example 13-3:
LO2 5. Refer to Example 13-3. Suppose that the new union con- tract limits the number of temporary workers in any month to 28 (i.e., 20 percent of permanent workers). Recall from Example 13-2 that up to 400 units can be produced dur- ing overtime per month. Using trade-off analysis and trial-and-error, find the minimum cost plan in this case. (Hint: Hire 10 temps for three months and 15 temps for two months starting in month 3.) EXAMPLE 13-1 The planner for a company that makes garden tractors is about to prepare an aggregate produc- tion plan that will cover the next six months. She has collected the following information: Month 1 2 3 4 5 6 Total Forecast demand 2.000 2,000 3,000 4,000 5,000 2,000 18,000 Permanent workforce 140 Production per month = 2,800 units or 20 per worker Initial inventory = 1,000 units Desired ending inventory at the end of sixth month = 1,000 units Costs Labour Regular time permanent = $100 per tractor Overtime = $150 per tractor Temporary = $100 per tractor Hire cost = $500 per temporary worker or $25 (= $500/20 units) per unit (charged to the first month of employment); assume that temporary workers have the same productivity as permanent workers. Inventory = $10 per tractor per month (charged on the average inventory level) Back order = $150 per tractor per month The planner now wants to evaluate a production plan that calls for level output/workforce (with the current level of permanent workforce, 140), using inventory to absorb the uneven forecast demand but allowing some back order. Chapter 13 Aggregate Operations Planning and Master Scheduling 505 SOLUTION The total regular-time output of permanent workers per month is 2,800 units. The filled worksheet for this level output/workforce strategy is shown below. The calculations are explained after the worksheet. Month 1 2 3 4 5 6 Total 2.000 2.000 3,000 4.000 5,000 2.000 18.000 Forecast demand Output 2,800 2,800 2,800 2.800 2.800 2.800 16,800 800 800 200 1,200 2,200 800 -1,200 Regular permanent Temporary Overtime Output - Forecast Inventory Beginning Ending Average Back order 1,800 2,600 2,400 0 1,000 1,800 1,400 1,200 o 2,400 1,200 0 2,600 2,200 2,500 1,800 600 0 8,500 0 0 0 0 1,000 200 1,200 2.800 2,800 2.800 2.800 2.800 2.800 2.800 16.800 800 800 -200 -1.200 -2,200 800 -1,200 1,800 2,600 2.400 1,200 0 1,000 1,800 1,400 2.400 0 0 2,600 2,200 1,200 1,800 2,500 600 8,500 200 0 0 0 0 1.000 200 1.200 Output Regular permanent Temporary Overtime Output - Forecast Inventory Beginning Ending Average Back order Costs Labour Regular perm (at $100/ unit) Temporary (at $100/unit) Overtime (at $150/unit) Hire temporary (at $25/ unit) Inventory (at $10/unit/ month) Back order (at $150/unit/ month) Total cost $280,000 $280,000 $280,000 $280,000 $280,000 $280,000 $1,680,000 14,000 22.000 25,000 18.000 6,000 0 85.000 0 0 0 0 150,000 30,000 180,000 $294.000 $302.000 $305,000 $298.000 $436.000 $310,000 $1,945.000 Starting with month 1, (Output - Forecast), = 2,800 2,000 = 800, X = Beginning inventory + (Output - Forecast), - Back ordero = 1.000 + 800 -0 = 1,800 Therefore, Ending inventory= 1,800 and Back order, = 0. Also, Average inventory1 = (Beginning inventory + Ending inventory1)/2 = (1,000 + 1,800)/2 = 1,400 Then, Beginning inventory2 = Ending inventory = 1,800. Continue with the same formulas, but note that in month 5, X = Beginning inventorys + (Output - Forecast)- Back order = 1.200 + (-2,200) - 0= -1,000 Therefore, Ending inventorys = 0 and Back orders =-(-1,000) = 1,000. Also, in month 6, X = Beginning inventory. + (Output - Forecast). - Back order's = 0 + 800 - 1,000 = -200 Therefore, Ending inventory6 = 0 and Back order = 200. The costs were calculated as follows. Regular permanent cost in each month equals 2,800 units x $100 per unit, or $280,000. Inventory holding cost = Average inventory x $10 per unit. Back-order cost is $150 