUMBER OF PRODUCTION RATE CHANGES WITH RELATIVELY SMALL MAGNITUDES ALMOST ALWAYS IT IS A MIXED STRATEGY CHARACTERIZED BY : The 4 th 4 th Strategy

32 Aggregate Planning TWO STARTING ASSUMPTIONS PASSIVE ASSUMES THE PRODUCT DEMAND PATTERN CANNOT BE ALTERED AGGRESSIVE ASSUMES THE PRODUCT DEMAND PATTERN CAN BE ALTERED IF NECESSARY

33 Product Demand Pattern PRE - STABILIZATION PRE - STABILIZATION THE DEMAND RATE MAY VARY DRAMATICALLY FROM THE NORMAL PRODUCTION RATE, MAKING AGGREGATE PLANNING DIFFICULT AND LESS COST EFFECTIVE ( time ) ( units ) Normal Production Output Variable Product Demand

34 Product Demand Pattern Normal Production Output ( units ) ( time ) Variable Product Demand STABILIZATION THE LEVELING OF DEMAND TO APPROACH THE NORMAL PRODUCTION RATE MAKES AGGREGATE PLANNING MUCH EASIER AND MORE COST EFFECTIVE

35 Some Demand Leveling Tactics Heavy advertising, price discounts, coupons, and contests during periods of low demand. Little or no advertising and price increases during periods of high demand. Production of countercyclic, similar products. Reservation systems.

36 Aggregate Planning with QM for Windows

37 We Select Aggregate Planning From The Menu WE SELECT AGGREGATE PLANNING FROM THE MENU

38 WE DESIRE TO SET UP A NEW PROGRAM To

39 We Select The 1 st Menu WE SELECT THE FIRST OPTION

40 THE DIALOGUE BOX APPEARS

41 WE SELECT SIX PERIODS FOR THE AGGREGATE PLAN AND LABEL THEM JANUARY THROUGH JUNE IF DEMAND EXCEEDS PRODUCT SUPPLY IN ANY MONTHLY PERIOD, IT IS ASSUMED THOSE SALES ARE LOST FOREVER

42 THE DATA TABLE APPEARS

43 Inventory Cushion Strategy EXAMPLE OBJECTIVE TO MAINTAIN A CONSTANT-SIZE WORK FORCE AND UNIFORM PRODUCTION RATE OVER THE SPECIFIED PLANNING PERIOD. ASSUMPTIONS DEMAND FORECAST OF 6,200 QUASI-UNITS OVER THE NEXT SIX MONTHS ONE-HUNDRED-TWENTY- FOUR AVAILABLE PRODUCTION DAYS QUASI-UNIT INVENTORY CARRY COST IS $5.00 PER MONTH EACH WORKER PRODUCES FIVE QUASI-UNITS PER DAY EACH WORKER IS PAID $120.00 PER DAY

44 CALENDAR MONTH ACTUAL PRODUCTION QUASI-UNIT FORECAST CUSHION NET CHANGE ENDING INVENTORY JAN1100900+200200 FEB900700+200400 MAR1050800+250650 APR10501200- 150500 MAY11001500- 400100 JUN10001100- 1000 6200 units 6200 units 1850 units 22 DAYS AVAILABLE X 50 UNITS DAILY 18 DAYS AVAILABLE X 50 UNITS DAILY FIRM MUST PRODUCE 50 UNITS DAILY IN ORDER TO MEET THE SIX-MONTH DEMAND ( 6,200 / 124 DAYS ) 21 DAYS AVAILABLE X 50 UNITS DAILY Inventory Cushion Strategy

45 1.6 hours of labor per quasi-unit X $15.00 / hour average labor rate Quasi-UnitDemandForecasts Quasi-UnitInventory Carry Cost perMonth PlannedMonthlyProduction

46 Inventory Cushion Strategy EXAMPLE TOTAL COSTS: $158,050.00 INVENTORY CARRY COSTS…………...….$9,250.00 ( 1,850 units x $5.00/unit ) OVERTIME, HIRE/FIRE, SUBCONTRACTING..............$0.00 ( fixed work force ) PRODUCTION RATE CHANGE COSTS………$0.00 ( uniform production rate ) LABOR COSTS..$148,800.00…..( 50/5 = 10 workers x $120.00/day x 124 days)

