Supply Contracts with Total Minimum Commitments

Multi-Product Case

Zeynep YILDIZ

Outline

• Introduction• Traditional inventory models• Literature review for single product• Advantages of multi-product

contracts• Literature review for multi-products• Model• Results and conclusion

Introduction

“According to Davis (1993) discussing HP’s supply chain management problem: The real problem with such a confusing network is the uncertainty that plagues it. This uncertainty-observed on a daily basis as late deliveries, machine breakdowns, order cancellations, and the like –leads to increased inventories. In fact, inventory exists more or less as a simple insurance against uncertainty”

Moinzadeh and Nahmias

Traditional Inventory Models

• Type of supply contracts with no commitments-Graves et al.

• The classical newsvendor inventory problem• Optimality of a base stock policy (Arrow, Karlin &

Scarf, 1958)• Total quantity purchased is unknown to the supplier• Order previous period’s demand

• Uncertainty of demand directly passes to the supplier • (s,S) policy, the variance of orders is even higher

than with a base stock policy (Moinzadeh & Nahmias)

• Bullwhip effect

Supply Contracts

“ However, in practice, many contracts impose some restrictions on the buyer. Usually, this takes form of commitments by the buyer to purchase certain minimum quantities”

Bassok and Anupindi, 1997

Literature review-Single Product

• Bassok and Anupindi (1997) analyzed a single product contract where the supplier offers discounts for a total minimum quantity commitment

• They extended this single product contract to allow for upper limits on the total volume purchases that qualify for discounted prices

Bassok and Anupindi, 1998

Literature review-Single Product

• Anupindi and Akella (1997) studied a class of contracts that require the buyer to commit, at the beginning of the planning horizon, to purchase a certain minimum quantity in every period of the horizon

• Reduce variance in order process to supplier

• Moinzadeh and Nahmias (1996) studied continuous review model in which the buyer makes a firm commitment to purchase a certain minimum quantity at regular time intervals

Bassok and Anupindi, 1998

Advantages of Multi-product ContractsSupplier Base• Increase total volume of business and market share• Increase market presence for higher priced products• Increase competitiveness • Ensure firm business for a finite horizon• Reduce order processing costs – packing(unpacking),

administrative and shipping costs• Increase savings from manufacturing costs

• less setups • larger production runs (especially if highly product specific)

• Product/process commonality in production• FMS

Advantages of Multi-product ContractsBuyer Base• Small number of suppliers and closer

cooperation • Improved quality• Improved service

• Greater assurance of supply• Pooling purchases-higher discount rates• Reduction in order processing costs• Associated risks

• Committing more than required-flexibility• Increase in inventory holding costs-discounts

Literature review-Multi Product

• Sadrian and Yoon (1994)• Problem of flexible procurement plan and changes in

forecasted demand and budget• Procurement Decision Support System• Multi-product multi-supplier contracts with discounts on

total dollar amount of sales volume• Deterministic environment• All product discount• “minimize the total discounted purchasing cost, how

many of each product should be purchased from which supplier and under which purchasing strategy, subject to fully satisfying the demands, providing the required flexibility in the purchasing plan, distributing the quantities to be purchased among several suppliers in acceptable proportions, and possibly limiting the total number of suppliers”

• Optimum product-supplier combinations under demand, commitment, market share and maximum number of suppliers constraints

Literature review-Multi Product

• Bassok and Anupindi (1998)• Multi-period multi-product dynamic program• Price discounts for total minimum dollar volume

commitments with flexibility• Stochastic environment• Optimal solution is complex

– Constrained dynamic program– Allocation problem and decision in this period affects future

decisions• Property of the optimal policy• Approximations, order policy assumptions and uniform

discount policy• Upper bound and lower bound for multi-product small

gap• Computational study

– Demonstrate error because of approximations is small– Managerial insights (effects of commitment, #of products on

costs )• Mean dollar volume commitments works quite well

Literature review-Multi Product

• (Qt,Mt)= (0,0) if It≥S∞

(S∞-It,0) if S∞-KtL<It<S∞

(KtL,0) if St-

KtL<It<S∞-Kt

L

(St-It,0) if St-Kt

U<It≤St-KtL

(KtU,0)

if Stm-Kt

U<It≤St-KtU

(KtU, St

m-KtU-It) o.w.

Where F(S∞)=/+h,and, St is the base stock level in period t for a standard (N-t+1)period newsboy problem with per unit order cost at c and,St

m is the base stock level for a standard newsboy problem with per unit order cost cr

Bassok&Anupindi,1998

Literature review-Multi Product

• Bassok and Anupindi (1998)• Multi-period multi-product dynamic program• Price discounts for total minimum dollar volume

commitments with flexibility• Stochastic environment• Optimal solution is complex

– Constrained dynamic program– Allocation problem and decision in this period affects future

decisions• Property of the optimal policy• Approximations, order policy assumptions and uniform

discount policy• Upper bound and lower bound for multi-product small

gap• Computational study

– Demonstrate error because of approximations is small– Managerial insights (effects of commitment, #of products on

costs )• Mean dollar volume commitments works quite well

Model-Basic Assumptions

• Demand• Deterministic or Stochastic• Correlation

=0 Sadrian&Yoon, Bassok&Anupindi >0 decrease in savings from pooling <0 increase in savings and synergy

• Lead-time• Instantaneous deliveries• Fixed lead-times

• Setup cost• Negligible• Fixed

Model-Basic Assumptions

• Zero salvage value can be relaxed (Bassok and Anupindi,1998)

• Flexibility• With adjustments(upwards, upwards&downwards)• Without adjustments

• Discount Schedule• All units• Restricted to commitments• Identical or product specific (Katz,

Sadrian&Tendick,1994)

• Purchasing of extra units• Random market prices (proposed in

Bassok&Anupindi,1998)

Model-Intuitions

• Useful for real cases• Relaxation of some assumptions

result in more complex and intractable model

• Complex models cannot be useful to implement and run

• Small gap between actual model and assumptions (Bassok & Anupindi,1998)

Results

• Coefficient of variation of demand• Decreases with increasing flexibility

• Commitments• Increases with the CV of demands for the same

level of flexibility

• Percentage of error• Additional cost due to committing the mean dollar

volume compared to optimum does not exceed 0.60%

• Flexibility• Increases with CV of demands but very small

compared to no flexibilityBassok and Anupindi,1998

Conclusions

• Discounts for commitments on product basis (Katz, Sadrian&Tendick, 1994)

• Multiple suppliers and product specific constraints (Bassok&Anupindi, 1998)

Multi-Product Multi-Supplier Model

• Sadrian and Yoon

Multi-Product Multi-Supplier Model

• Sadrian and Yoon

Multi-Product Multi-Supplier Model• Sadrian and Yoon

CjDj and assume demand is stochasticwith fj(.) and Fj(.)and use the mean value

Introduce initialinventory andinventory balance equation

Thank YouQ&A