Pricing and Market Mix Optimization in Freight Transportation Markets

1. Pricing and Market Mix Optimization in Freight Transportation Markets Michael F. Gorman, Ph.D. Presented to Charles River Associates March, 2004

2. Agenda • Problem Positioning • Introduction and Motivation • Problem Formulation • Solution Algorithm • Illustrative Example • Sample Results • Conclusions

3. Problem Positioning: Continuum of Pricing Decisions • Looking at intermediate (quarterly or annual) pricing decisions • Not short-term, tactical decisions (like yield management) • Not long-term, strategic product and market positioning Short-Term Price Adjustments Short-term Capacity Utilization Revenue and Yield Management Strategic Pricing Product Positioning Market Evaluation Tactical Pricing Quarterly/Annual Adjustments Product Mix Strategic, Tactical and Short-term pricing decisions are related topics and tools, but different questions are being addressed.

4. Introduction:Freight Markets Descriptions and Definitions • Freight Market: a movement of freight from an origin to a destination within a designated equipment type • Equipment type: trailers, containers of different lengths • Markets are represented by directed arcs in a network • Characterized a price and quantity • Locations: origin or destination (or both) of a one or more freight markets • Every market affects the equipment balance at two nodes • Inequalities in markets into and out of origin and destination nodes create equipment imbalances • Markets are interrelated through common locations • Every node is origin and/or destination to multiple markets • Pricing focuses on freight markets; imbalances occur at nodes

5. Introduction: Problem Motivation • Problem: • BNSF was exploring ways to reduce the cost of empty equipment repositioning that resulted from network flow imbalances. • A new perspective: • Rather than attack the equipment repositioning problem, we attacked the source of the imbalance: the markets that caused them. • By considering empty repositioning as part of the pricing problem, we hoped to reduce the structural imbalance that existed in the network. • Early observations: • Empty repositioning costs are largely and “externality”, or a cost that is not considered by the decision maker (pricing manager). • Because equipment imbalances are a function of many markets, no one market feels “responsible” for the costs. • The externality creates suboptimal prices and product mix (quantities produced of each product) across the network of products. • By introducing these costs directly to the profit equation (internalizing them), pricing managers improve price and produce mix • The network of products are interrelated through the repositioning component of their cost functions, and cannot be considered in isolation.

6. Introduction: Elements of the Problem • Large Scale • Must consider entire network of interrelated products • Combinatorial optimization problem • Non linear profit function • Non-differentiable and discontinuous due to the nature of the cost function • Demand Curve Uncertainties • Future demand is unknown • Customer responsiveness to price changes is unknown • Combines numerous decision support techniques • Primary economics • Simulation • Optimization

7. Problem Formulation • The firm's objective is to maximize network profit (Pn) across all markets (Mijk): • P n= SiSjSk(TRijk - TCijk) - Sk(TRCk). • Where the direct market profits for equipment k, from node i to node j, are given by • P d ijk = TRijk - TCijk. Direct market profit function • TRijk = Pijk*Qijk. Total revenue function • TCijk = (qijk) Total cost function • q ijk = f(pijk, ijk, ijk) Demand curve • Network market profits must include total repositioning cost (TRC): • TRC = Sk Min TRCk = SkSiSj RCijk Rijk • subject to SiRjik -SiRijk = Iik for all j • Cost Allocation: Allocating repositioning costs back to individual markets: • MECijk = Dik-Djk. • Pijk = TRijk - TCijk + qijk *MECijk

8. P P ijk ijk p1 p p D’ D D” D’ D D” high Q low high low Q Figure 2b. Demand Shift Sampling (ijk) Figure 2a. Demand Elasticity Sampling (ijk). Allowing for Uncertainty:Sampling due to Demand Curve Uncertainty

9. Expected Parameter Value Low End of Expected Value Rang High End of Expected Value Rang Best Case Value Worst Case Value Figure 1. Trapezoidal Probability Density Function Used For Demand Parameters ijk and ijk. Probability Density Function For Demand Elasticity and Demand Shift

