1. 1 AGGREGATE PLANNING Production Planning and Control 2 Haeryip Sihombing Fakulti Kejuruteraan Pembuatan Universiti Teknologi Malaysia Melaka

2. 2 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

3. 3 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.

4. 4 Capacity Decisions Hierarchy Linkages Facilities Planning Aggregate Planning Scheduling Time Frame Facilities Planning Aggregate Planning Scheduling Time

5. 5 Aggregation refers to the idea of focusing on overall capacity, rather than individual products or services. Aggregation is done according to: Products Products Labor Labor Time Time

6. 6 Production Planning Long Range Planning Long Range Planning Strategic planning (1-5 years) Strategic planning (1-5 years) Medium Range Planning Medium Range Planning Employment, output, and inventory levels (2-18 months) Employment, output, and inventory levels (2-18 months) Short Range Planning Short Range Planning Job scheduling, machine loading, and job sequencing (0-2 months) Job scheduling, machine loading, and job sequencing (0-2 months)

7. 7 Aggregate production planning involves managing... Work force levels - the number of workers required for production. Work force levels - the number of workers required for production. Production rates - the number of units produced per time period. Production rates - the number of units produced per time period. Inventory levels - the balance of unused units carried forward from the previous period. Inventory levels - the balance of unused units carried forward from the previous period.

8. 8 Common objectives of production planning... MINIMIZE: MINIMIZE: cost, 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 MAXIMIZE: MAXIMIZE: profits, customer service

9. 9 Methods of Influencing Demand Price Incentives Price Incentives Reservations Reservations Backlogs Backlogs Complementary Products or Services Complementary Products or Services Advertising/promotion Advertising/promotion

10. 10 Methods of Influencing Supply Hiring/firing workers Hiring/firing workers Overtime/slack time Overtime/slack time Part time/temporary labor Part time/temporary labor Subcontracting Subcontracting Cooperative arrangements Cooperative arrangements Inventories Inventories

11. 11 Aggregate Production Planning Variable Costs Hiring/firing costs Hiring/firing costs Overtime/slack time costs Overtime/slack time costs Part time/temporary labor costs Part time/temporary labor costs Subcontracting costs Subcontracting costs Cooperative arrangements costs Cooperative arrangements costs Inventory carrying costs Inventory carrying costs Backorder or stock out costs Backorder or stock out costs

12. 12 Aggregate Production Planning Strategies Chase strategy Chase strategy production rates or work force levels are adjusted to match demand requirements over planning horizon production rates or work force levels are adjusted to match demand requirements over planning horizon Level strategy Level strategy constant production rate or work force level is maintained over planning horizon constant production rate or work force level is maintained over planning horizon Mixed strategy Mixed strategy both inventory level changes and work force level changes occur both inventory level changes and work force level changes occur

13. 13 Aggregate Production Planning Techniques Trial-and-error method Trial-and-error method Mathematical techniques Mathematical techniques

14. 14 Trial-and-Error Method Examples of alternative strategies: Vary work force levels Vary work force levels Level work force, vary inventories and backorders Level work force, vary inventories and backorders Level work force, use subcontracting Level work force, use subcontracting Level work force, use overtime and subcontracting Level work force, use overtime and subcontracting

15. 15 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

16. 16 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.

17. 17 Aggregate Planning in Services For service companies, aggregate planning results in staffing plans that call for changing the number of employees or subcontracting.

18. 18 The END

19. 19 Production Planning Environment Competitor’s 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

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

21. “Intermediate-range plan” (6 month – 2 years) updated quarterly “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 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 Decision: changes in work force, additional machines, subcontracting, overtime Level of detail: Aggregated Level of detail: Aggregated Degree of uncertainty: Medium Degree of uncertainty: Medium Planning Production

22. “Short-range plan” (1 week – 6 month) updated daily or weekly “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 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 Decision: overtime and undertime, possibility of not fulfilling all demand, subcontracting, delivery dates for suppliers, product quality Level of detail: Very Detailed Level of detail: Very Detailed Degree of uncertainty: Low Degree of uncertainty: Low Planning Production

