INSET 3 Costs in Aggregate Planning
4.4.3 Production Plans
A production plan specifies the number of units to be produced per day. For this example, the number remains constant within a given month but may change from one month to the next month. Table 4.6 gives the expected demand per day for each month in the planning horizon. The calculation of expected demand per day was explained in Inset 2. The expected demand per day fluctuates between a mini- mum of 16.67 in October to a maximum of 29.55 in August. The average demand per day for the 12-month period is 23.33 (see Inset 2).
We will now develop different plans that specify the number of units produced in each month. We are going to develop the following four production plans: (1) Level, (2) Chase, (3) Hybrid, and (4) a plan that uses overtime. Three of these were also discussed in Example 1.
We assume that the plant is currently set up to produce 21 units per day.
4.4.3.1 Level Plan
In the Level plan, the production per day is constant at the average demand level of the 12-month period which is 23.33 units per day. Table 4.8 gives the calculations for this plan. The production in any given month is calculated by multiplying the number of days in that month by the production rate per day (23.33). The produc- tion numbers are rounded to the nearest integer. For example, in January, 513 is used for the number of units produced whereas 23.33 × 22 = 513.26.
However, the demand in January is only 400 units. Therefore, 113 (513 − 400) units go into inventory. In February, 467 (23.33 × 20) units are produced, whereas the expected demand is for 440 units. Therefore, 27 units (467 − 440) are added to inventory and the inventory level at the end of February is 140. In March, the production is 513 (23.33 × 22) which is 87 units less than the demand (600 units).
Therefore, 87 units are withdrawn from inventory to meet the demand for March.
The inventory level at the end of March becomes 53. In this way, the calculations continue.
In some months, the production is less than the demand and there are not enough units in inventory to meet the demand. In such situations, shortages occur.
For example, in July we produce 117 units less than the demand (443 − 560). Only 37 units are available in inventory at the end of June. These 37 units are used to off- set the shortage of July. However, there still will be a shortage of 80 (117 − 37) units.
In August, the shortage increases to 217, then decreases to 154 units in September, and then decreases to 14 units in October. At the end of November, there are 73 units in inventory. Finally, we see zero inventories at the end of December. Total
Table 4.8Level Plan: Production per Day= 23.33 units for a Manufacturer of Storable Product MonthJan.Feb.Mar.Apr.MayJun.Jul.Aug.Sep.Oct.Nov.Dec.Total Expected demand (units)4004406003964806105606504503503805405856 Working days222022212022192222212020251 Production per day (units)23.3323.3323.3323.3323.3323.3323.3323.3323.3323.3323.3323.33 Total production (units)5134675134904675134435135134904674675856 Inventory addition/shortages (units)11327−8794−13−97−117−1376314087−73 Ending inventory/shortages (units)1131405314713437−80−217−154−14730 Ending inventory (units)11314053147134370000730697 Ending shortages (units)000000802171541400465 Inventory cost ($)2825350013253675335092500001825017,425 Shortage cost ($)000000800021,70015,40014000046,500 Change in production level2.3300000000000 Change in production level—up2.33000000000002.33 Change in production level—down0000000000000 Cost of changing production level up ($)46600000000000466 Cost of changing production level down ($)0000000000000 The number of workers required in the Level plan is constant in each month and is calculated as follows: number of workers= (total demand in 12 months× production time per unit)/(number of days in 12 months× hours worked per day) = (5856× 4)/(251× 8)= 11.67. Alternatively, number of workers= (production in units per day× production time per unit)/hours worked per day= (23.33× 4)/8= 11.67.
production is equal to demand. Nevertheless, with the Level plan, both overstock and out-of-stock situations occur in spite of ending the year with zero inventories.
The inventory and shortage costs in each month are calculated by multiplying the end of month inventory or shortage by the respective costs. Alternatively, we can total the end-of-month inventories (sum of all end-of-month inventories = 697) and also total the end-of-month shortages (sum of all end-of-month shortages = 465).
