1. Nahmias 3.7. A rolling production schedule implies that the schedule may be revised at the start of a new planning period. However, this does not mean that there is no value in knowing future demands. First, schedules may be frozen for several periods so that changes are not possible. Second, if high demands are anticipated, production may have to be ramped up well in advance to meet those demands. 2. Nahmias 3.12 (a) Net demand for July is 1250 - 500 + 300 = 1050. Adding 300 units to July’s demand guarantees that the ending inventory for each month never drops below 300. December’s demand has been increased by 400 so that we have 400 units left at the end of december. A person employed for a month produces 20*8/5 = 32 units. Column workers gives us the number of employees required to meet the net demand. Its calculated by dividing net demand by 32 and rounding up. Because of the rounding up we actually produce a little more than required. This is captured in the column Inv(-300). Because of this extra production we could actually reduce the employees required and this is shown in the column Workers(adj), e.g. in Sept, the net inventory surplus allows us to reduce one worker. Using this we can calculate the new production plan that gives the min number of workers without inccuring negative inventory. Table 1: Production plan 1 Workers Prod Inv−300 33 1056 6 35 1120 26 30 960 36 29 896 40 32 1024 64 49 1568 82

Month July Aug Sept Oct Nov Dec

Forc Dem 1250 1100 950 900 1000 1150

Net Dem 1050 1100 950 900 1000 1550

Workers (adj) 33 35 29 28 31 49

Prod (adj) 1056 1120 928 896 992 1568

(b) The ratio is the cum. net demand divided by the cumulative number of units produced by one worker rounded to the next largest integer (multiples of 32). The maximum of these numbers, 35, is the minimum constant workforce required to guarantee that no shortages occur. Assuming 35 workers each month gives the production and inventory levels in the last two columns. The net forecast demand for July is adjusted by adding 300 and subtracting 500. That for December is adjusted by adding 400. 3. Nahmias 3.32 Assumptions: (1) All demands are met at the end of the quarter. (2) We don’t pay inventory cost during the quarter of production of a good; we pay for it every quarter after that. (3) For the year end inventory, we pay one quarter inventory cost. (a) Constant work force. Referring to Table 3, the minimum constant workforce is 465 workers. The cost of the resulting plan is: 1

Month July Aug Sept Oct Nov Dec

Forc Dem 1250 1100 950 900 1000 1150

Table 2: Production plan 2 for Question 2 Net Dem Cum Net Dem Cum# Workers 1050 1050 32 33 1100 2150 64 34 950 3100 96 33 900 4000 128 32 1000 5000 160 32 1550 6550 192 35

Prod 1120 2240 3360 4480 5600 6720

Inv(-300) 70 90 260 480 600 170

Quarter 1 2 3 4

Table 3: Constant Workforce: Production plan for Question 3 U nit/W orker N etDemand CumN etDem M inimumW orkF orce 1 300 300 300 1 630 930 465 1 220 1150 384 1 180 1330 333

The total cost of this plan is: 1200(465 - 280) + 1000(940+20) = 1,182,000. Note that we must add back in the 20,000 required to be on hand in the fourth quarter. (b) Varying workforce. Referring to table 5, total cost = 1200(350) + 2500(450) + 20000 = 1,565,000 (c) If we use the minimum number of workers required through period 3 of 1150/3 = 384, it will satisfy the conditions stated. Total cost = 1200(384 - 280) + 1000(312) + 2000(162) = 760,800. Better than policies in parts (a) or (b). Note that the total inventory 312 = 84+2+206+20, assuming that the excess demand is back-ordered and ﬁlled up by the end of quarter 2. 4. Nahmias 3.34 (a) Variables: Wt = No. of workers employed during quarter t It = Inventory in quarter t (measured in 1000s) Pt = Production in quarter t (measured in 1000s) Ht = No. of workers hired at the beginning of quarter t Ft = No. of workers ﬁred at the beginning of quarter t...