The Loan Desk That Had to Be Rearranged
Biratnagar-based Srijana Savings and Credit Cooperative is reorganising its loan desk after a merger. Four loan officers - Anil, Bina, Chandra and Dipa - must each occupy one of four counters, numbered 1 to 4 from the entrance, and each must handle exactly one product: home loan, SME loan, agriculture loan or education loan. The branch rules are:
- Dipa sits at counter 1; Anil sits at counter 2.
- Bina handles agriculture.
- The education desk must be at an end counter.
- The SME officer and the home-loan officer must sit at adjacent counters.
- Chandra sits at an end counter.
- The home-loan officer must not sit at counter 1.
The board also requires that no officer handle a product where a close relative has a pending loan. Dipa's brother has a pending SME loan of Rs 18 lakh, which she disclosed after the desk plan was published. Her manager wants the plan fixed 'without reopening anything else', and the audit committee wants a written note on why the plan cannot simply be left as it is.
1. Derive the full assignment of officer, counter and product, showing the deduction step by step.
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Deduction
- Rule 1 fixes Dipa at counter 1 and Anil at counter 2.
- Rule 5 requires Chandra at an end counter; counter 1 is taken by Dipa, so Chandra = counter 4, and therefore Bina = counter 3.
- Rule 2 gives Bina = agriculture.
- Rule 3 puts education at counter 1 or 4. Test counter 1: Dipa would hold education, leaving home and SME for Anil (2) and Chandra (4). Rule 4 requires home and SME to be adjacent, but counters 2 and 4 are not adjacent. Counter 1 fails, so education = counter 4 = Chandra.
- Home and SME remain for Dipa (1) and Anil (2), which are adjacent, satisfying rule 4. Rule 6 forbids home at counter 1, so Anil = home loan at counter 2 and Dipa = SME loan at counter 1.
Solution: counter 1 Dipa, SME; counter 2 Anil, home; counter 3 Bina, agriculture; counter 4 Chandra, education.
Check every rule: 1 yes, 2 yes, 3 yes (education at 4), 4 yes (1 and 2 adjacent), 5 yes, 6 yes. The solution is unique, which matters for the next task: any fix must break a named rule.
2. The board also asks about a separate collateral policy: (i) every loan above Rs 25 lakh requires collateral; (ii) some SME loans exceed Rs 25 lakh; (iii) no agriculture loan exceeds Rs 20 lakh. Which of these follow necessarily? A. Some SME loans require collateral. B. Some collateral-backed loans are not SME loans. C. No agriculture loan requires collateral.
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Only A follows.
A follows. Premise (ii) says some SME loans exceed Rs 25 lakh; premise (i) says every loan above Rs 25 lakh requires collateral. Those particular SME loans therefore require collateral. This is a straightforward chain of universal plus particular.
B does not follow. Nothing in the premises says collateral exists only for loans above Rs 25 lakh, but neither does anything guarantee that at least one collateral-backed loan is non-SME. To conclude 'some' from all-negative evidence is to assume existence; the premises are consistent with every collateral-backed loan being an SME loan.
C does not follow, and this is the tempting error. Premise (iii) tells us agriculture loans are at or below Rs 20 lakh, which places them outside the trigger in premise (i). But (i) is a one-way rule: it does not say that loans of Rs 25 lakh or less cannot require collateral. An agriculture loan may be collateralised under some other rule or by contract. Negating the antecedent proves nothing.
The pattern worth memorising: 'all A above X require Y' licenses an inference from above-X to Y, and nothing in the reverse direction on either side.
3. Dipa cannot handle the SME desk. Identify the minimum change required, state which constraint must be relaxed, and specify the controls the cooperative should attach.
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Diagnosis. As published, Dipa holds the SME desk at counter 1 while her brother has a pending SME loan. Even with no decision-making power over that specific file, the appearance and the practical access are unacceptable under the board's rule.
Minimum change. Dipa and Anil swap products: Dipa takes the home loan at counter 1, Anil takes the SME loan at counter 2. Check the rules: 1 yes (positions unchanged), 2 yes, 3 yes (education still at 4), 4 yes (home and SME remain at adjacent counters 1 and 2), 5 yes. Only rule 6 (home-loan officer not at counter 1) is breached.
Why nothing else works. Counters are fixed by rules 1 and 5, Bina is locked to agriculture by rule 2, and Chandra is locked to education by the deduction in the first task. The only remaining degree of freedom is which of Dipa and Anil holds home or SME, and rule 6 is exactly the constraint that pins it. So the honest answer to the manager is: something must be reopened, and the cheapest thing to reopen is rule 6, which is a customer-flow preference rather than a control.
Controls to attach: written recusal from the brother's file, a second signatory on all SME approvals routed through Anil, a log of any file Dipa physically handles, a review after one quarter, and a note to the audit committee recording that rule 6 was relaxed on conflict-of-interest grounds.