per unit times the number of back orders. The total cost for this plan (either sum the row totals of costs or sum the column totals of costs) is $1,945,000. EXAMPLE 13-2 > The planner has decided to investigate the use of overtime to make up for the shortage in Example 13-1. It is the policy of the company that the maximum amount of overtime output per month be 400 units. Develop an aggregate production plan in this case and compare it to Plan 1. SOLUTION The 1,200 units to be produced during overtime must be scheduled on and before month 5 (when shortage or back order starts to occur). Scheduling it earlier would increase inventory holding costs ($10/unit/month); scheduling it later would incur back-order cost ($150/unit/month), which is larger. That is why the maximum permitted overtime production should be used in months 3-5. The completed worksheet is given below. Month 1 2 3 4 5 6 Total Forecast 2.000 2.000 3,000 4,000 5,000 2.000 18,000 2,800 2,800 2,800 2,800 2,800 2,800 16,800 400 400 400 1,200 800 800 200 -800 -1,800 800 Output Regular permanent Temporary Overtime Output - Forecast Inventory Beginning Ending Average Back order 1.000 1,800 2,600 2,800 2,000 200 1,800 2,600 2,800 2.000 200 1,000 1,400 2,200 2,700 2,400 1,100 600 10,400 0 0 0 0 0 0 0 Costs Labour $280,000 $280,000 $280,000 $280,000 Regular perm (at $100/ unit) $280,000 $280,000 $1,680,000 0 0 O 0 0 0 O 60,000 60,000 60,000 180,000 Temporary (at $100/unit) Overtime (at $150/unit) Hire temporary (at $25/ unit) Inventory (at $10/unit month) Back order (at $150/unit/ month) Total cost 14.000 22.000 27,000 24,000 11,000 6,000 104,000 0 0 0 0 0 0 O $294,000 $302,000 $367,000 $364,000 $351,000 $286,000 $1,964,000 Note that, even though Plan 2 has a larger total cost than Plan 1, it produces 1,200 more units and leaves 1,000 units at the end of month 6. ( EXAMPLE 13-3 The third option is to use temporary workers during months of high demand. Suppose that tem- porary workers will be working during a second shift and enough of them are available. Develop an aggregate production plan in this case (Plan 3). SOLUTION Dividing the number of short units (1,200) by the output rate of 20 per temporary worker, you find that 60 temporary worker- months are needed (e.g., 60 temporary workers for one month each, or 30 temporary workers for two months each, or 20 tempo- rary workers for three months each, etc.). Therefore, the planner tried each of the alternatives: hiring 60 temporary workers in month 5 on a one-month contract, or hiring 30 temporary workers starting in month 4 on a two-month contract, or hiring 20 temporary workers starting in month 3 on a three- month contract, and so on. The plan with the lowest total cost, including inventory holding cost, hires temporary workers starting in month 4 on a two-month contract. The completed worksheet for this plan is given below. Note that the hiring cost applies to only the first month of employment Month 1 3 4 5 6 2 2,000 Total 18,000 2,000 3,000 4,000 5,000 2,000 2,800 2,800 2,800 2,800 2,800 2,800 600 16.800 1,200 600 800 800 -200 -600 -1.600 800 0 1,000 2,600 2.400 1,800 200 1,800 2,600 1,800 200 Forecast Output Regular permanent Temporary Overtime Output forecast Inventory Beginning Ending Average Back order Costs Labour Regular perm (at $100/unit) Temporary (at $100/unit) Overtime (at $150/unit) Hire temporary (at $25/unit) Inventory (at $10/unit/month) Back order (at $150/unit/month) Total cost 1,000 2,400 2,500 0 1,400 0 1,800 2,100 0 2,200 0 9,800 1,000 600 0 o 0 $280,000 $280,000 $1,680,000 $280,000 $280,000 60,000 $280,000 $280,000 60,000 120.000 0 15,000 14,000 98,000 0 o 15,000 0 0 22,000 25,000 21,000 10,000 6,000 0 0 0 0 0 $302,000 $305,000 $376,000 $350,000 $286,000 0 0 $294,000 $1.913,000 Overall, the total cost for this plan ($1,913,000) seems to be the lowest possible, so Plan 3 seems to be