10. Derivation: Implied Elasticity of Demand

11. Price Recommendation Demand Simulation Profit Calculation Software Modules:Conceptualization Outputs: - Expected profits - Repositioning flows and costs - Cost allocations Inputs: - Market information - Repositioning information Empty LP Nested Modules in the Intermodal Pricing Model

12. Solution Algorithm:Overview • A heuristic solution technique was employed. • Computational gradient approach • Other solution techniques were explored and discarded • Exhaustive enumeration - OK for small problems • Powell’s Non-linear search in N variables - slow and poor-performing • We also explored ways to improve the algorithm • Tabu-search constructs to broaden search and improve solutions - added to complexity of algorithm with no appreciable improvement in solution • We found our heuristic was suitable for many reasons: • Quick to get high-confidence solution • Practical impossibility of ensuring optimality • Vast uncertainties in the problem reduced the need for exact or optimal solutions • In practice, model output was used as direction-setting indicators for price changes, so exact “optimal” numbers were unnecessary

13. Solution Algorithm:Outline • Initialize parameters: • Set starting prices for search, initial price change step size, step size adjustment • Establish sampling for demand shift and elasticity • Each iteration • Given prices, calculate quantities • Given quantities, calculate repositioning costs • Given repositioning costs, calculate network profits • Adjust prices • Change prices in all markets by +/- step size • Estimate impact on profit based on elasticity and duals • Sample demand elasticity • Generate expected profits based on a random sampling of demand elasticities • If no profit improvement is found, reduce step size • If step size is less than minimum step, end search

14. Imbalance: + = Surplus - = Deficit Market Flow Empty Flow -500 CH 150 650 +150 50 900 LA 300 350 FW 300 200 200 +350 Simple Illustrative ExampleNetwork Flows

15. Illustrative ExampleStarting Prices, Repositioning, and Profits

16. Illustrative ExampleFinal Pricing, Repositioning, and Profits

17. Illustrative ExampleChange in Pricing, Repositioning, and Profits In this example, somewhat surprisingly we should raise rates CH-FW, close the FW-CH market. Results show over 80% reduction in repositioning and 20% increase in network profit.

18. Results:Measures of Success • Increased profitability and margins of services • High-confidence basis for price changes • Improved equipment utilization • Reduction in structural equipment imbalances by location • Reduced equipment repositioning • Organizational change surrounding new pricing philosophy

19. Losses: 1.4% reduction in revenues 4.1% decrease in “direct” profit, ignoring repositioning costs Gains 61% reduction in repositioning costs 15% increase in “network” profits, inclusive of repositioning) Sample BNSF Results: Recommended Network Profitability In nearly every analysis conducted, “losses” in individual markets were required to achieve the greater gains in the network-wide profit function.

20. Sample Results: Confidence in Price Change Recommendations In this example, recommended prices create an average of 3% improvement in profits. Regardless of demand conditions, price changes are directionally correct almost 95% of the time.

21. Sample Results:Trends of Improved Equipment Utilization BNSF’s Loaded miles have increased, while empty miles have decreased

22. Extensions and Complications • Price change constraints • We implemented “maximum change” constraints • Allows for analysis of selected subset of markets; testing of “pricing initiatives” • Equipment and capacity constraints • We have estimated uncapacitated demand; in practice limits exist on the ability to grow certain lanes or equipment flows in total • Non-linear marginal costs • We recognize the low-volume impacts on marginal costs • Sensitivity analysis indicates our AVC estimates overstate MC • Products have interrelated demands; Pricing action in one market affects another • Equipment can be substituted • Geographically approximate origins and destinations can be substituted • We built a “cross-product” substitution matrix (elasticity) to capture this phenomena • Rarely used due to the size of the matrix and complexity of estimation • Experimentally found to have little affect on recommended prices

23. References for this work: "Pricing and Product Mix Optimization in Transportation Markets", Transportation Research Forum, 41 (1), January, 2002. “Intermodal Pricing Model Creates a Network Pricing Perspective at BNSF”, Interfaces, 31 (4) 37-49, July-August, 2001. “Estimation of Implied Price Elasticity of Demand Through Current Pricing Practices”, submitted to Applied Economics.