ITPEC IP April 2019 Question 66

Topic: Corporate Activities

A product is manufactured by allocating a set of materials to three factories A, B, and C. The profit of the factories according to the allocated units of the material is as shown in the table below. How much (in million yen) is the maximum profit that is obtained by allocating a total of four units of the material to the factories?

Unit: million yen

Allocated units of material0 units1 unit2 units3 units4 units
Profit of factory A02578
Profit of factory B01357
Profit of factory C03468

Explanation

10 million yen — give 3 units to factory A and 1 unit to factory C.

List the ways to split 4 units among A, B and C and add the profits from the table. The best splits:

Units to A, B, CProfit
3, 0, 17 + 0 + 3 = 10
2, 0, 25 + 0 + 4 = 9
2, 1, 15 + 1 + 3 = 9
4, 0, 0 or 0, 0, 48

Every other split, including every split that gives B two or more units, earns 8 or less, so the maximum is 10.

Answer (c)

Why not others:

  • •(a) 8 — what all 4 units at A or all 4 at C give, and several mixed splits too; better splits exist
  • •(b) 9 — the best split of 2 units each to A and C (or 2, 1, 1), missing that A's third unit adds 2 while C's second adds only 1
  • •(d) 11 — needs 5 units, for example 3 at A and 2 at C, 7 + 4; with only 4 units no split reaches it

Key rule: For allocating a few indivisible units, enumerate the splits that use exactly the total and compare them; the increments are not always diminishing (A's second unit adds 3, more than its first), so a quick guess can miss the optimum.

Corporate Activities · AI-written, may contain errors

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