Lecture 2 – Inventory Management, Risk Pooling, and Distribution
Why do you need inventory?
▪ Uncertainty in customer demand
▪ Uncertainty in supplies
▪ Delivery lead times
▪ Incentives for larger shipments
Economic order quantity: Q ¿ =
√ 2 KD
h
K = fixed cost every time an order is placed
h = holding cost per unit held
Demand uncertainty:
▪ The forecast is always wrong
▪ The longer the forecast horizon, the worse the forecast
▪ Aggregate forecasts are more accurate
Single period models: one ordering opportunity only before demand occurs
Continuous demand, Normal distribution
-> Q=μ+ z∗σ
Optimal service level = Marginal profit / (Marginal profit + Marginal cost)
(Q,R) policy = whenever inventory level falls to a reorder level ‘R’, place an
order for ‘Q’ units R=L∗AVG+ z∗STD∗√ L
R=AVG∗AVGL+ z √ AVGL∗STD 2∗AVG 2∗STDL2Q=
opportunities:
√ 2 K∗AVG Multiple order
h
1. Short intervals
-> during each inventory review, if the inventory position falls below ‘s’, order
enough to raise the inventory position to ‘S’
2. Longer intervals
-> always order after an inventory level review to fill the inventory to a pre-
determined target inventory level (the base-stock level)
Forms of risk-pooling:
1. Location pooling
= keep inventory at a central location to satisfy demand in multiple markets,
demand variability will be reduced
2. Standardized design (product pooling)
-> product category demand is easier to predict
3. Delayed differentiation
= delay of product differentiation until closer to the sale of the product
4. Production flexibility (capacity pooling)
= product multiple products/models in one factory
The newsvendor problem: “how many newspapers to buy in the morning to
maximize profits?”
,Surplus cost (overage cost): Co = purchase cost – salvage value
Shortage cost (underage cost): Cu = sales price – purchase cost
Critical ratio: CR = Cu / (Co + Cu)
F(Q*) = CR
-> F is the cdf of demand
Lecture 4 : Supply Contracts & The Value of Information
Optimal Service Level (CSL*) = CU / (CU + CO)
Make To Order contracts (MTO):
1. Buy-Back Contract
, = supplier agrees to buy back leftover inventory for a predetermined price
▪ Requires suppliers to have an effective reverse logistics system and may
increase logistics cost
▪ Buyers have an incentive to push the product not under the buyback contract
2. Wholesale Price Contract
= buyer shares some of its revenue with the supplier in return for a discount on
the wholesale price
▪ Requires suppliers to monitor the buyer’s revenue and thus increases
administrative cost
▪ Buyers have an incentive to push competing products with higher profit
margins
-> both contracts will increase total profits shared by the two parties
3. Quantity-Flexibility Contract
= supplier provides a full refund for returned items as long as the number of
returns is no larger than a certain quantity
4. Sales Rebate Contract
= a rebate is paid by the supplier for any item sold above a certain quantity
Make To Stock contracts (MTS):
1. Pay-Back Contract
= the buyer agrees to pay some agreed-upon price for the inventory produced by
the supplier but not purchased
▪ Buyer’s risk increases
▪ supplier has an incentive to produce more units
2. Cost-Sharing Contract
= the buyer shares some of the production cost with the supplier, in return for a
discount on the wholesale price
▪ Reduces effective production cost for the supplier
▪ Supplier has an incentive to produce more units
Contracts with Asymmetric Information:
1. Capacity Reservation Contract
= buyer pays to reserve a certain level of capacity at the supplier
▪ A menu of prices for different capacity reservations provided by supplier
▪ Buyer signals true forecast by reserving a specific capacity level
2. Advance Purchase Contract
= supplier charges special price before building capacity
▪ When demand is realized, price charged is different
▪ Buyer’s commitment to paying the special price reveals the buyer’s true
forecast
Operational Causes for the Bullwhip effect:
1. Demand forecasting
-> estimates about demand are regularly modified
2. Lead time
-> small changes in demand estimates are magnified
3. Batch ordering
-> a large order followed by several periods of no order
4. Price fluctuations
-> discounts or promotions cause forward buying
Why do you need inventory?
