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## Inventory Control

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**Lecture Topics**• Week 1 Introduction to Production Planning and Inventory Control • Week 2 Inventory Control – Deterministic Demand • Week 3 Inventory Control – Stochastic Demand • Week 4 Inventory Control – Stochastic Demand • Week 5 Inventory Control – Stochastic Demand • Week 6 Inventory Control – Time Varying Demand • Week 7 Inventory Control – Multiple Echelons**Lecture Topics (Continued…)**• Week 8 Production Planning and Scheduling • Week 9 Production Planning and Scheduling • Week 12 Managing Manufacturing Operations • Week 13 Managing Manufacturing Operations • Week 14 Managing Manufacturing Operations • Week 10 Demand Forecasting • Week 11 Demand Forecasting • Week 15 Project Presentations**Inventory in the US Economy**$98.60 billion $122.10 Farm billion (7.9%) Other (9.8% $424.60 billion Manufacturing (33.9%) $290.40 billion Wholesale (23.2%) $316.00 billion Retail (25.2%)**Inventory Types**• Raw Materials • Work-in-process (WIP) • Finished goods inventory (FGI)**Inventory Location**• Manufacturing Facility • In-Transit • Warehouse • Retailer • Customer**Benefits of Inventory**• Reduces ordering, setup & transportation costs (economies of scale) • Buffer against demand fluctuations • Buffer against supply fluctuations • Supply shortages • Variability in supply lead times**Benefits of Inventory (continued…)**• Buffer against price fluctuations • Benefits from quantity discounts • Protects production capacity • Allows production smoothing • Reduces managerial complexity (eliminates the need for coordination) • Can increase demand**The Cost of Inventory**• Tied up capital • Warehousing cost • Deterioration • Obsolescence • Demand shortfall • Quality defects • Changes in raw material prices • Changes in product design specifications**What if demand uncertainty and variability are eliminated?**• What if replenishment lead times are made insignificant?**What if demand uncertainty and variability are eliminated?**• What if replenishment lead times are made insignificant? • What if ordering & setup costs are made negligible?**What if demand uncertainty and variability are eliminated?**• What if replenishment lead times are made insignificant? • What if ordering & setup costs are made negligible? • What if production capacity is never a constraint?**Characteristics of Inventory Systems**• Demand • Constant • Time varying • Stochastic • Supply lead time • Deterministic • Stochastic • Load-dependent • Review • Continuous review • Periodic review**Characteristics of Inventory Systems**(continued…) • Excess Demand • Backordering • Lost sales • Impatient customers • Item substitution • Capacity • Unlimited • Limited • Deterministic • Stochastic**Characteristics of Inventory Systems**(continued…) • Number of items & customer classes • Single item • Multiple items • Single customer class • Multiple customer classes • Inventory quality • Perishability • Obsolescence • Imperfect yield**Cost Measures**• Holding cost (h) • Capital cost • Taxes and insurance • Deterioration, spoilage, obsolescence • Ordering (setup) cost (A) • Purchasing (production) costs (c) • Shortage cost (p) • backordering cost • lost sale cost**Example**• h = ic (cost per unit per time period) • 28% = cost of capital • 2% = taxes and insurance • 6% = storage cost • 1% = breakage cost • 37% = total interest charge (i) per year • If c = $100, then h = ic = $37**Assumptions of the Basic Model**• Demand occurs continuously over time with a constant rate • Inventory can be replenished instantaneously • There are no capacity limits or limits on the size of replenishment orders • A replenishment order incurs a fixed ordering (or setup) cost • Multiple products can be analyzed independently of each other • No backorders are allowed**Notation**• D: demand rate (units per unit time) • c: unit purchase/production cost (dollars per unit) • A: fixed cost to place an order (dollars) • h: holding cost (dollars per unit per unit time); if the holding cost consists entirely of interest on money tied up in inventory, then h=ic where i is an interest rate. • Q: the size of the order (a decision variable)**Inventory versus Time**Q Inventory Time**Inventory versus Time**Q Inventory Q/D 2Q/D 3Q/D 4Q/D Time**Costs**• Holding cost:**Costs**• Holding cost: • Ordering/setup cost:**Costs**• Holding cost: • Ordering/setup cost: • Production cost: c per unit**Total Cost**• Total cost per unit time:**Total Cost**• Total cost per unit time: • Total cost per unit:**Economic Order Quantity**hQ/2 Y(Q) AD/Q Q**Optimal Cost**• Optimal average cost per unit: • Optimal average cost per unit time (e.g., per year):**Sensitivity to Order Quantity**• Ordering and holding cost from using Q’: • Ratio**Sensitivity to Order Quantity**• Ordering and holding cost from using Q’: • Ratio • Example • Q' = 2Q*, then the ratio of the actual to optimal cost is (1/2)[2 + (1/2)] = 1.25**Sensitivity to Order Quantity**Large deviations from the optimal order quantity lead to relatively small deviations from the optimal cost.**Order Quantity versus Order Interval**• Order Interval: Let T represent time between orders, then • Total cost: • Optimal Order Interval:**Some Limitations of the EOQ Model**• Demand is deterministic and constant • Instantaneous replenishments • Ordering costs are constant and independent of order size • No accounting for interactions among multiple items • No backordering**Extensions**• Non-zero order replenishment lead times • Non-zero safety stocks • Finite supply capacity • Backordering**Non-Zero Replenishment Lead Times**• If L is the lead time and r is the reorder point, then r = DL • A non-zero lead time has no effect on Q* or Y(Q)**Non-Zero Safety Stocks**• If ss is the safety stock, then • A non-zero safety stock affects Y(Q) but has no effect on Q*