CPE 613 Lecture 16 Study Notes

Problem Formulation in Process Design: Scheduling and Retrofit

• Multi-level optimization in the chemical industries.

  • Focus on various problems that need addressing in the current market.

  • Brief discussion of scheduling and planning examples to illustrate key concepts.

• Question of the Day: Discuss the relevance of these concepts in real-world applications.

Scheduling and Planning

• Process systems engineering encompasses the crucial areas of planning and scheduling.

  • Key problems to solve within this domain include:

  • Batch Planning:

    • Determine what to make, how much to produce, and when production should occur.

  • Logistics:

    • Ascertain the volume of goods to ship and their destinations.

  • Business Planning:

    • Decide locations for new plants and the number of plants to build.

Business and Engineering

• The term Operations Research refers to the optimization of supply chains from a business perspective.

• Inclusion of multiple objectives leads to Enterprise-Wide Optimization:

  • Goals include:

  • Minimization of Overall Carbon Footprint:

    • Focus on reducing environmental impact.

  • Seizing Business Opportunities:

    • Explore options to expand product offerings.

  • Achieving Economies of Scale:

    • Boost cost-efficiency through increased production volume.

Example: Scheduling and Planning

• Engineers are critical for formulating scheduling and planning problems effectively.

• Example case: A petrochemical company operates two refineries located in Los Angeles and Houston.

  • Distribution Goal: Jet fuel is to be shipped to four tank farms for further delivery to airports situated in Denver, Kansas City, Nashville, and Pittsburgh.

Scheduling Example: Data

• Production Capacities:

  • Los Angeles Refinery: 150,000 bbl/week.

  • Houston Refinery: 200,000 bbl/week.

• Demand at Each Tank Farm:

  • Denver: 100,000 bbl/week.

  • Kansas City: 80,000 bbl/week.

  • Nashville: 60,000 bbl/week.

  • Pittsburgh: 90,000 bbl/week.

Scheduling Example: Costs

• Shipping Costs in dollars per 100,000 bbl shipped:

  • To/from Tank Farms:

  • From Los Angeles:

    • Denver: 4

    • Kansas City: 7

    • Nashville: 9

    • Pittsburgh: 11

  • From Houston:

    • Denver: 5

    • Kansas City: 4

    • Nashville: 5

    • Pittsburgh: 8

Scheduling: Problem Statement

• Objective Question: How many shipments of jet fuel (in increments of 100,000 bbl) should be dispatched from each refinery to each tank farm?

  • Goal: Meet all demand while minimizing the total shipping cost.

• Steps to formulate the problem:

  • Define All Variables: Assign terms for easy reference.

  • Write an Objective Function: Create a mathematical expression for cost minimization.

  • Write Down All Constraints: Include limitations based on production capacity and demand.

Variable Definitions

• Definitions of Variables:

  • Refineries:

  • Refinery 1: Los Angeles (i=1)

  • Refinery 2: Houston (i=2)

  • Tank Farms:

  • Tank Farm 1: Denver (j=1)

  • Tank Farm 2: Kansas City (j=2)

  • Tank Farm 3: Nashville (j=3)

  • Tank Farm 4: Pittsburgh (j=4)

  • Define $x_{ij}$ as the number of 100,000 bbl shipments from refinery i to tank farm j.

  • Define $c_{ij}$ as the known cost to ship a 100,000 bbl shipment from refinery i to tank farm j.

Objective Function

• Objective Function: Minimize C

  • Mathematical Expression:

  • $C = ext{Total Cost} = ext{Sum of } (c_{ij} imes x_{ij})$ where the sum is taken over all i, j combinations.

Constraints

• Constraints to be satisfied:

  • Production Limits:

  • $x_{11} + x_{12} + x_{13} + x_{14} \leq 150$ (for Los Angeles)

  • $x_{21} + x_{22} + x_{23} + x_{24} \leq 200$ (for Houston)

  • Satisfy Demands:

  • $x_{11} + x_{21} \geq 100$ (for Denver)

  • $x_{12} + x_{22} \geq 80$ (for Kansas City)

  • $x_{13} + x_{23} \geq 60$ (for Nashville)

  • $x_{14} + x_{24} \geq 90$ (for Pittsburgh)

  • Non-Negativity Constraints:

  • $x_{ij} \geq 0 ext{ for all } i=1,2 ext{ and } j=1,2,3,4$

Mathematical Formulation

• Characterization of the Problem:

  • This problem is classified as a Linear Program (LP),

  • It is characterized by having a linear objective function and linear constraints.

  • Uniqueness: There is a single, globally optimal solution due to the nature of LPs.

Conclusions

• Optimization theory offers valuable methods for addressing large-scale scheduling and planning challenges in the chemical industry.

• Emphasizing enterprise-wide approaches can lead to enhanced efficiencies that are unattainable through smaller, localized solutions.

• Even problems involving multiple objectives can be systematically approached using optimization techniques.