23. Production Planning Example Small company makes one product – plastic cases to hold CD’s. Small company makes one product – plastic cases to hold CD’s. Two different types of mold are used to produce top & bottom. Two different types of mold are used to produce top & bottom. Two halves are manually put together, placed in the boxes & shipped. Two halves are manually put together, placed in the boxes & shipped. The injection molding machines can make 550 pieces per hour. The injection molding machines can make 550 pieces per hour. A worker can finish 55 cases in 1 hour (10 workers / machine) 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. 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. 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 Each worker can not work more than 8 hours per day 4,000/8 = 500 pieces per hour have to be produced. 4,000/8 = 500 pieces per hour have to be produced. Plan: 1 machine, 10 workers, 8 hours/day, 5 days/week Plan: 1 machine, 10 workers, 8 hours/day, 5 days/week

24. 24 The Hierarchy of Production Planning Decisions

25. 25 Goal: To plan gross work force levels and set firm-wide production plans. 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.) 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.) Introduction to Aggregate Planning

26. Constant production rate can be satisfied with constant capacity. 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. 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. 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. If supplier and customers are also close, frequent deliveries of raw material and finished goods will keep inventory low. How realistic is this example? How realistic is this example? Strategies to cope with fluctuating demand? Strategies to cope with fluctuating demand? -- change the demand-- produce at constant rate anyway -- vary the production rate-- use combination of above strategies

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

28. Introduction to Aggregate Planning: Influencing Demand Shift demand from peak periods to non-peak periods / create new demand for non-peak periods 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 company’s – differential pricing system) Creating new demand can be done through advertising or incentive programs (automobile industry: rebates; telephone company’s – differential pricing system) Smoothing demand Smoothing demand

29. Introduction to Aggregate Planning: Influencing Demand Produce several products with peak demand in different periods Produce several products with peak demand in different periods Products should be similar, so that manufacturing them is not too different Products should be similar, so that manufacturing them is not too different Snowmobiles and jetskis – same engines, similar body work Snowmobiles and jetskis – same engines, similar body work Lawn-mowers – snowblowers; baseball – football equipment Lawn-mowers – snowblowers; baseball – football equipment

30. 30 Medium Range Planning: Aggregate Production Planning Establish production rates by major product groups Establish production rates by major product groups by labor hours required or units of production 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) Attempt to determine monthly work force size and inventory levels that minimizes production related costs over the planning period (for 6-24 month)

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

32. Overview of the Problem Suppose that D 1, D 2,..., D T are the forecasts of demand for aggregate units over the planning horizon (T periods.) The problem is to determine both work force levels (W t ) and production levels (P t ) to minimize total costs over the T period planning horizon.

33. Important Issues Smoothing. Refers to the costs and disruptions that result from making changes from one period to the next. 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. 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. 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. Treatment of Demand. Assume demand is known. Ignores uncertainty to focus on the predictable/systematic variations in demand, such as seasonality.

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

35. 35 Cost of Changing the Size of the Workforce

36. 36 Holding and Back-Order Costs Back-orders Positive inventory Slope = C P Slope = C i $ Cost Inventory

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

38. Example of fictitious aggregate units. (Example.1) One plant produced 6 models of washing machines: Model # hrs. Price % sales A 55324.228532 K 42424.934521 L 98985.139517 L 38005.242514 M 26245.452510 M 38805.872506 Question: How do we define an aggregate unit here?

39. Example continued Notice: Price is not necessarily proportional to worker hours (i.e., cost): why? 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. 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.

40. 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. 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.

41. Step 1: Determine “net” demand. (subtract starting inv. from per. 1 forecast and add ending inv. to per. 8 forecast.) MonthNet Predicted Cum. Net Days Demand Demand Demand Demand 1(Jan)220220 22 2(Feb)280500 16 3(Mar)460960 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

42. 42

43. Step 2. Graph Cumulative Net Demand to Find Plans Graphically

44. Constant Work Force Plan Suppose that we are interested in determining a production plan that doesn’t change the size of the workforce over the planning horizon. How would we do that? Suppose that we are interested in determining a production plan that doesn’t 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. 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.

