AGGREGATE PLANNING PROBLEMS

Transcription

1 AGGREGATE PLANNING PROBLEMS Problem 1. ABC Corporation has developed a forecast for a group of items that has the following seasonal demand pattern. Quarter Demand Cumulative Demand Plot the demand as a histogram. Determine the production rate required to meet average demand, and plot the average demand forecast on the graph. 2. Plot the actual cumulative forecast requirements over time and compare them with the available average forecast requirement. Indicate the excess inventories and backorders of the graph. 3. Suppose that the firm estimates that it costs Rs.1 per unit to increase the production rate, Rs.15 to decrease the production rate, Rs.5 per quarter to carry the items on inventory, and an incremental cost of Rs.8 per unit if subcontracted and Rs. 2 per unit if overtime is used. Compare the cost incurred if pure strategies are used. 4. Given these costs, design a mixed strategy solution for this problem. The histogram and the cumulative requirements graph show how the forecast deviates from the average requirements (see Figures 1 and 2). Using pure strategies, it is possible to come up with several plans as follows: rate units/quarter 3 Forecast requirements Average Forecast requirements Figure 1 Histogram of forecast and average requirements Planning Problems 25

2 Cumulative demand units Backorders Actual forecast requirements Average forecast requirements Plan 1: Varying the Workforce Size. Demand can be met exactly by varying the workforce size. The plan involves hiring and firing as necessary. The production rate will equal the demand. The cost of this plan is Rs.138,, as computed in Table 1. Notice that inventory and backorders are both zero in each quarter, and the resulting costs for those are zero. Plan 2: Changing Levels. Suppose that a firm wants to avoid frequent hiring and layoffs. It might choose a production level equal to its average demand and meet the variations in demand by holding inventory. The cost of such a plan is computed in Table 2. The plan incurs a maximum shortage of 27 units during period 5. Since a certain amount of uncertainty is involved in any forecast, the firm might decide to carry the inventory from the beginning of period 1. Adjusted inventories and cost of carrying inventories are shown. The total cost of the plan is Rs.96,5. Notice, however, that if the item in question is high-fashion apparel the firm might not want to carry unnecessary inventory, even though Plan 2 is less costly than Plan 1. Quarter Table 1: Varying the Workforce size to meet the demand Demand forecast Periods (Quarters) Figure 2 Cumulative and average forecast graph Cost of Increasing Level: Hiring (Rs.) Cost of Decreasing Level: Layoff (Rs.) Total cost of Plan (Rs.) ,5 7, , - 23, , - 2, , 33, , 27, ,5 1, , - 17, Total 138, Planning Problems 26

3 Table 2 Changing Levels to Meet the Demand Quarter Demand Forecast Cumulative Demand Level Cumulative Adjusted with 27 at Beginning of Period 1 Cost of Holding Inventories (Rs. 1s) Plan 3: ing. A firm might prefer to produce an amount equal to its lowest requirements and meet the rest of the demand by subcontracting. The cost of such a plan amounts to Rs.18,, as computed in Table 3. Again, it may not always be feasible or desirable to subcontract. This decision leads us to a fourth plan, involving a mixed strategy. Plan 4: Mixed Strategy. As a compromise, a firm might combine the pure strategies, thus designing a mixed strategy. This mixed strategy varies production capacity slightly up or down as aggregated demand varies. Drastic changes in production capacity are curtailed, and frequent hiring and lay off situations are avoided. For example, based on past experience and available personnel, management may decide to maintain a constant production rate of 2 per quarter and permit 25% overtime when the demand exceeds the production rate. To meet any further demand, the firm chooses to hire and layoff workers. Remember, in a mixed strategy, a host of other alternatives exist. Trial-anderror computations of such a plan can be carried out step by step, as shown in Table 4. Table 3 ing Costs Quarter Demand Forecast Units Units Incremental Cost at Rs.8 per Unit (Rs.) , , , , , , ,6 18,8 Planning Problems 27

4 Quarter Units of Demand Forecast Regular Time Units Additional Units Needed After Regular Time Table 4 Mixed strategy Additional units Needed After Regular Time + Cost of (Rs.) Cost of (Rs.) Cost of Changing Workforce (Rs.) Total Cost (Rs.) (-3) c , a (-6) 3 3, b (9) 1 9, 1, (35) 1 26, 27, (13) 1 33, 34, ,5 19, (-7) 35 3, (-2) 1 1 2, Rs.11,5 a Note that the inventory in period 2 is sixty units b If the existing inventory of sixty units is used, an increase of only ninety units is required. c Negative quantities in parenthesis indicate inventories, and positive quantities in parentheses denote the quantities to be produced by changing the capacity. Note:- All the above cost calculated are incremental cost. They are the cost incurred over the regular time production cost. That is, the total cost of a plan is the plan cost shown above plus the cost incurred if all the requirements are produced in regular time. Planning Problems 28

