Part 1The Donald Fertilizer Company produces industrial chemical fertilizers. The projected manufacturing requirements (in gallons) for the next four quarters are 90,00090,000, 90,00090,000, 60,00060,000, and 140,000140,000 respectively. A level workforce is desired, relying only on anticipation inventory as a supply option. Stockouts and backorders are to be avoided, as are overtime and undertime.Part 2a. Determine the quarterly production rate required to meet total demand for the year, and minimize the anticipation inventory that would be left over at the end of the year. Beginning inventory is 00.The quarterly production rate is [input]enter your response here gallons. (Enter your response as an integer.)

Question

Part 1The Donald Fertilizer Company produces industrial chemical fertilizers. The projected manufacturing requirements​ (in gallons) for the next four quarters are 90,00090,000​, 90,00090,000​, 60,00060,000​, and 140,000140,000 respectively. A level workforce is​ desired, relying only on anticipation inventory as a supply option. Stockouts and backorders are to be​ avoided, as are overtime and undertime.Part 2a. Determine the quarterly production rate required to meet total demand for the​ year, and minimize the anticipation inventory that would be left over at the end of the year. Beginning inventory is 00.The quarterly production rate is [input]enter your response here gallons. ​(Enter your response as an​ integer.)

BlackTom AI · Accepted Answer

We start by restating the problem in our own words to ensure understanding: we have four quarters with demand 90,000; 90,000; 60,000; and 140,000 gallons respectively. Beginning inventory is 0. We want a constant production rate each quarter (P), use only anticipation inventory to smooth production, avoid stockouts and extra overtime, and minimize the ending inventory after the fourth quarter.

Next, we set up the inventory balance equations quarter by quarter with I0 = 0 and I_t representing inventory after quarter t:
- I1 = I0 + P − D1 = P − 90,000. To avoid a stockout in Q1, I1 must be >= 0, which implies P >= 90,000.
- I2 = I1 + P − D2 = (P − 90,000) + P − 90,000 = 2P − 180,000. To avoid a stockout in Q2, I2 must be >= 0, which implies 2P >= 180,000, so P >= 90,000.
- I3 = I2 + P − D3 = (2P − 180,000) + P − 60,000 = 3P − 240,000. To avoid a stockout in Q3, I3 must be >= 0, which implies 3P >= 240,000, so P >= 80,000.
- I4 = I3 + P − D4 = (3P − 240,000) + P − 140,000 = 4P − 380,000. To avoid a stockout in Q4 and finish the year with nonnegative ending inventory, I4 must be >= 0, which implies 4P >= 380,000, so P >= 95,000.

From these inequalities, the tightest (largest) lower bound on P is 95,000. This bound ensures there are no stockouts in any quarter while keeping the ending inventory as low as possible, since any larger P would only increase ending inventory without eliminating stockouts (which are already prevented at P = 95,000).

To summarize the reasoning without committing to a numeric conclusion in advance: evaluate each quarter’s nonnegativity constraint on inventory with a constant production P, determine the minimum P that satisfies all four constraints, and the controlling constraint will be the maximum of the individual minimums required per quarter. In this case, the controlling constraint is driven by the final quarter’s demand relative to cumulative production, yielding the effective production rate that minimizes the ending inventory while avoiding stockouts across the year.

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