Operations Research - Gunadarma

Operations Research

Buku

• “Introduction to Operations Research” by F.S. Hillier and G.J. Lieberman, 8th

• Edition, McGraw‐Hill, 2005

Objectives

• This course is an introduction to the principles and the practice of Operations Research, and its role in human decision making. In particular, the course focuses on mathematical programming techniques such as linear programming (the Simplex Method, concepts of duality and sensitivity analysis), network optimization (transportation and assignment problems), and dynamic, nonlinear, and integer programming.

What is Linear Programming?

• Linear Programming provides methods for allocating limited resources among competing activities in an optimal way.– Linear → All mathema cal func ons are linear– Programming → Involves the planning of ac vi es

• Any problem whose model fits the format for the linear programming model is a linear programming problem.

Example 3.1 – Wyndor Glass Co.

• The company produces glass products and owns 3 plants. Management decides to produce two new products.

• Product 1– 1 hour production time in Plant 1– 3 hours production time in Plant 3– $3,000 profit per batch

• Product 2– 2 hours production time in Plant 2– 2 hours production time in Plant 3– $5,000 profit per batch


Production time available each week– Plant 1: 4 hours– Plant 2: 12 hours– Plant 3: 18 hours


Plant

Production Time per Batch (Hours) Production TimeAvailable per Week

(Hours)Product

1 2

1 1 0 4

2 0 2 12

3 3 2 18

Profit per batch $3,000 $5,000

Maximize Z = 3x1 + 5x2, Subject to:x1 ≤ 4 2x2 ≤ 12 3x1 + 2x2 ≤ 18 and x1 ≥ 0, x2 ≥ 0


General Linear Programming Problems

• Allocating resources to activities

Example General

Production capacities of plants3 plants

Resourcesm resources

Production of products2 ProductsProduction rate of product j, xj

Activitiesn activitiesLevel of activity j, xj

Profit Z Overall measure of performance Z



Other Forms of Linear Programming Problems

• Minimize rather than maximize objective function– Minimize Z = c1x1 + c2x2 + ... + cnxn

• Some function constraints with greater‐than‐or‐equal‐to (≥)– ai1x1 + ai2x2 + ... + ainxn ≥ bi for some value of i

• Some functional constraints in equation form– ai1x1 + ai2x2 + ... + ainxn = bi for some value of i

• Deleting non‐negativity constraints– xj unrestricted in sign for some value of j

Linear Programming Solutions

• Solution – Any specification of values for the decision variables (xj)

• Feasible solution – A solution for which all constraints are satisfied

• Infeasible solution – A solution for which at least one constraint is violated

• Feasible region – The collection of all feasible solutions

• Optimal solution – A feasible solution that has the most favorable value of the objective function





Linear Programming Assumptions

• Proportionality – The contribution of each activity to Z or a constraint is proportional to the level of activity xj– Z = 3x1 – 1 + 5x2

• Additivity – Every function is the sum of the individual contributions of the activities– Z = 3x1 + 5x2 + x1x2

• Divisibility – Decision variables are allowed to have any value, including non‐integer values

• Certainty – The value assigned to each parameter is assumed to be a known constant

Another Example• 3.1‐6 – The Whitt Window Company is a company withonly three employees which makes two different kindsof hand‐crafted windows: a wood‐framed and analuminum‐framed. They earn $60 profit for each wood‐framed window and $30 for each aluminum‐framedwindow. Doug makes the wood frames, and can make6 per day. Linda makes the aluminum frames, and canmake 4 per day. Bob forms and cuts the glass, and canmake 48 square feet of glass per day. Each wood‐framed window uses 6 square feet of glass and eachaluminum‐framed window uses 8 square feet of glass.

• How many windows of each type should be producedper day to maximize profit?

Operations Research - Gunadarma

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