45. Monthly Production = 1940/8 = 242.2 or rounded to 243/month. But: there are stockouts.

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

47. 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 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

48. 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 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. 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.

49. To overcome these shortcomings: Assume number of workdays per month is given Assume number of workdays per month is given K factor given (or computed) where K factor given (or computed) where K = # of aggregate units produced by one worker in one day

50. 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: 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/(38*40) = 0.3421 K= 520/(38*40) = 0.3421 = average number of units produced by one worker in one day. = average number of units produced by one worker in one day.

51. 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 = 22+16+23 = 61. Find 960/61 = 15.7377 units/day implies 15.7377/.3421 = 46 workers required. 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 = 22+16+23 = 61. Find 960/61 = 15.7377 units/day implies 15.7377/.3421 = 46 workers required.

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

53. Addition of Costs Holding Cost (per unit per month): $8.50 Holding Cost (per unit per month): $8.50 Hiring Cost per worker: $800 Hiring Cost per worker: $800 Firing Cost per worker: $1,250 Firing Cost per worker: $1,250 Payroll Cost: $75/worker/day Payroll Cost: $75/worker/day Shortage Cost: $50 unit short/month Shortage Cost: $50 unit short/month

54. Cost Evaluation of Constant Work Force Plan Assume that the work force at end of Dec was 40. Assume that the work force at end of Dec was 40. Cost to hire 6 workers: 6*800 = $4800 Cost to hire 6 workers: 6*800 = $4800 Inventory Cost: accumulate ending inventory: (126+98+0+...+687) = 2093. Add in 100 units netted out in Aug = 2193. Hence Inv. Cost = 2193*8.5=$18,640.50 Inventory Cost: accumulate ending inventory: (126+98+0+...+687) = 2093. Add in 100 units netted out in Aug = 2193. Hence Inv. Cost = 2193*8.5=$18,640.50 Payroll cost: Payroll cost: ($75/worker/day)(46 workers )(167days) = $576,150 Cost of plan: $576,150 + $18,640.50 + $4800 = $599,590.50 ~ $600K Cost of plan: $576,150 + $18,640.50 + $4800 = $599,590.50 ~ $600K

55. 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. 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.

56. 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. 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.

57. 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. 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.

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

59. 59 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. 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.

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

61. Overview of the Problem D 1, D 2,..., D T - the forecasts of demand for aggregate units over the planning horizon (T periods) Determine: W t - work force levels P t - production levels I t – inventory levels H t – number of workers hired in this period F t – number of workers fired in this period O t – overtime production in units U t – “undertime”, worker idle time in units S t – number of units subcontracted from outside to minimize total costs over the T period planning horizon

62. Example of fictitious aggregate units: One plant produced 6 models of washing machines: Model # hrs. Price % sales A 55324.228532 K 42424.934521 L 98985.139517 L 38005.242514 M 26245.452510 M 38805.872506 Question: How do we define an aggregate unit here? Price/#hours67.8670.4177.4581.7397.22125.0

63. Example (continued) Notice: Price is not necessarily proportional to worker hours (i.e., cost): why? Notice: Price is not necessarily proportional to worker hours (i.e., cost): why? One method for defining an aggregate unit: 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. Forecasts for demand for aggregate units can be obtained by taking a weighted average (using the same weights) of individual item forecasts.

64. The washing machine plant is interested in determining work force and production levels for the next 8 months The washing machine plant is interested in determining work force and production levels for the next 8 months Forecasted demands for Jan-Aug. are: 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 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 Find monthly production levels Example (continued)

65. Step 1: Determine “net” demand. (subtract starting inventory from period 1 forecast and add ending inventory to period 8 forecast) Month ForecastedNet PredictedCum. Net Demand Demand Demand Demand Demand Demand 1(Jan)420420-200=220 220 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)125 125+100=225 1940  Starting inventory - 200 and final inventory - 100 units

66. Step 2. Graph Cumulative Net Demand to Find Plans Graphically Determine a production plan that doesn’t change the size of the workforce over the planning horizon. What to do? 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.