▪ Uncertainty in customer demand
▪ Uncertainty in supplies
▪ Delivery lead times
▪ Incentives for larger shipments
Economic order quantity: Q ¿ =
√ 2 KD
h
K = fixed cost every time an order is placed
h = holding cost per unit held
Demand uncertainty:
▪ The forecast is always wrong
▪ The longer the forecast horizon, the worse the forecast
▪ Aggregate forecasts are more accurate
Single period models: one ordering opportunity only before demand occurs
Continuous demand, Normal distribution
-> Q=μ+ z∗σ
Optimal service level = Marginal profit / (Marginal profit + Marginal cost)
(Q,R) policy = whenever inventory level falls to a reorder level ‘R’, place an
order for ‘Q’ units R=L∗AVG+ z∗STD∗√ L
R=AVG∗AVGL+ z √ AVGL∗STD 2∗AVG 2∗STDL2Q=
opportunities:
√ 2 K∗AVG Multiple order
h
1. Short intervals
-> during each inventory review, if the inventory position falls below ‘s’, order
enough to raise the inventory position to ‘S’
2. Longer intervals
-> always order after an inventory level review to fill the inventory to a pre-
determined target inventory level (the base-stock level)
Forms of risk-pooling:
1. Location pooling
= keep inventory at a central location to satisfy demand in multiple markets,
demand variability will be reduced
2. Standardized design (product pooling)
-> product category demand is easier to predict
3. Delayed differentiation
= delay of product differentiation until closer to the sale of the product
4. Production flexibility (capacity pooling)
= product multiple products/models in one factory
The newsvendor problem: “how many newspapers to buy in the morning to
maximize profits?”
,Surplus cost (overage cost): Co = purchase cost – salvage value
Shortage cost (underage cost): Cu = sales price – purchase cost
Critical ratio: CR = Cu / (Co + Cu)
F(Q*) = CR
-> F is the cdf of demand
Lecture 4 : Supply Contracts & The Value of Information
Optimal Service Level (CSL*) = CU / (CU + CO)
Make To Order contracts (MTO):
1. Buy-Back Contract
, = supplier agrees to buy back leftover inventory for a predetermined price
▪ Requires suppliers to have an effective reverse logistics system and may
increase logistics cost
▪ Buyers have an incentive to push the product not under the buyback contract
2. Wholesale Price Contract
= buyer shares some of its revenue with the supplier in return for a discount on
the wholesale price
▪ Requires suppliers to monitor the buyer’s revenue and thus increases
administrative cost
▪ Buyers have an incentive to push competing products with higher profit
margins
-> both contracts will increase total profits shared by the two parties
3. Quantity-Flexibility Contract
= supplier provides a full refund for returned items as long as the number of
returns is no larger than a certain quantity
4. Sales Rebate Contract
= a rebate is paid by the supplier for any item sold above a certain quantity
Make To Stock contracts (MTS):
1. Pay-Back Contract
= the buyer agrees to pay some agreed-upon price for the inventory produced by
the supplier but not purchased
▪ Buyer’s risk increases
▪ supplier has an incentive to produce more units
2. Cost-Sharing Contract
= the buyer shares some of the production cost with the supplier, in return for a
discount on the wholesale price
▪ Reduces effective production cost for the supplier
▪ Supplier has an incentive to produce more units
Contracts with Asymmetric Information:
1. Capacity Reservation Contract
= buyer pays to reserve a certain level of capacity at the supplier
▪ A menu of prices for different capacity reservations provided by supplier
▪ Buyer signals true forecast by reserving a specific capacity level
2. Advance Purchase Contract
= supplier charges special price before building capacity
▪ When demand is realized, price charged is different
▪ Buyer’s commitment to paying the special price reveals the buyer’s true
forecast
Operational Causes for the Bullwhip effect:
1. Demand forecasting
-> estimates about demand are regularly modified
2. Lead time
-> small changes in demand estimates are magnified
3. Batch ordering
-> a large order followed by several periods of no order
4. Price fluctuations
-> discounts or promotions cause forward buying