Multiply these numbers by their respective costs. Therefore, the total inventory cost is $17,425 (697 × 25) and the total shortage cost is $46,200 (462 × 100). The total production during regular time is 5856. Therefore, the regular time labor cost is
$468,480 (5856 × 4 × 20). The total material cost is $585,600 (5856 × 100). The production level changes from 21 to 23.33 in January. After that the production level remains constant at 23.33. Therefore, the cost of changing production level is $466 (2.33 × 200). The total annual cost of this plan is $1,118,171. Note: rounding errors can make small changes in the various totals alluded to in this paragraph.
4.4.3.2 Chase Plan
An alternative to the Level plan is the Chase plan in which the production level per day in each month is equal to the demand per day in that month. The production per day changes from 1 month to the next. This is called “Chase plan” because pro- duction chases (matches) demand. The details of this plan are given in Table 4.9. In this plan, there are no inventories or shortages. However, the production level has to be adjusted in each month. In January, the production level is 18.18 units per day. Therefore, the production level is reduced by 2.82 (21 − 18.18) units, assum- ing that the plant is currently set up to produce 21 units per day. In February, the production level increases to 22 and the level has to be adjusted upward by 3.82 (level in February − level in January = 22 − 18.18) units.
As shown in Table 4.9, the production levels have to be adjusted upwards in March (5.27), May (5.14), June (3.73), July (1.75), August (0.07), November (2.33), and December (8.00). The numbers in parentheses show the amount of upward adjustments. In January, April, September, and October, the production levels have to be adjusted down by 2.82, 8.42, 9.09 and 3.79, respectively. The total upward adjustment is 30.11 with a cost of $6023 (30.11 × 200). The total down- ward adjustment is 24.11 with a cost of $7234 (24.11 × 300). The total production during regular time is 5856. Therefore, the regular time labor cost is $468,480 (=
5856 × 4 × 20) and the total material cost is $585,600 (= 5856 × 100). The total annual cost of this plan is $1,067,336. Note: rounding errors can make small changes in the various totals alluded to in this paragraph.
4.4.3.3 Mixed or Hybrid Plan
The Level and Chase plans are at two extremes. The Level plan keeps the workforce intact. There are no firings and employees feel secure. With chasing, it is important
Table 4.9Chase Plan: Production per Day is Variable Matching Expected Demand for a Manufacturer of Storable Product MonthJan.Feb.Mar.Apr.MayJun.Jul.Aug.Sep.Oct.Nov.Dec.Total Expected demand (units)4004406003964806105606504503503805405856 Working days222022212022192222212020251 Production per day (units)18.1822.0027.2718.8624.0027.7329.4729.5520.4516.6719.0027.00 Total production (units)4004406003964806105606504503503805405856 Inventory addition/ shortages (units)There are no inventories or shortages in this plan Change in production level (units)−2.823.825.27−8.425.143.731.750.07−9.09−3.792.338.00 Change in production level—up (units)0.003.825.270.005.143.731.750.070.000.002.338.0030.11 Change in production level—down (units)2.820.000.008.420.000.000.000.009.093.790.000.0024.11 Number of workers9.0911.0013.649.4312.0013.8614.7414.7710.238.339.5013.50 Chase plan: Production rate= demand rate in each month. The number of workers is calculated using the formula given in Inset 4. For example, in May the required number of workers is (24× 4)/8= 12.
to explain to the workforce how its interests are represented by this strategy which hires and fires according to the customer demand system. These considerations lead to the idea that we could have a “Mixed or Hybrid” plan in which the production rate is kept constant at a given level for a few months and then changed to another level. This adds some stability for the benefit of the employees while producing sav- ings on seasonal shifts in demand.
The production level can be changed several times during the planning horizon.
In the Hybrid plan proposed in this section, the production level is constant at 23.04 units per day for the first 6 months which is the average demand for the first 6 months (January–June); then it changes to 26.35 which is the average demand for the next 3 months (July, August, and September) and finally the production level changes to 20.82 which is the average demand for the last 3 months. This plan entails inventories, shortages, and changing production levels up and down. The calculations are done in the same way as for the Level and Chase plans. This plan is shown in Table 4.10. The total annual cost of this plan is $1,093,359 which includes the regular time labor cost ($468,480), material cost ($585,600), inventory carry- ing cost ($17,650), shortage cost ($18,900), increasing production level ($1070), and decreasing production level ($1659). Note: rounding errors can make small changes in the various totals alluded to in this paragraph.