5 Problem 2. You are supplied with a monthly demand forecast, an organizational policy of requiring 1% of a month s forecast as safety stock, and the number of operating days available each month. There is no inventory available at the beginning of the first month, January. The following table contains the demand requirements. January February March April May June 1. Beginning inventory 1, 1,5 3, 2,7 3, 2. Forecasted demand 1, 15, 3, 27, 3, 16, 3. Safety Stock 1, 1,5 3, 2,7 3, 1,6 4. requirements (2+3-1) 11, 15,5 31,5 26,7 3,3 14,6 5. Operating days Cumulative forecasted 1, 25, 55, 82, 112, 128, demand 7. Cumulative production requirements 8. Cumulative operating days The costs for the organization are as follows: 11, 26,5 58, 84, 115, 129, Manufacturing cost/unit holding cost a Hourly wage rate Stockout cost per unit Hourly overtime wage rate ing cost/unit Labour hours/unit Layoff cost/worker Hiring and training cost/worker Rs.1 Rs.2./unit-month Rs.8. Rs.5. 15%, or Rs.12. Rs.14 4 hours Rs.5 Rs.4 a 2% of manufacturing cost per month. Three potential plans for the production are: 1. Produce to exact production requirements by varying the size of the work force on regular hours. Assume there are 25 workers available in January. 2. Maintain a constant work force of 518 workers. Assume no subcontracting is available and inventory will fluctuate with stockouts filled from the following month's production. 3. Produce with a fixed work force of 5 on regular time and subcontract all excess demand over the period of production. will increase when production exceeds demand; no stockouts are permitted. Each of the three plans is tabulated in Table 5, and a comparison is made in Table 6. Plan 2 with the constant work force and production rate with variable inventory and stockouts result s the lowest incremental cost, Rs.12,176. There are obviously many other plans that could be evaluated. Planning Problems 29

6 Month Unit required Table 5 Three possible strategies Plan Available Workers No. of No. of Hiring cost hours hours per required workers workers [(col.5)xrs. required month per [(col.2)/ hired a laid off a 4] [(col.1)x4] worker [(no. (col.3)] Layoff cost [(col.6)xrs. 5] of days)x8] Jan. 11, 44, Feb. 15,5 62, Rs. 63,2 Mar. 31,5 126, Rs. 136,8 Apr. 26,7 16, Rs. 57, May 3,3 121, Rs. 21,2 Jun. 14,6 58, Rs. 162, Rs. 221,2 Rs. 219, Month Plan Unit required hours available [(no. of days ) x 8x518] Units produced [(col.2)/4] Ending b holding cost c Stockout cost d Jan. 11, 91,168 22,792 11,792 Rs. 23,584 Feb. 15,5 78,736 19,684 15,976 31,952 Mar. 31,5 87,24 21,756 6,232 12,464 Apr. 26,7 87,24 21,756 1,288 2,576 May 3,3 91,168 22,792 (-6,22) 31,1 Jun. 14,6 82,88 2,72 (-1) 5 Rs. 7,576 Rs. 31,6 Month Plan Unit required hours available [(no. of days ) x 8x5] Units produced [(Col.2)/4] Units ed e Ending ing cost [(Col.4xR s4)] holding cost d [(Col.5xRs 2)] Jan. 11, 88, 22, 11, Rs. 22, Feb. 15,5 76, 19, 14,5 29, Mar. 31,5 84, 21, 4, 8, Apr. 26,7 84, 21, 1,7 Rs. 6,8 May 3,3 88, 22, 8,3 Rs. 33,2 Jun. 14,6 8, 2, 5,4 1,8 Rs. 4, Rs. 69,8 a lf the difference (col. 4) t+1 - (col.4) t is positive, it represents the number of workers hired. If negative, it represents the number laid off. b Ending inventory is (beginning inventory) + (units produced) - (unit production required). C lnventory holding cost is the positive ending inventory balance times 2. d Stockout cost is the negative ending inventory balance times 5. - e Ending inventory is (beginning inventory) + (units produced) - (unit production required). When a negative number is obtained for ending inventory, it represents the units subcontracted, and ending inventory is recorded as zero. Planning Problems 3