67. Monthly Production = = 1940 / 8 = 242.5 (rounded to 243/month) Any shortfalls in this solution? Demand is backlogged

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

69. 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: 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 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 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: Some thoughts: Final inventory is 620 units, not 100 units Final inventory is 620 units, not 100 units Cost of carrying inventory in each period Cost of carrying inventory in each period

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

72. Example #2 (based on example #1) The plant has 38 workers who produced 630 units in a period of 40 days The plant has 38 workers who produced 630 units in a period of 40 days K= 630/(38*40) = 0.414  average number of units produced by one worker in one day K= 630/(38*40) = 0.414  average number of units produced by one worker in one day Assume we are given the following # working days per month: Assume we are given the following # working days per month: jan 22apr 20jul18 feb 16may 21aug10 mar 23 jun17

73. Constant Work Force Production Plan: 38 workers, K=.414 Month # wk Prod. Cum Cum Nt End Inv Month # wk Prod. Cum Cum Nt End Inv days Dem Level Prod Dem days Dem Level Prod Dem Jan 22 220 346 346 220 126 Jan 22 220 346 346 220 126 Feb 16 280 252 598 500 98 Feb 16 280 252 598 500 98 Mar 23 460 362 960 960 0 Mar 23 460 362 960 960 0 Apr 20 190 315 1275 1150 125 Apr 20 190 315 1275 1150 125 May 21 310 330 1605 1460 145 May 21 310 330 1605 1460 145 Jun 22 145 346 1951 1605 346 Jun 22 145 346 1951 1605 346 Jul 21 110 330 2281 1715 566 Jul 21 110 330 2281 1715 566 Aug 22 125 346 2627 1940 687 Aug 22 125 346 2627 1940 687 +100 +100

74. Addition of Costs Holding Cost (per unit per month): $ 8.50 Holding Cost (per unit per month): $ 8.50 Hiring Cost per worker: $ 800.00 Hiring Cost per worker: $ 800.00 Firing Cost per worker: $ 1,250.00 Firing Cost per worker: $ 1,250.00 Payroll Cost ( per worker/day): $ 75.00 Payroll Cost ( per worker/day): $ 75.00 Shortage Cost (unit short/month): $ 50.00 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 Assume that the work force at end of Dec was 32 Cost to hire 6 workers: 6*800 = $4,800 Cost to hire 6 workers: 6*800 = $4,800 Inventory Cost  accumulate ending inventory: (126+98+0+125+145+346+567+687) = 2,095 Inventory Cost  accumulate ending inventory: (126+98+0+125+145+346+567+687) = 2,095 (100 units at the end of the august in included in 687 units inventory) (100 units at the end of the august in included in 687 units inventory) Hence Inventory Cost = 2095*8.5=$17,809.37 Payroll cost: Payroll cost: ($75/worker/day)(38 workers )(167days) = $475,950 Cost of plan: 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 The modified plan calls for hiring 6 workers in Jan (to 38) hiring 6 workers in Jan (to 38) reducing the workforce to 23 (from 38) at start of April reducing the workforce to 23 (from 38) at start of April cost of hiring is $ 4,800.00 $ 4,800.00 cost of hiring is $ 4,800.00 $ 4,800.00 cost of layoffs is $ 18,750.00 $ 0.00 cost of layoffs is $ 18,750.00 $ 0.00 payroll cost is $ 356,700.00 $ 475,950.00 payroll cost is $ 356,700.00 $ 475,950.00 holding costs are $ 2,528.93 $ 17,809.37 holding costs are $ 2,528.93 $ 17,809.37 shortage costs are $ 7,770.40 $ 0.00 shortage costs are $ 7,770.40 $ 0.00 The total cost of the modified plan is $ 390,548.33 The total cost of the modified plan is $ 390,548.33 Original plan had cost of $ 498,559.37 Original plan had cost of $ 498,559.37

