Unit 4: Replacement & inventory models
Operation Research notes · PTU syllabus (BCOM 602-18)
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Unit summary
Machines wear out and stocks must be replenished — both involve cost trade-offs. This unit covers replacement of items that deteriorate with time, individual and group replacement of items that fail suddenly, inventory costs, and deterministic inventory models with uniform and non-uniform demand and with infinite or finite production rates.
After this unit you can
- Determine the optimal replacement age for deteriorating items
- Compare individual and group replacement policies
- Explain inventory costs and the EOQ model
- Apply EOQ models with shortages and finite production (EPQ)
PTU syllabus topics
- Replacement of items that deteriorate with time
- individual and group replacement policy for items that fail suddenly
- inventory costs
- deterministic demand models with uniform and non-uniform demand and finite/infinite production rates
EOQ
√(2DS / H)
D = demand, S = order cost, H = holding cost
Number of orders
D / EOQ
Total inventory cost
(D/Q)S + (Q/2)H
Replace when
Average annual cost is at its minimum
Topic 1
Replacement of items that deteriorate with time
- Maintenance cost rises with age while capital cost per year falls; replace when the average annual total cost is minimum.
Average annual cost for n years
[C − S + Σ maintenance costs for n years] ÷ n
Replacement rule
Replace at the end of year n when the next year's maintenance cost exceeds the current average annual cost
| Year | Maintenance | Cumulative maintenance | C − S (₹12,000 − resale) | Total cost | Average cost |
|---|---|---|---|---|---|
| 1 | 1,000 | 1,000 | 12,000 − 9,000 = 3,000 | 4,000 | 4,000 |
| 2 | 1,500 | 2,500 | 12,000 − 7,000 = 5,000 | 7,500 | 3,750 |
| 3 | 2,000 | 4,500 | 12,000 − 6,000 = 6,000 | 10,500 | 3,500 |
| 4 | 2,500 | 7,000 | 12,000 − 5,500 = 6,500 | 13,500 | 3,375 |
| 5 | 3,500 | 10,500 | 12,000 − 5,000 = 7,000 | 17,500 | 3,500 |
- Average cost is minimum (₹3,375) at 4 years → replace every 4 years.
- With time value of money: compare the weighted average (discounted) annual costs.
Topic 2
Items that fail suddenly: individual and group replacement
- Items like bulbs and tubes fail completely; failure probabilities are known from life tables.
- Individual replacement: replace each item as it fails.
- Group replacement: replace all items at fixed intervals (whether failed or not) and individually replace failures in between — cheaper when individual replacement cost is much higher than group cost per unit.
Expected life
Σ (life × probability of failure at that age)
Average failures per period (individual)
N ÷ Expected life
Individual policy cost per period
(N ÷ mean life) × individual replacement cost
Group policy cost per period at t
[N × group cost + Σ failures up to t × individual cost] ÷ t
- Rule: group replacement at the end of period t if the cost of individual replacement in the next period is greater than the average cost per period up to t.
Example
1,000 bulbs; individual replacement ₹10 each, group replacement ₹4 each; mean life 2.5 months → individual policy ≈ 1,000 ÷ 2.5 × 10 = ₹4,000 per month. If group replacement every 2 months costs ₹3,200 per month (including in-between failures), group replacement is better.
Topic 3
Inventory costs and the basic EOQ model
Ordering (set-up) cost
Placing and receiving orders, machine set-up
Carrying (holding) cost
Storage, insurance, interest, obsolescence
Shortage (stock-out) cost
Lost sales, goodwill, backorder costs
Purchase (unit) cost
Price paid, affected by discounts
Model I: uniform demand, infinite (instantaneous) replenishment, no shortages
Economic order quantity
Q* = √(2DS ÷ H)
Number of orders
D ÷ Q*
Cycle time
Q* ÷ D
Minimum total variable cost
√(2DSH)
Example
D = 9,000 units a year, S = ₹200 per order, H = ₹4 per unit per year. Q* = √(2 × 9,000 × 200 ÷ 4) = √9,00,000 ≈ 949 units; orders ≈ 9.5 a year; total variable cost = √(2 × 9,000 × 200 × 4) ≈ ₹3,795.
Topic 4
Other deterministic models
Model II: finite production rate (EPQ), no shortages
EPQ
Q* = √[2DS ÷ (H (1 − d ÷ p))] — d demand rate, p production rate
Maximum inventory
Q* (1 − d ÷ p)
Model III: uniform demand with planned shortages (backorders)
Optimal order quantity
Q* = √(2DS ÷ H) × √[(H + B) ÷ B] — B shortage cost per unit per year
Maximum shortage
Q* × H ÷ (H + B)
Non-uniform demand and quantity discounts
- Non-uniform demand within a cycle (demand varying across periods): use period-by-period methods or treat average demand; with known patterns, use the total demand over the cycle in place of uniform demand.
- Quantity discounts: compute EOQ at each price; if EOQ is not feasible for that price break, use the minimum quantity of the break; choose the quantity with the lowest total cost (purchase + ordering + carrying).
Exam tip
Always state the assumptions of each inventory model before solving — they carry marks.
Key terms
- Replacement policy
- Rule for when to replace equipment to minimise cost
- Group replacement
- Replacing all items at fixed intervals plus individual failures in between
- EOQ
- Order size minimising ordering and carrying costs
- EPQ
- Economic production quantity with finite production rate
- Backorder
- Unmet demand satisfied later from the next replenishment
Quick revision
- Deteriorating items: replace when average annual cost is minimum.
- Sudden failure: compare individual vs group replacement costs.
- Inventory costs: ordering, carrying, shortage, purchase.
- EOQ = √(2DS/H); EPQ adds (1 − d/p); shortages add √[(H + B)/B].
- Quantity discounts: compare total cost at each price break.
Important exam questions
Practice questions written to the PTU exam pattern for this unit's syllabus: short answers (Section A style) and long answers (Sections B and C style).
Short-answer questions
- Q1.When should a deteriorating machine be replaced?
- Q2.What is group replacement?
- Q3.State the assumptions of the EOQ model.
- Q4.Write the EPQ formula.
- Q5.What is a backorder?
- Q6.How are quantity discounts handled in inventory models?
Long-answer questions
- Q1.Explain the replacement policy for items that deteriorate with time with an illustration.
- Q2.Explain individual and group replacement policies for items that fail suddenly.
- Q3.Explain inventory costs and derive the EOQ formula.
- Q4.Explain inventory models with finite production rate and shortages.
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