47 Strategy Total Cost Total Labor Cost Total Inventory Carry Costs

48 Quasi-Unit Demand Regular Time Quasi-Unit Production

49 The Inventory Cushion ( peaks at mid-term of plan )

50 Skeleton Force Strategy EXAMPLE OBJECTIVE TO BUILD A PERMANENT LABOR FORCE AROUND A SPECIFIC LEVEL OF DEMAND, TOLERATING PAID IDLE TIME DURING LOWER DEMAND PERIODS AND INCURRING COSTS OF OVERTIME LABOR OR SUBCONTRACTING DURING HIGHER DEMAND PERIODS. THE SPECIFIC LEVEL OF DEMAND IS USUALLY THE LOWEST LEVEL OF DEMAND

51 Skeleton Force Strategy EXAMPLE ASSUMPTIONS A PERMANENT LABOR FORCE BUILT AROUND THE LOWEST DEMAND PERIOD IN ORDER TO SAVE ON RETIREMENT BENEFITS, HEALTH INSURANCE, PAID VACATIONS, LEAVE, ETC. FEBRUARY HAS THE LOWEST FORECASTED QUASI-UNIT DEMAND (700 units) FEBRUARY HAS EIGHTEEN (18) AVAILABLE PRODUCTION DAYS FIRM WANTS TO SUBCONTRACT PRODUCTION TO AN OUTSIDE COMPANY WHENEVER QUASI-UNIT DEMAND EXCEEDS THE PERMANENT LABOR FORCE CAPABILITY SUBCONTRACTED QUASI-UNITS COST THE FIRM $10.00 EACH

52 Skeleton Force Strategy EXAMPLE CALCULATIONS DAILY FEBRUARY PRODUCTION…....700/18 days = 39 units daily IN-HOUSE PRODUCTION…..39 units/day x 124 days = 4,836 units SUBCONTRACTED PRODUCTION…….6,200 – 4,836 = 1,364 units REQUIRED WORKERS……..…39/5 units per worker per day = 7.8

53 Quasi-UnitSubcontractCost In February, in-house production was 39 units per day for 18 days = 702 quasi-units which resulted in overproduction of 2 quasi-units In March, in-house production was 39 units per day for 21 days = 819 quasi-units which resulted in overproduction of 19 quasi-units In April, in-house production was 39 units per day for 21 days = 819 units. The shortfall seems to be ( 1200 - 819 ) = 381 units, but it was reduced to 360 quasi-units, due to the 21 quasi-unit surplus generated in February and March Over Production in Regular Time

54 Skeleton Force Strategy EXAMPLE COSTS: $129,704.00 LABOR COST...$116,064.00 ( 7.8 workers x $120.00 per day x 124 days) SUBCONTRACT COST………..$13,640.00 (1,364 units x $10.00 per unit) OVERTIME, HIRE/FIRE, TRAINING COSTS……..$0.00 (rejected options) INVENTORY CARRY COST……..$0.00 (all demand satisfied exactly via house production or subcontracting)

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56 RT - Regular Time Production Sub - Subcontracted Production Subcontracting Over Production In Regular Time

57 In-House Production Shortfall ( Subcontracted )

58 Chase or Matching Strategy EXAMPLE OBJECTIVE TO HIRE OR TERMINATE PERSONNEL AS NEEDED IN ORDER TO MATCH PRODUCTION TO DEMAND, PERIOD-BY-PERIOD, RESULTING IN THE ELIMINATION OR DRASTIC REDUCTION OF BOTH INVENTORY CARRY AND STOCKOUT COSTS.

59 Chase or Matching Strategy EXAMPLE ASSUMPTIONS EACH QUASI-UNIT REQUIRES 1.6 HOURS OF DIRECT LABOR PERSONNEL TERMINATION COST IS PRORATED AT $15.00 PER MANUFACTURED QUASI-UNIT CANCELLED PERSONNEL RECRUITING AND TRAINING COST IS PRORATED AT $10.00 PER MANUFACTURED QUASI-UNIT ADDED EACH WORKER EARNS $15.00 PER HOUR ON AVERAGE.