78. Cost Evaluation of Modified Plan with Two Workforce Adjustment: The modified plan calls for The modified plan calls for hiring 6 workers in January hiring 6 workers in January firing 8 workers at start of April firing 8 workers at start of April firing 12 workers at start of June firing 12 workers at start of June Two One None Two One None cost of hiring is $ 4,800.00 $ 4,800.00 $ 4,800.00 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 cost of layoffs is $ 25,000.00 $ 18,750.00 $ 0.00 payroll cost is $ 353,850.00 $ 356,700.00 $ 475,950.00 payroll cost is $ 353,850.00 $ 356,700.00 $ 475,950.00 holding costs are $ 3,452.87 $ 2,528.93 $ 17,809.37 holding costs are $ 3,452.87 $ 2,528.93 $ 17,809.37 shortage costs are $ 0.00 $ 7,770.40 $ 0.00 shortage costs are $ 0.00 $ 7,770.40 $ 0.00 The total cost : $ 387,102.87 $ 390,548.33 $ 498,559.37 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 Month # wk Prod. Cum Cum Nt End Inv days Dem Level Prod Dem days Dem Level Prod Dem Jan 22 220 346 346 220 126 Jan 22 220 346 346 220 126 Feb 16 280 252 598 500 98 Feb 16 280 252 598 500 98 Mar 23 460 362 960 960 0 Mar 23 460 362 960 960 0 Apr 20 190 315 1275 1150 125 Apr 20 190 315 1275 1150 125 May 21 310 330 1605 1460 145 May 21 310 330 1605 1460 145 Jun 22 145 346 1951 1605 346 Jun 22 145 346 1951 1605 346 Jul 21 110 330 2281 1715 566 Jul 21 110 330 2281 1715 566 Aug 22 125 346 2627 1940 687 Aug 22 125 346 2627 1940 687 +100 +100

80. Cost Reduction in Constant Work Force Plan

81. Zero Inventory Plan (Chase Strategy) Idea: 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) 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. Then take net demand each month and dividing by this quantity. The resulting ratio is rounded up and possibly adjusted downward.

82. n 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. 83 Hybrid Strategies Use a combination of options: Use a combination of options: Build-up inventory ahead of rising demand & use backorders to level extreme peaks Build-up inventory ahead of rising demand & use backorders to level extreme peaks Finished goods inventories: Anticipate demand Finished goods inventories: Anticipate demand Back orders & lost sales: Delay delivery or allow demand to go unfilled Back orders & lost sales: Delay delivery or allow demand to go unfilled Shift demand to off-peak times: Proactive marketing Shift demand to off-peak times: Proactive marketing Overtime: Short-term option Overtime: Short-term option Pay workers a premium to work longer hours Pay workers a premium to work longer hours

84. 84 Hybrid Strategies Undertime: Short-term option Undertime: Short-term option Slow the production rate or send workers home early (lowers labor productivity, but doesn’t tie up capital in finished good inventories) Slow the production rate or send workers home early (lowers labor productivity, but doesn’t tie up capital in finished good inventories) Reassign workers to preventive maintenance during lulls Reassign workers to preventive maintenance during lulls Subcontracting: Medium-term option Subcontracting: Medium-term option Subcontract production or hire temporary workers to cover short- term peaks Subcontract production or hire temporary workers to cover short- term peaks Hire & fire workers: Long-term option Hire & fire workers: Long-term option Change the size of the workforce Change the size of the workforce Layoff or furlough workers during lulls Layoff or furlough workers during lulls

85. 85 Another APP Example _________________________ 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 QuarterSales Forecast (lb) Spring80,000 Summer50,000 Fall120,000 Winter150,000

86. 86 Level Production Strategy SalesProduction SalesProduction Quarter ForecastPlanInventory Spring80,000100,00020,000 Summer50,000100,00070,000 Fall120,000100,00050,000 Winter150,000100,0000 400,000140,000 Cost = 140,000 pounds x 0.50 per pound = $70,000

