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Optimization – Part I Applications of Optimization To Operations Management

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Applications of Optimization

To Operations Management

- For this session, the learning objectives are:
- Learn what a Linear Program is.
- Learn how to formulate a Linear Program and solve it using Excel’s Solver.
- Using Solver to solve a Make-or-Buy Problem.
- Using Solver to solve a Transshipment Problem (Product Distribution).

- Optimization involves the maximization or minimization of an objective subject to a set of constraints.
- Every copy of Microsoft Excel includes Solver, which enables you to solve the following types of optimization problems:
- a Linear Program,
- an Integer Linear Program,
- a Nonlinear Program.

The next page summarizes the use of Excel’s Solver.

DuPunt, Inc. manufactures three types of chemicals. For the upcoming month, DuPunt has contracted to supply its customers with the following amounts of the three chemicals:

DuPunt’s production is limited by the availability of processing time in two chemical reactors. Each chemical must be processed first in Reactor 1 and then in Reactor 2. The following table provides the hours of processing time available next month for each reactor and the processing time required in each reactor by each chemical:

Because of the limited availability of reactor processing time, DuPunt has insufficient capacity to meet its demand with in-house production. Consequently, DuPunt must purchase some chemicals from vendors having excess capacity and resell them to its own customers. The following table provides each chemical’s in-house production cost and outside purchase cost:

DuPunt’s objective is to fill its customers’ orders with the cheapest combination of in-house production and outside purchases. In short, DuPunt must decide how much of each chemical to produce in-house (i.e., “make”) and how much of each chemical to purchase outside (i.e.,”buy”).

Formulation of the Make-or-Buy Problem as a Linear Program

Define the following 6 decision variables:

Minimize Total Costs

Contracted Sales

Reactor Availabilities

Nonnegativity Constraints

- Consider a firm that for simplicity produces a single product. The firm has 3 plants (Tokyo, Hong Kong, and Bangkok), 2 warehouses (Seattle and Los Angeles), and 4 customers (Chicago, New York, Atlanta, and Dallas) geographically dispersed as diagrammed below. The firm ships its product from a plant to a warehouse and then on to a customer. In the diagram below:
- The number to the left of each plant represents the plant’s supply.
- The number to the right of each customer represents the customer’s demand.
- The number appearing along an arrow from a plant to a warehouse or from a warehouse to a customer represents the corresponding unit shipping cost. For example, the unit shipping cost from Bangkok to Seattle is $25 per unit.
- The firm wants to distribute its product at minimum cost.

72 = Total Supply

Total Demand = 70

Formulation of the Transshipment Problem as an LP

Let AZ denote the amount shipped from location A to location Z. As examples, TL denotes the amount shipped from Tokyo to Los Angeles, and SN denotes the amount shipped from Seattle to New York.

Min Total Shipping Costs

Supply Constraints

Transshipment Constraints

Demand Constraints

Nonnegativity Constraints

Solving the Transshipment LP Using Excel’s Solver

The Spreadsheet Before Optimization

Solving the Transshipment LP Using Excel’s Solver

The Spreadsheet After Optimization

INTRODUCTION TO THE BLENDING PROBLEM

In many businesses and industrial environments, the goal is to find the optimal “recipe” for blending a variety of “ingredients” to obtain a product that meets lower and/or upper limits on a variety of characteristics. The table below summarizes several applications.

The exercise below is designed to review the “basics” of formulating a linear program and solving it using Solver.

- Harrus Feeding Company’s Blending Problem
- The Harrus Feeding Company (HFC) operates a feedlot to which cattle are brought for the final fattening process. Since HFC’s cattle population averages about 100,000, it is important for HFC to feed the cattle in a way that meets their nutritional requirements at minimum cost.
- The mixture HFC feeds the cattle is blend of four feedstuffs: corn, wheat, barley, and hay. The table below provides the relevant dietary and cost data per pound of each feedstuff, along with a steer’s daily nutritional requirement. For example, for each pound of corn a steer consumes, it receives 2 grams of fat, 20 grams of protein, 4 milligrams of iron, and 200 calories.
- Assuming a steer’s daily consumption of feedstuffs must be exactly 24 pounds, formulate and solve a linear program for determining the dietary blend that satisfies HFC’s daily requirements at minimum cost.
- How would you modify your formulation if a steer’s daily consumption of feedstuffs must be in the range of 23-25 pounds?
- How would you modify your formulation if there were no daily limit on the pounds of feedstuffs that a steer must consume?
- Can the formulations in part (a) and part (c) result in distinct optimal solutions? Can you anticipate a potential problem with the optimal solution to the linear program in part (c)?

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