Res 351: The Art Of Optimization Is Simply Defined ✓ Solved
Res 351 Eagthe Art Of Optimizationoptimization Is Simply Defined As
Res 351 Eagthe Art Of Optimizationoptimization Is Simply Defined As
RES 351 EAG The Art of Optimization* Optimization is simply defined as the best resource allocation given a set of definable objectives. There are many types of models that apply in this aspect of management science. Some “classes” of this are product mix, ingredients mix (blend), personnel assignment, transportation and warehousing, inventory purchases, and networks. If you notice, these factors have a lot to do with supply chains and logistics management. Therefore, there is much application that can be done in assisting a business environment.
Perhaps this is not directly on a “glorious” web page of E-Business/M-Commerce, but it sure might make a difference in the “behind the scenes” in regular Business or E-Business management. In truth, regardless of storefront, any business needs solid decisions. Managers need to maximize profits at minimal cost and use the resources to the best competitive advantage. Business managers often know how to define business objectives in written language. The key to modeling this business acumen is to write out the statement in plain English, then translate it into the mathematical equivalent.
The case below is a simple entrepreneur making some clay bowls and mugs given a week of production. First, start out with an objective: For a week of production, the company can make four dollars profit ($4) per bowl produced and five dollars ($5) per mug. Declaring two types of products means two mathematical unknowns. Since one does not know how many to make yet, the goal is to decide. Typically this would mean “x” bowls and “y” mugs.
To keep in a ready-to-expand form, use X1 and X2. This can expand to mean Xi where i could represent any number of products so you can see the flexibility now introduced (e.g., i=1,2,3,4…). Now, what does this mean from a mathematical standpoint? Let maximum profit = Z (total profit to solve).
Decision variables:
- X1 = number of bowls made
- X2 = number of mugs made
This starts out: Z= $4×X1 + $5×X2.
Next, our model needs to know limits or constraints on the objective. Constraints are simply when the given resource will be used to capacity.
In the storage shed, the entrepreneur notices there are 120 lbs of clay, and there is one work week (40 hours) to make a production run.
The constraints in written language are:
1. It takes four pounds of clay per bowl and three pounds of clay per mug.
2. It takes one hour to make a bowl and two hours to make a mug.
The mathematical equivalent of the above statements on the left-hand side of the constraint equations are:
- For clay: 4×X1 + 3×X2
- For time: 1×X1 + 2×X2
The right side of the equations represents the resource limits:
- 120 lbs of clay
- 40 hours of work
Expressed mathematically with “≤” notation:
1. 4×X1 + 3×X2 ≤ 120
2. 1×X1 + 2×X2 ≤ 40
Thus, the entire simple model becomes:
Maximize Z = 4×X1 + 5×X2
Subject to:
- 4×X1 + 3×X2 ≤ 120
- 1×X1 + 2×X2 ≤ 40
To implement this model in Excel, you need to declare cells for input values and formulas for the optimizer (Solver).
For example:
- Decision variables (X1 and X2) are input into cells, say C2 and D2.
- The target cell, representing total profit Z, is in B2, with the formula: =4C2 + 5D2
- Constraints are calculated via formulas in other cells:
- Clay constraint: =4C2 + 3D2 ≤ 120
- Time constraint: =1C2 + 2D2 ≤ 40
The Solver add-in can then be used to determine the optimal values for C2 and D2 that maximize B2 while respecting the constraints.
Installation of Solver:
- Go to Excel options.
- Navigate to Add-ins.
- Click “Go” at the bottom.
- Check the box for Solver Add-in and click “OK.”
- The Solver option then appears in the Data tab.
Using Solver:
- Set the target cell (B2) to Max.
- Set changing cells (C2 and D2).
- Add the constraints:
- Cell calculating clay constraint ≤ 120
- Cell calculating time constraint ≤ 40
- Click “Solve” to find the optimal solution.
This process illustrates the practical application of linear programming in resource allocation to maximize profit in manufacturing scenarios.
Understanding this methodology enables better decision-making in managing limited resources efficiently across various business models, including supply chain optimization, production scheduling, and logistics planning.
References:
1. Winston, W. L. (2004). Operations Research: Applications and Algorithms. Brooks/Cole.
2. Hillier, F. S., & Lieberman, G. J. (2010). Introduction to Operations Research. McGraw-Hill.
3. Taha, H. A. (2017). Operations Research: An Introduction. Pearson.
4. Ravindran, A., Phillips, D. T., & Solberg, J. J. (2008). Operations Research: Principles and Practice. John Wiley & Sons.
5. Nemhauser, G. L., & Wolsey, L. A. (1988). Integer and Combinatorial Optimization. John Wiley & Sons.
6. Özoguz, A. (2019). Linear Programming and Its Applications. IntechOpen.
7. Pinedo, M. (2016). Scheduling: Theory, Algorithms, and Systems. Springer.
8. Hillier, F., & Lieberman, G. (2015). Operations Research. McGraw-Hill Education.
9. Bazaraa, M. S., Jarvis, J. J., & Sherali, H. D. (2010). Linear Programming and Network Flows. Wiley.
10. Li, L., & Gans, N. (2017). Supply Chain Network Design and Optimization. Springer.