7 Cost item Table 6 Comparison of the three plans Plan 1 Plan 2 Plan 3 (constant work (exact production; force; vary vary work force) inventory and stockouts) (constant work force; vary inventory and subcontract) Hiring cost Rs. 221,2 Layoff cost 219, holding cost Rs. 7,576 Rs. 69,8 Stockout cost 31,6 cost Rs. 4, Rs. 44,2 Rs. 12,176 Rs. 19,8 Problem 3. An organization uses overtime, inventory, and subcontracting to absorb fluctuations in demand. A production plan for 12 months is devised and updated each month. The expected demand for the next 12 periods is as follows: Time period Unit demand (1 3 ) The following costs and constraints are relevant: Maximum regular production/period Maximum overtime production/period Regular production cost production cost ing cost holding cost/period 19, units 4, units Rs.3/unit Rs.35/unit Rs.37/unit Rs.l/unit What is the optimum production plan for the next 12 months (assume beginning inventory is zero, desired ending inventory is zero, and no stockouts can be tolerated)? From the costs and constraints it is apparent that it is desirable to produce to inventory on regular time for five holding periods before overtime production is economic. production should be used for two holding periods before subcontracting is desirable. to inventory on regular time should be used for seven holding periods before subcontracting is desirable. If units are not used in the period in which they are produced, holding costs are incurred. With these cost tradeoffs in mind, the optimum production plan will be devised in the following table: Table 7 Development of production plan Strategy variables a Period Demand Beginning Regular Ending inventory production production inventory b (1), 7(3), 2(4) (2), 4(3) (3) 2(4), 2(5) (4) 4(4) (5) 4(5) 5(5) Planning Problems 31

8 (6), 1(11) (7), 7(11) (8), 6(1), 3(11) (9), 1(1) (1) (11) (12) a The quantity to be produced is entered in the row associated with the production time period. The parentheses immediately after the production quantity indicate the time period in which it will be demanded. b The beginning and ending inventory columns are determined after all the strategy variables are assigned. Table 8 plan Period Strategy variable Regular production production Total Problem 4. An organization with a stable work force uses inventory, overtime, and subcontracting to meet demand requirements. No shortages are permitted, and demand must be satisfied through in-house production or subcontracting. The following data are available for the upcoming periods: Period Expected demand (units) Regular capacity (units) capacity (units) capacity (units) 1. 1, ,8 3,55 1,7 1,2 The beginning inventory at the start of period 1 is 2 units and the desired ending inventory for period 6 is 1 units. The relevant cost data are as follows: Regular cost/unit cost/unit cost/unit holding cost/period Rs.1 Rs.125 Rs.13 Rs.2/unit Determine the production plan that will satisfy demand at minimum cost. Use the transportation method of linear programming. The problem is attacked by starting with a linear programming tableau similar to the tableau shown in Fig.3. (The table shown in Fig. 3 is suitable for three-period problem) The last column in the table (total capacity available) is filled in, as is the Planning Problems 32

9 last row (demand). On an incremental cost basis, the regular cost per unit of Rs.1 is assigned a zero value; the overtime cost becomes Rs.25/unit, and the subcontract cost becomes Rs.3/unit. The appropriate incremental costs are added to each feasible cell in the table. The next step is to assign available capacity to meet the demand requirement at the least cost. When all the demand has been satisfied without violating the capacity constraints, the problem is solved. The optimum production plan is contained in Table 9. The following should be noted while preparing the linear programming tableau: 1. Total capacity exceeds demand, so a "slack" demand of unused capacity is added to achieve the required balance of supply and demand.. 2. The beginning inventory of 2 units is available at no additional cost if used in period 1. Holding cost is Rs.2/unit if units are retained until period 2, Rs.4/unit until period 3, and so on. 3. Regular cost per unit is assigned a zero incremental cost if used in the month produced; otherwise a holding cost of Rs.2/unit-period is added on for each month the units are retained. 4. cost per unit is assigned a Rs.25 incremental cost if used in the month produced; otherwise a holding cost of Rs.2/unit-period is incurred as in the regular situation. 5. cost per unit is assigned a Rs.3 incremental cost if used in the period produced. If unused in the period, a holding cost of Rs.2/unit-period is incurred. 6. The desired ending inventory (1 units) must be available at the end of period 6, and it has been added to the period 6 demand of 9 units to obtain the 1 units. 7. Since no stockouts are permitted, production in any month to satisfy a preceding month's demand is not a feasible alternative. Infeasible cells in the table are crosshatched. 8. Unused capacity is assigned a zero value in this problem. If there were an opportunity cost to unused capacity on regular time, it would normally be assessed. For example, unused capacity on regular time might result in idle labour. In this situation the labor cost contribution to the product would be the cost of unused capacity. Planning Problems 33