87. 87 Chase Demand Strategy (Zero Inventory) SalesProductionWorkersWorkersWorkers SalesProductionWorkersWorkersWorkers QuarterForecastPlanNeededHiredFired Spring80,00080,00080-20 Summer50,00050,00050-30 Fall120,000120,00012070- Winter150,000150,00015030- 10050 Cost = (100 workers hired x $100) + (50 workers fired x $500) = $10,000 + 25,000 = $35,000 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

88. 88 APP By Linear Programming Min Z = $100 (H 1 + H 2 + H 3 + H 4 ) + $500 (F 1 + F 2 + F 3 + F 4 )+ $0.50 (I 1 + I 2 + I 3 + I 4 ) Subject to P 1 - I 1 = 80,000(1)Demand P 1 - I 1 = 80,000(1)Demand I 1 + P 2 - I 2 = 50,000(2)constraints I 2 + P 3 - I 3 = 120,000(3) I 3 + P 4 - I 4 = 150,000(4) P 1 - 1,000 W 1 = 0(5)Production P 2 - 1,000 W 2 = 0(6)constraints P 3 - 1,000 W 3 = 0(7) P 4 - 1,000 W 4 = 0(8) W 1 - H 1 + F 1 = 100(9) Work force W 1 - H 1 + F 1 = 100(9) Work force W 2 - W 1 - H 2 + F 2 = 0(10) constraints W 3 - W 2 - H 3 + F 3 = 0(11) W 4 - W 3 - H 4 + F 4 = 0(12) where H t = # hired for period t F t = # fired for period t I t = inventory at end of period t W t = workforce at period t P t = # units produced at period t

89. 89 Optimal Solutions to Aggregate Planning Problems Via Linear Programming D t – the forecasts of demand for aggregate units for period t, t = 1 … T n t – number of units that can be made by one worker in period t n t – number of units that can be made by one worker in period t C t P – cost to produce one unit in period t C t W – cost of one worker in period t C t H – cost to hire one worker in period t C t L – cost to layoff one worker in period t C t I – cost to hold one unit in inventory in period t C t B – cost to backorder one unit in period t W t – number of workers available in period t P t – number of units produced in period t I t – number of units held in the inventory at the end of period t H t – number of workers hired in period t F t – number of workers fired in period t known info decision variables

90. 90 Optimal Solutions to Aggregate Planning Problems Via Linear Programming LP: s.t constraints All variables are continuously divisible – is it a problem? All variables are continuously divisible – is it a problem? Solution: Produce 214.5 of aggregated units Hire 56.38 workers IP: IP: s.t constraints* Some variables are continuously divisible, some are real number only – problem? Some variables are continuously divisible, some are real number only – problem?

91. 91 Linear Programming: Objective Function and Constraints s.t. production constraint labour constraint inventory constraint production salaryhiring layoffs inventorybacklogs Number of constraints is 3T, number of unknown is 5T W 0, I 0, B 0 – initial workforce, initial inventory/backlog

92. 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 D it – the maximum demand for product i for period t d it – the minimum sales allows of product i for period t a ij – time required on workstation j to produce one unit of product i c jt – capacity of workstation j in period t in the same units as a ij r i – net profit from one unit of product i h i – cost to hold one unit of product i for one period in the inventory X it – amount of product i produced in period t S it – amount of product i sold in period t I it – number of units of product i held in the inventory at the end of period t

93. 93 Linear Programming: Product Mixed Planning Objective Function and Constraints s.t. sales constraint capacity constraint inventory constraint profit inventory This model can be used to obtain information on demand feasibilitybottleneck locationproduct mix

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

95. 95 Homework Assignment Read “Production & Operations Analysis” by S.Nahmias chapter 3, sections 1 – 4 Read “Production & Operations Analysis” by S.Nahmias chapter 3, sections 1 – 4 Problems: Problems: 3.5 3.5 3.9 – 3.11 3.9 – 3.11 3.14 – 3.16 3.14 – 3.16

96. The END