4.4.3.4 Overtime in a Chase Plan with Level Production and Overtime
In this plan, we show the usage of overtime for aggregate production planning. We use overtime in a combination of the Level plan and the Chase plan. In the Level plan, the production is kept constant at a specified level. In Example 2, the Level plan used 23.33 as the production rate per day. It was the average production rate per day over the planning horizon. However, when overtime is used, we can set the production level at a lower number. For this problem, we have set the level produc- tion at 16.67 units per day. It could have been any other reasonable number. The number 16.67 is equal to the minimum of the production per day in 12 months.
The minimum production per day, 16.67, is for October (see Table 4.6).
Regular time production in a given month is obtained by multiplying 16.67 by the number of days in that month. The production numbers are rounded to the nearest integer. The difference between the demand and production in any month gives the number of units that have to be produced during over time. For example, the production in January is 367 (16.67 × 22) units. Therefore, overtime produc- tion will be 33 (400 − 367). The calculations for this plan are given in Table 4.11.
In this plan, 4184 units are produced during regular time and 1672 units are produced during overtime for a total of 5856 units. Therefore, the regular time production cost is $334,720 (4184 × 20 × 4) and the overtime cost is $200,640 (1672 × 30 × 4). The total material cost is $585,600 which is the same as in the other plans. The production level per day had to be decreased by 4.33 which is the
Table 4.10 Mixed (Hybrid) Plan MonthJan.Feb.Mar.Apr.MayJun.Jul.Aug.Sep.Oct.Nov.Dec.Total Expected demand (units)4004406003964806105606504503503805405856 Working days222022212022192222212020251 Production per dayAverage Demand per Day (Jan. to Jun.)Average Demand per Day (Jul. to Sep.)Average Demand per Day (Oct. to Dec.) 23.0423.0423.0423.0423.0423.0426.3526.3526.3520.8220.8220.82 Total production (units)5074615074844615075015805804374164165856 Inventory addition/ shortages (units)10721−9388−19−103−59−701308736−124 Ending inventory/ shortages (units)107128351221030−59−1300871240 Ending inventory (units)107128351221030000871240706 Ending shortages (units)000000591300000189 Inventory cost ($)2675320087530502575000021753100017,650 Shortage cost ($)000000590013,000000018,900 Change in production level (units)2.0400000300− 600 continued
Table 4.10 (continued) Mixed (Hybrid) Plan MonthJan.Feb.Mar.Apr.MayJun.Jul.Aug.Sep.Oct.Nov.Dec.Total Change in production level—up (units)2.040.000.000.000.000.003.310.000.000.000.000.005.35 Change in production level—down (units)0.000.000.000.000.000.000.000.000.005.530.000.005.53 Cost of changing production level—up ($)407.870.000.000.000.000.00661.970.000.000.000.000.001070 Cost of changing production level—down ($)0.000.000.000.000.000.000.000.000.001658.860.000.001659 Number of workers11.5211.5211.5211.5211.5211.5213.1713.1713.1710.4110.4110.41
Table 4.11 Overtime in a Chase Plan with Level Production MonthJan.Feb.Mar.Apr.MayJun.Jul.Aug.Sep.Oct.Nov.Dec.Total Expected demand (units)4004406003964806105606504503503805405856 Working days222022212022192222212020251 Production per day (units)16.6716.6716.6716.6716.6716.6716.6716.6716.6716.6716.6716.67 Regular time production (units)3673333673503333673173673673503333334184 Overtime production (units)3310723346147243243283830472071672 Total production (units)4004406003964806105606504503503805405856 Ending inventory/shortages (units)000000000000 Ending inventory (units)0000000000000 Ending shortages (Units)0000000000000 Inventory cost ($)0000000000000 Shortage cost ($)0000000000000 Change in production level (units)− 4.3300000000000 Change in production level—up (units)0000000000000 Change in production level—down (units)4.33000000000004.33 Cost of changing production level up ($)0000000000000 Cost of changing production level down ($)1299000000000001299 Number of workers8.348.348.348.348.348.348.348.348.348.348.348.34
difference between the current production level of 21 and the production level in January of 16.67. The cost of changing the production level is $1299.
The total cost of this plan is $1,122,259. Note: rounding errors can make small changes in the various totals alluded to in this paragraph.