optimal. LO2 5. Refer to Example 13-3. Suppose that the new union con- tract limits the number of temporary workers in any month to 28 (i.e., 20 percent of permanent workers). Recall from Example 13-2 that up to 400 units can be produced dur- ing overtime per month. Using trade-off analysis and trial-and-error, find the minimum cost plan in this case. (Hint: Hire 10 temps for three months and 15 temps for two months starting in month 3.) EXAMPLE 13-1 The planner for a company that makes garden tractors is about to prepare an aggregate produc- tion plan that will cover the next six months. She has collected the following information: Month 1 2 3 4 5 6 Total Forecast demand 2.000 2,000 3,000 4,000 5,000 2,000 18,000 Permanent workforce 140 Production per month = 2,800 units or 20 per worker Initial inventory = 1,000 units Desired ending inventory at the end of sixth month = 1,000 units Costs Labour Regular time permanent = $100 per tractor Overtime = $150 per tractor Temporary = $100 per tractor Hire cost = $500 per temporary worker or $25 (= $500/20 units) per unit (charged to the first month of employment); assume that temporary workers have the same productivity as permanent workers. Inventory = $10 per tractor per month (charged on the average inventory level) Back order = $150 per tractor per month The planner now wants to evaluate a production plan that calls for level output/workforce (with the current level of permanent workforce, 140), using inventory to absorb the uneven forecast demand but allowing some back order. Chapter 13 Aggregate Operations Planning and Master Scheduling 505 SOLUTION The total regular-time output of permanent workers per month is 2,800 units. The filled worksheet for this level output/workforce strategy is shown below. The calculations are explained after the worksheet. Month 1 2 3 4 5 6 Total 2.000 2.000 3,000 4.000 5,000 2.000 18.000 Forecast demand Output 2,800 2,800 2,800 2.800 2.800 2.800 16,800 800 800 200 1,200 2,200 800 -1,200 Regular permanent Temporary Overtime Output - Forecast Inventory Beginning Ending Average Back order 1,800 2,600 2,400 0 1,000 1,800 1,400 1,200 o 2,400 1,200 0 2,600 2,200 2,500 1,800 600 0 8,500 0 0 0 0 1,000 200 1,200 2.800 2,800 2.800 2.800 2.800 2.800 2.800 16.800 800 800 -200 -1.200 -2,200 800 -1,200 1,800 2,600 2.400 1,200 0 1,000 1,800 1,400 2.400 0 0 2,600 2,200 1,200 1,800 2,500 600 8,500 200 0 0 0 0 1.000 200 1.200 Output Regular permanent Temporary Overtime Output - Forecast Inventory Beginning Ending Average Back order Costs Labour Regular perm (at $100/ unit) Temporary (at $100/unit) Overtime (at $150/unit) Hire temporary (at $25/ unit) Inventory (at $10/unit/ month) Back order (at $150/unit/ month) Total cost $280,000 $280,000 $280,000 $280,000 $280,000 $280,000 $1,680,000 14,000 22.000 25,000 18.000 6,000 0 85.000 0 0 0 0 150,000 30,000 180,000 $294.000 $302.000 $305,000 $298.000 $436.000 $310,000 $1,945.000 Starting with month 1, (Output - Forecast), = 2,800 2,000 = 800, X = Beginning inventory + (Output - Forecast), - Back ordero = 1.000 + 800 -0 = 1,800 Therefore, Ending inventory= 1,800 and Back order, = 0. Also, Average inventory1 = (Beginning inventory + Ending inventory1)/2 = (1,000 + 1,800)/2 = 1,400 Then, Beginning inventory2 = Ending inventory = 1,800. Continue with the same formulas, but note that in month 5, X = Beginning inventorys + (Output - Forecast)- Back order = 1.200 + (-2,200) - 0= -1,000 Therefore, Ending inventorys = 0 and Back orders =-(-1,000) = 1,000. Also, in month 6, X = Beginning inventory. + (Output - Forecast). - Back order's = 0 + 800 - 1,000 = -200 Therefore, Ending inventory6 = 0 and Back order = 200. The costs were calculated as follows. Regular permanent cost in each month equals 2,800 units x $100 per unit, or $280,000. Inventory holding cost = Average inventory x $10 per unit. Back-order cost is $150 per