Making room for better decisions.
Two kinds of task compete for the same resources. A small model makes the trade-off visible and finds the best feasible mix under stated assumptions.
Worked example
Problem
Choose how many A and B tasks to complete. Each task consumes machine and crew capacity and contributes a fixed score. Tasks are indivisible, so the counts must be whole numbers.
| Quantity | A | B |
|---|---|---|
| Machine units | 3 | 1 |
| Crew units | 1 | 2 |
| Score | 9 | 7 |
Available capacity: 12 machine units and 8 crew units. The starting plan uses 2 A tasks and 2 B tasks. Both plans are compared using the same inputs and constraints.
Result
| Measure | Start | Best |
|---|---|---|
| A tasks | 2 | 3 |
| B tasks | 2 | 2 |
| Machine use | 8 | 11 |
| Crew use | 6 | 7 |
| Score | 32 | 41 |
The best allocation uses 3 A tasks and 2 B tasks, leaving 1 machine unit and 1 crew unit unused. Spare capacity can remain when tasks are indivisible.
Method
Let a and b be the counts of tasks A and B. Maximize the total score while keeping both resource uses within capacity.
Maximize 9a + 7b.
Machine: 3a + b ≤ 12.
Crew: a + 2b ≤ 8.
a and b are nonnegative integers.
The calculation enumerates every pair within the resource bounds, rejects infeasible pairs and selects the highest score. There are 17 feasible allocations in this example. Because every feasible choice is checked, the best score is an exact optimum for this fixed model.
Why it matters
The score rises from 32 to 41 with the same resource capacity. Comparing feasible combinations reveals a better task mix without adding machines or crew.
Methodology and limitations
- Scores and resource requirements are fixed assumptions. Different priorities or inputs can change the preferred allocation.
- The model omits timing, sequencing, uncertainty and other operating constraints. Enumeration is practical here because the example is small.