60 EXAMPLE: If sales were forecasted to be 100 units lower in the next period, then the prorated employee termination costs would be ( 100 x $15.00 ) $1500.00 If sales were forecasted to be 100 units higher in the next period, then the prorated employee hiring and training costs would be ( 100 x $10.00 ) $1000.00 PRORATED TERMINATION COST PRORATED HIRE/TRAIN COST

61 MONTH DEMAND FORECAST PRODUCTION COSTS FORECAST CHANGE FIRE HIRE/FIRE COSTS TOTAL COSTS JAN900$21,600.$21,600. FEB700$16,800. (200)x$15 $3,000.$19,800. MAR800$19,200. 100 x $10 $1,000.$20,200. APR1200$28,800. 400 x $10 $4,000.$32,800. MAY1500$36,000. 300 x $10 $3,000.$39,000. JUN1100$26,400. (400)x$15 $6,000.$32,400. $148,800.$8,000.$9,000.$165,800. 900 UNITS x 1.6 HOURS x $15.00 = $21,600.00 700 UNITS X 1.6 HOURS X $15.00 = $16,800.00 TOTAL MONTHLY COST = PRODUCTION + HIRE( FIRE ) COSTS The Chase Strategy

62 Chase or Matching Strategy EXAMPLE COSTS : $165,800.00 REGULAR TIME LABOR COST……………………………..$148,800. HIRE / TERMINATION COSTS………………………………..$17,000 INVENTORY CARRY AND STOCKOUT COSTS………………... $ 0. 6200 UNITS x 1.6 HOURS/UNIT = 9,920 HOURS x $15.00/HOUR $8,000.00 HIRE COSTS + $9,000.00 TERMINATION COSTS PRODUCTION MATCHES DEMAND EXACTLY PERIOD-BY-PERIOD

63 $17,000.00 total broken down into hire/fire and termination

64 Period Demand = Period Production

65 CUMULATIVE PRODUCTION EQUALS CUMULATIVE DEMAND ( ONE IN THE SAME LINE )

66 Aggregate Plan Examples POSTSCRIPT I.If only three strategies or plans were generated and skeleton force evaluated, the firm would select the skeleton force since it has the lowest projected total costs. II.Direct material cost, direct machine hour cost, and applied overhead per quasi-unit were identical under all three plans. Hence, they are non-relevant costs and routinely omitted from the analysis.

67 The Relevant Range of Activity 0 1000 2000 3000 4000 5000 6000 7000 8000 CUMULATIVE PRODUCTION ( IN UNITS ) COSTS SEMI-VARIABLE COSTS VARIABLE COSTS Relevant Range 1 Relevant Range 2 Relevant Range 3 X 6200 FIXED COSTS PRODUCTION OF 6200 QUASI-UNITS FALLS JUST WITHIN THE RELEVANT COST RANGE OF 6000-8500 UNITS. THEREFORE, UNIT DIRECT LABOR, DIRECT MATERIALS, AND OVERHEAD REMAIN THE SAME 6000 8500

68 Aggregate Plan Examples POSTSCRIPT III.The mix of resourceslabor force size, production rate, and inventory level as well as subcontracting ----and their timing allowed the development of 3 unique aggregate plan proposals and associated costs. IV.In the skeleton force strategy, the firm elected to augment the capacity of its small permanent labor force by subcontracting exclusively. Variations of this strategy could, and usually are generated us- ing overtime hours, temporary labor hours, and subcontracting exclusively or in combination.

69 Aggregate Plan Examples POSTSCRIPT V.Corporate policies may impose limitations on the use of subcontracting, overtime hours, inventory levels, production rates, machines, available days for production, and so on.

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71 Aggregate Planning with QM for Windows Transportation Algorithm

72 Transportation Algorithm Approach to Aggregate Planning When aggregate planning is viewed as an allocation of capacity to meet forecasted demand. Produces an optimal plan for minimizing costs ! Can specify regular, overtime, and subcontracting production in each time period. Can specify inventory carryover from period to period. Will not handle non-linear costs such as hiring and layoff. Applied Management Science for Decision Making, 1e © 2011 Pearson Prentice-Hall, Inc. Philip A. Vaccaro, PhD

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77 Forecasted Quasi-Unit Demand 700 Quasi-Units can be produced on regular time each month 50 Quasi-Units can be produced on overtime each month The number of quasi-units that can be subcontracted out each month Beginning Inventory ( from May ) Per Unit Labor Costs Inventory Carry Costs per Quasi-Unit per Month

78 June demand of 800 is met by 50 units of beginning inventory, 700 units of June regular production, and 50 units of June subcontracted production. July demand of 1000 is met by 50 units of beginning inventory, 50 units of June overtime production, 700 units of July regular production, 50 units of July overtime production, and 150 units of subcontracted July production. August demand of 750 is met by 700 units of August regular production, and 50 units of August overtime production.

