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Dynamic Corporate Transformation

Vol. II, Ch. 7 · Part 2. Dynamic Enterprise Optimization · Week 11

Dynamic Programming for Enterprise Systems

Learning outcomes

After completing this chapter, the reader should be able to:

  1. Explain Bellman's Principle of Optimality and its role in sequential enterprise decision-making.
  2. Formulate recursive enterprise optimization problems from declared multistage instances.
  3. Construct enterprise value functions and read them as priced inventories of future opportunity.
  4. Develop Bellman equations for finite- and infinite-horizon enterprise problems.
  5. Analyze multistage enterprise decision processes through their recursive structure.
  6. Interpret recursive enterprise policies as greedy rules against summarized futures.
  7. Solve finite-horizon enterprise problems by backward induction.
  8. Formulate infinite-horizon discounted enterprise models and certify their well-posedness.
  9. Implement value iteration, policy iteration, and approximate dynamic programming computationally.
  10. Prepare enterprise optimization models for Hamilton–Jacobi–Bellman analysis.

Reading guide

Work through the chapter in section order; the full development, proofs, and worked examples are in the book — this page indexes them and does not replace them.

  1. Motivation for Dynamic Programming

    Motivation for Dynamic Programming

  2. Sequential Enterprise Decision Problems

    Sequential Enterprise Decision Problems

  3. Bellman's Principle of Optimality

    Bellman's Principle of Optimality

  4. Enterprise Value Functions

    Enterprise Value Functions

  5. Bellman Equations

    Bellman Equations

  6. Finite-Horizon Dynamic Programming

    Finite-Horizon Dynamic Programming

  7. Infinite-Horizon Dynamic Programming

    Infinite-Horizon Dynamic Programming

  8. Computational Algorithms

    Computational Algorithms

  9. Enterprise Applications

    Enterprise Applications

  10. Approximate Dynamic Programming

    Approximate Dynamic Programming

  11. Preparation for Hamilton–Jacobi–Bellman Theory

    Preparation for Hamilton–Jacobi–Bellman Theory

  12. Chapter Summary

    Chapter Summary

  13. Worked Examples

    Worked Examples

  14. Exercises

    Exercises

  15. Notes and Sources

    Notes and Sources

AXIOM

This chapter is instrumented by:

Launch the module, load the chapter model, modify inputs, run the optimization, and compare against the worked examples in the book.

Exercises

12 exercises, grouped A concept checks · B mathematical · C computational · D enterprise applications. Starred (★) exercises are on the advanced track. Full solutions appear in the Instructor's Manual, Chapter 7.

A. Concept checks

  1. 7.1
    Explain why the Principle of Optimality can fail for objectives that are not stage-separable (e.
  2. 7.2
    For each row of Table (see book), identify what the value function's slope prices, and name the governance decision that number would inform.

B. Mathematical exercises

  1. 7.3
    Derive the affine recursion of Example (see book) from (see book), verify a0=5.023a_0 = 5.023 and V0(10)=90.98V_0(10) = 90.98, and show the slope recursion is the explicit Euler discretization of the Chapter 6 current-value adjoint.
  2. 7.4
    Re-solve Example (see book) on meshes Δ∈{1,1/2,1/4,1/8}\Delta \in \{1, 1/2, 1/4, 1/8\} and confirm first-order convergence of V0ΔV_0^{\Delta} to the continuous optimum 91.6091.60.
  3. 7.5
    In Example (see book), derive the threshold structure analytically: show the keep-minus-replace value gap is increasing in age, so the optimal policy is a threshold, and compute how a∗a^{*} moves with the replacement cost.
  4. 7.6
    Formulate the infinite-horizon version of Example (see book) with stationary demand growth handled by a detrended state, and exhibit its stationary expansion policy.

C. Computational exercises

  1. 7.7
    Prove the Berge propagation step used in Theorem (see book): under (A1), if WW is continuous and bounded then BW\Bop W is continuous, and the greedy correspondence is upper hemicontinuous with nonempty compact values.
  2. 7.8
    Prove that value iteration's iterates from V0≡0V_0 \equiv 0 are monotone nondecreasing when R≥0R \ge 0, and combine with Theorem (see book)(i) to conclude uniform convergence from below with computable two-sided brackets.

D. Enterprise applications

  1. 7.9
    Implement VI and PI on Example (see book); reproduce 279279 sweeps versus 22 iterations, plot the Figure (see book) envelope, and time both as β↑1\beta \uparrow 1 over {0.9,0.95,0.99}\{0.9, 0.95, 0.99\}.
  2. 7.10
    Implement fitted value iteration for Example (see book) with a quadratic-in-xx architecture; confirm the fit is exact (bound of Prop.
  3. 7.11
    Formulate Meridian's restructuring program (Example 5.
  4. 7.12 ★
    Develop average-reward dynamic programming for going-concern enterprises where discounting is contested: state the average-reward Bellman equation with gain and bias, prove its verification counterpart to Theorem (see book) under unichain assumptions, and compare the resulting replacement threshold with the discounted one across β↑1\beta \uparrow 1.

Downloads

All three companions consume the same seeded engine (26207), so their numbers agree by construction — the MFMF convention, carried forward.