10 Figure 3 A sample table Supply from Beg. Demand for Period 1 Period 2 Period 3 o h 2h Unused Capacity Total Capacity Available (supply) Regular r r + h r + 2h I R Regular Regular v v + h v + 2h s s + h s + 2h r r + h v v + h s s + h r v s O 1 S 1 R 2 O 2 S 2 R 3 O 3 S 3 h Carrying cost per unit per period r Incremental cost for regular time production v Incremental cost for over time production s - Incremental cost for subcontracted quantity I e Ending inventory, I Initial inventory R i, O i, S i Regular time, over time, and subcontracting capacities respectively for period i D i Demand of period i Period Demand D 1 D 2 D 3 + l e C Grand Total Regular production (units) Table 9 Optimum production plan production (units) production (units) Total production cost a Rs. 85, , , , , ,5 3, Rs. 499,9 a Total production cost does not include holding cost, which is 4(7 * ) + 6(1 * ) + 2(5 * ) + 2(1 * ) = Rs.46. Thus total cost is Rs.499,9 + Rs.46 = Rs.5,36. *These quantities can be obtained from linear programming table. Planning Problems 34

11 Problem 5. The PQR Company produces several products and two of their products (A and B) that are similar in terms of labour content and skills required. Company management wishes to level the number of employees needed each day so no hiring or layoff will be required during the year for these products. A complication to this problem is that the number of working days in each quarter varies. Demand Working Days Quarter Product A Product B 1 9,8 14, , 3, , 19, , 25, 58 Beginning inventory: 2,4 units for product A 9 units for product B holding cost: Rs 3 per unit per quarter (either product) No back orders allowed No variations in size of workforce allowed Output rate = 25 units of either product per day per employee Develop a production plan that will meet the demand forecast and yield minimum inventory at the end of quarter 4. Devise an optimum production plan, if the company wants to employ a constant workforce of 2 workers per quarter and overtime. The overtime cost is Rs 2 above the normal wage per hour. The company policy is to allow a maximum of 4 hours overtime per worker per day. Assume an 8 hour work day. No shortages are allowed. If there is a very large cost for not using the regular time capacity what may be the best production plan? Solution Since the products A and B are similar in terms of labour content and skills required, the aggregate demand can be calculated for sales & operations planning. The first part of this problem requires the determination of the constant level of workforce. Demand Aggregate Demand Quarter Product A Product B Working Days 1 9,8 14, , 3, , 19, , 25, Total Average demand after considering initial inventory = = 38 4 Average number of working days = 244/4 = 61 days 38 Number of workers required to meet the average demand = = workers Planning Problems 35

12 plan and inventory status corresponds to the level strategy Beginning inventory at the planning horizon = 33 Quarters Demand Ending inventory = = = = Solution for the next part Capacity Calculation Number of employees = 2 Output rate = 25 units of either product per day per employee Over time allowed = 4 hours per day per employee 8 hours work day Quarters Regular time Over time capacity capacity Optimum plan Number of units produced per worker per hour = 25/8 = production cost per unit (Incremental) = 2/3.125 = Rs 6.4 holding cost: Rs 3 per unit per quarter (either product) If overtime capacity available it is economical to use over time of the current period than carrying the item from the previous period. Beginning inventory at the planning horizon = 33 Quarters Demand Regular time capacity Over time capacity Regular time production (period of use) Over time production (period of use) (1) (2) 14 (2) (3) 2 (3) 125 (4) (4) 145 (4) Solution for the last part Cost of not using regular time capacity is very large Optimum plan Use regular time capacity fully first and carry inventory, and if the demand is not met, use over time. Beginning inventory at the planning horizon = 33 Quarters Demand Regular time capacity Over time capacity Regular time production (period of use) Over time production (period of use) (1) 13 (2) (2) 1 (2) (3) 2 (3) 125 (4) (4) 145 (4) Planning Problems 36