79 This shows the costs and headings to be inserted in each cell for the normal transportation tableau, if there had been one. Note that cells that are not allowed to be filled, have a prohibitive cost of $9,999.00 ! Also note that the dummy column for unused capacity in each month is missing in the original transportation tableau ! For each month that a quasi-unit remains in inventory, an additional $2.00 carry (holding) cost is added to its original cost

80 Shows the total number of quasi-units that should be made under regular-time, overtime, and subcontracting over the life of the 3-month aggregate plan. Total quasi-unit production over the 3-month aggregate plan The only carry costs were for the initial inventory of 100 quasi-units

81 Here, we are using the regular transportation algorithm module to solve the same aggregate planning problem

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85 For each month that a quasi-unit is kept in inventory, a $2.00 carry cost will be added to its original production cost

86 June demand of 800 was met by 100 units of beginning inventory (from May) and 700 units of regular time production in June itself. July demand of 1,000 was met by 50 units made overtime in June, 50 units subcontracted in June, 700 units of regular time production in July itself, 50 units made overtime in July, and 150 units subcontracted in July. August demand of 750 was met by 700 units of regular time production in August and 50 units made overtime in August.

87 The 1 st feasible solution The 2 nd feasible solution

88 The 3 rd feasible and optimal solution

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91 Aggregate Planning

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95 Sample Data + Basic Template

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101 The Simplex Method An alternative to the transportation method of linear programming. Must be employed where non-linear costs are involved, such as hiring and layoff. Minimum and maximum constraints can be put on the desired amounts of regular labor, overtime labor, subcontracting, backordering, and many other factors on a monthly, bi-monthly, quarterly, or semi-annual basis.

102 The Simplex Method Constraint formulation is quite complex. The optimal solution virtually always must be obtained via computer. Dozens or hundreds of variables are involved.

103 The Simplex Method A production manager must develop an aggregate plan for the next two quarters of the year. The plant produces computer terminals. 700 terminals need to be shipped to customers in the 1 st quarter and 3,200 in the 2 nd quarter. It is the firms policy to ship orders in the quarter in which they are ordered. It takes 5 hrs labor to produce each terminal, and only 9,000 hours of straight-time labor is available in each of the two quarters. Overtime can be used, but the firm limits overtime in each quarter to 10% of straight-time labor available. Labor costs $12.00 per hour at the straight-time rate and $18.00 per hour at the overtime rate. EXAMPLE

104 The Simplex Method If a terminal is produced in one quarter and shipped in the next quarter, a carrying cost of $50.00 is incurred. Requirement: How many terminals should be produced on straight-time and overtime in each of the two quarters to minimize straight-time labor, overtime labor, and carrying costs? The market requirements, straight-time labor availability, and overtime policy must be adhered to.

105 The Simplex Method X 1 = terminals made on straight-time in 1 st quarter and shipped in 1 st quarter. X 2 = terminals made on overtime in 1 st quarter and shipped in the 1 st quarter. X 3 = terminals made on straight-time in 1 st quarter and shipped in the 2 nd quarter. X 4 = terminals made on overtime in 1 st quarter and shipped in the 2 nd quarter. X 5 = terminals made on straight-time in 2 nd quarter and shipped in the 2 nd quarter. X 6 = terminals made on overtime in 2 nd quarter and shipped in the 2 nd quarter. Q 1 = 1 st quarter, Q 2 = 2 nd quarter, Z = total cost of the plan Defining the Decision Variables

106 The Simplex Method Minimize Z = 60X 1 + 90X 2 + 110X 3 + 140X 4 + 60X 5 + 90X 6 Subject to: X 1 + X 2 => 700 Q 1 demand X 3 + X 4 + X 5 + X6 => 3,200 Q 2 demand 5X 1 + 5X 3 =< 9,000 Q 1 straight-time labor 5X 5 =< 9,000 Q 2 straight-time labor 5X 2 + 5X 4 =< 900 Q 1 overtime labor 5X 6 =< 900 Q 2 overtime labor The Model

107 Aggregate Planning with QM for Windows Simplex Linear Programming

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114 X 1 = 580 units made and shipped on straight- time in 1 st Qtr X 2 = 120 units made on overtime and shipped in 1 st Qtr X 3 = 1,220 units made on straight-time in 1 st Qtr and shipped during 2 nd Qtr X 5 = 1,800 units made and shipped on straight- time in 2 nd Qtr X 6 = 180 units made on overtime in 2 nd Qtr and shipped in 2 nd Qtr

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137 Aggregate Planning Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall, Inc. Philip A. Vaccaro, PhD


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