“Each of P1 to P4 should be 1 or greater” is simply premise (3), “Assume that there is at least one operator in every work pattern,” written as a constraint, so it is an appropriate condition to consider. Without this constraint, the cost-minimizing calculation could produce a solution that sets some pattern to zero operators, which would contradict the setting of the question. Since the question asks for the option that is NOT appropriate, ア, a valid condition, is not the answer.
○イ
正解The correct answer is イ, “The sum of P1 to P4 should be greater than the sum of Q1 to Q4.” This question asks whether you can organize a cost-minimization optimization problem into “the objective you want to minimize” and “the constraints that must be satisfied.” Looking at Figure 2, one operator works across several time periods. For example, the P1 operators in work pattern 1 work from 9:00 to 17:00, so they are counted in all three of time periods A, B, and C. In fact, the number of operators on duty in each period is A=P1, B=P1+P2, C=P1+P2+P3, and D=P2+P3+P4. What must therefore be satisfied is a period-by-period condition, namely that “in each time period, the number of operators on duty is at least the minimum required number” (P1≧Q1, P1+P2≧Q2, P1+P2+P3≧Q3, P2+P3+P4≧Q4). Simply adding up Q1 to Q4, which are required numbers for separate time periods, and comparing that total with the total of P1 to P4 is meaningless. In practice, because the same person is counted repeatedly across several periods, the required number for every period can be met even when the total of P is smaller than the total of Q; conversely, even when the total of P is larger than the total of Q, one particular period may still be short of people. Moreover, the cost is the total of “hourly wage x working hours x number of operators,” so setting up a condition that makes the total number of people larger runs in the opposite direction of the goal of minimizing cost. Therefore, イ is not an appropriate condition to consider.
×Incorrectウ
“The total of ‘hourly wage x working hours x number of operators’ for each work pattern should be minimized” is exactly the total labor cost derived from premises (1) and (2), that is, the objective function to be minimized in this optimization problem. It is the very core of what the question asks about, “conditions to consider in order to minimize cost,” so it is appropriate and is not the answer.
×Incorrectエ
“The number of operators on duty in each time period should satisfy the minimum required number” is the constraint obtained by matching the bottom row of Figure 2 (A=P1, B=P1+P2, C=P1+P2+P3, D=P2+P3+P4) with Q1 to Q4 in Table 1, and it is a condition that must be met for the operation to work at all. It is an appropriate condition, so it is not the answer. Note also that the comparison of totals in イ cannot serve as a substitute for this period-by-period condition.
ここがポイント
Organize an optimization problem by separating “the objective function you want to minimize (or maximize)” from “the constraints that must be satisfied.” In this question the objective function is the total labor cost (the total of hourly wage x working hours x number of operators), and the constraints are that the required number of operators is met in each time period and that each pattern has at least one operator.
When one person spans several time periods, as in a shift schedule, conditions on the number of people must be set up period by period. Simply comparing totals across time periods is meaningless because of double counting, and it is neither a necessary nor a sufficient condition.