unit times the number of back orders. The total cost for this plan (either sum the row totals of costs or sum the column totals of costs) is $1,945,000. EXAMPLE 13-2 > The planner has decided to investigate the use of overtime to make up for the shortage in Example 13-1. It is the policy of the company that the maximum amount of overtime output per month be 400 units. Develop an aggregate production plan in this case and compare it to Plan 1. SOLUTION The 1,200 units to be produced during overtime must be scheduled on and before month 5 (when shortage or back order starts to occur). Scheduling it earlier would increase inventory holding costs ($10/unit/month); scheduling it later would incur back-order cost ($150/unit/month), which is larger. That is why the maximum permitted overtime production should be used in months 3-5. The completed worksheet is given below. Month 1 2 3 4 5 6 Total Forecast 2.000 2.000 3,000 4,000 5,000 2.000 18,000 2,800 2,800 2,800 2,800 2,800 2,800 16,800 400 400 400 1,200 800 800 200 -800 -1,800 800 Output Regular permanent Temporary Overtime Output - Forecast Inventory Beginning Ending Average Back order 1.000 1,800 2,600 2,800 2,000 200 1,800 2,600 2,800 2.000 200 1,000 1,400 2,200 2,700 2,400 1,100 600 10,400 0 0 0 0 0 0 0 Costs Labour $280,000 $280,000 $280,000 $280,000 Regular perm (at $100/ unit) $280,000 $280,000 $1,680,000 0 0 O 0 0 0 O 60,000 60,000 60,000 180,000 Temporary (at $100/unit) Overtime (at $150/unit) Hire temporary (at $25/ unit) Inventory (at $10/unit month) Back order (at $150/unit/ month) Total cost 14.000 22.000 27,000 24,000 11,000 6,000 104,000 0 0 0 0 0 0 O $294,000 $302,000 $367,000 $364,000 $351,000 $286,000 $1,964,000 Note that, even though Plan 2 has a larger total cost than Plan 1, it produces 1,200 more units and leaves 1,000 units at the end of month 6. ( EXAMPLE 13-3 The third option is to use temporary workers during months of high demand. Suppose that tem- porary workers will be working during a second shift and enough of them are available. Develop an aggregate production plan in this case (Plan 3). SOLUTION Dividing the number of short units (1,200) by the output rate of 20 per temporary worker, you find that 60 temporary worker- months are needed (e.g., 60 temporary workers for one month each, or 30 temporary workers for two months each, or 20 tempo- rary workers for three months each, etc.). Therefore, the planner tried each of the alternatives: hiring 60 temporary workers in month 5 on a one-month contract, or hiring 30 temporary workers starting in month 4 on a two-month contract, or hiring 20 temporary workers starting in month 3 on a three- month contract, and so on. The plan with the lowest total cost, including inventory holding cost, hires temporary workers starting in month 4 on a two-month contract. The completed worksheet for this plan is given below. Note that the hiring cost applies to only the first month of employment Month 1 3 4 5 6 2 2,000 Total 18,000 2,000 3,000 4,000 5,000 2,000 2,800 2,800 2,800 2,800 2,800 2,800 600 16.800 1,200 600 800 800 -200 -600 -1.600 800 0 1,000 2,600 2.400 1,800 200 1,800 2,600 1,800 200 Forecast Output Regular permanent Temporary Overtime Output forecast Inventory Beginning Ending Average Back order Costs Labour Regular perm (at $100/unit) Temporary (at $100/unit) Overtime (at $150/unit) Hire temporary (at $25/unit) Inventory (at $10/unit/month) Back order (at $150/unit/month) Total cost 1,000 2,400 2,500 0 1,400 0 1,800 2,100 0 2,200 0 9,800 1,000 600 0 o 0 $280,000 $280,000 $1,680,000 $280,000 $280,000 60,000 $280,000 $280,000 60,000 120.000 0 15,000 14,000 98,000 0 o 15,000 0 0 22,000 25,000 21,000 10,000 6,000 0 0 0 0 0 $302,000 $305,000 $376,000 $350,000 $286,000 0 0 $294,000 $1.913,000 Overall, the total cost for this plan ($1,913,000) seems to be the lowest possible, so Plan 3 seems to be optimal
