2. Strategic Allocation of Resources
Chapter 8
Strategic Allocation
of Resources

3. Overview Introduction Recognizing Linear Programming (LP) Problems
Formulating LP Problems
Solving LP Problems
Real LP Problems
Interpreting Computer Solutions of LP Problems
Wrap-Up: What World-Class Companies Do

4. Introduction
Today many of the resources needed as inputs to operations are in limited supply.
Operations managers must understand the impact of this situation on meeting their objectives.
Linear programming (LP) is one way that operations managers can determine how best to allocate their scarce resources.

5. Linear Programming (LP) in OM
There are five common types of decisions in which LP may play a role
Product mix
Ingredient mix
Transportation
Production plan
Assignment

6. LP Problems in OM: Product Mix
Objective
To select the mix of products or services that results in maximum profits for the planning period
Decision Variables
How much to produce and market of each product or service for the planning period
Constraints
Maximum amount of each product or service demanded; Minimum amount of product or service policy will allow; Maximum amount of resources available

7. LP Problems in OM: Ingredient Mix
Objective
To select the mix of ingredients going into products that results in minimum operating costs for the planning period
Decision Variables
How much of each ingredient to use in the planning period
Constraints
Amount of products demanded; Relationship between ingredients and products; Maximum amount of ingredients and production capacity available

8. LP Problems in OM: Transportation
Objective
To select the distribution plan from sources to destinations that results in minimum shipping costs for the planning period
Decision Variables
How much product to ship from each source to each destination for the planning period
Constraints
Minimum or exact amount of products needed at each destination; Maximum or exact amount of products available at each source

9. LP Problems in OM: Production Plan
Objective
To select the mix of products or services that results in maximum profits for the planning period
Decision Variables
How much to produce on straight-time labor and overtime labor during each month of the year
Constraints
Amount of products demanded in each month; Maximum labor and machine capacity available in each month; Maximum inventory space available in each month

10. LP Problems in OM: Assignment
Objective
To assign projects to teams so that the total cost for all projects is minimized during the planning period
Decision Variables
To which team is each project assigned
Constraints
Each project must be assigned to a team; Each team must be assigned a project

11. Recognizing LP Problems
Characteristics of LP Problems in OM
A well-defined single objective must be stated.
There must be alternative courses of action.
The total achievement of the objective must be constrained by scarce resources or other restraints.
The objective and each of the constraints must be expressed as linear mathematical functions.

12. Steps in Formulating LP Problems
1. Define the objective.
2. Define the decision variables.
3. Write the mathematical function for the objective.
4. Write a 1- or 2-word description of each constraint.
5. Write the right-hand side (RHS) of each constraint.
6. Write <, =, or > for each constraint.
7. Write the decision variables on LHS of each constraint.
8. Write the coefficient for each decision variable in each constraint.

13. Example: LP Formulation
Cycle Trends is introducing two new lightweight bicycle frames, the Deluxe and the Professional, to be made from aluminum and steel alloys. The anticipated unit profits are $10 for the Deluxe and $15 for the Professional.
The number of pounds of each alloy needed per frame is summarized on the next slide. A supplier delivers 100 pounds of the aluminum alloy and 80 pounds of the steel alloy weekly. How many Deluxe and Professional frames should Cycle Trends produce each week?

14. Example: LP Formulation
Aluminum Alloy Steel Alloy
Deluxe
Professional

15. Example: LP Formulation
Define the objective
Maximize total weekly profit
Define the decision variables
x1 = number of Deluxe frames produced weekly
x2 = number of Professional frames produced weekly
Write the mathematical objective function
Max Z = 10x1 + 15x2

16. Example: LP Formulation
Write a one- or two-word description of each constraint
Aluminum available
Steel available
Write the right-hand side of each constraint
100 80
Write <, =, > for each constraint
< 100
< 80

17. Example: LP Formulation
Write all the decision variables on the left-hand side of each constraint
x1 x2 < 100
x1 x2 < 80
Write the coefficient for each decision in each constraint
+ 2x1 + 4x2 < 100
+ 3x1 + 2x2 < 80

18. LP Problems in General
Units of each term in a constraint must be the same as the RHS
Units of each term in the objective function must be the same as Z
Units between constraints do not have to be the same
LP problem can have a mixture of constraint types

19. Solving LP Problems
Graphical Solution Approach - used mainly as a teaching tool
Simplex Method - most common analytic tool
Transportation Method - one of the earliest methods
Assignment Method - occasionally used in OM

20. Graphical Solution Approach
1. Formulate the objective and constraint functions.
2. Draw the graph with one variable on the horizontal axis and one on the vertical axis.
3. Plot each of the constraints as if they were lines or equalities.
4. Outline the feasible solution space.
. . . more

21. Graphic Solution Approach
5. Circle the potential solution points -- intersections on the inner (minimization) or outer (maximization) perimeter of the feasible solution space.
6. Substitute each of the potential solution point values of the two decision variables into the objective function and solve for Z.
7. Select the solution point that optimizes Z.

22. Example: Graphical Solution Approach
3x1 + 2x2 < 80 (Steel)
Potential Solution Point
x1 = 0, x2 = 25, Z = $375.00 40 35
Optimal Solution Point
x1 = 15, x2 = 17.5, Z = $412.50 30 25
2x1 + 4x2 < 100 (Aluminum) 20
Feasible
Solution
Space
MAX 10x1 + 15x2 (Profit) 15 10
Potential Solution Point
x1 = 26.67, x2 = 0, Z = $266.67 5 x1

23. Simplex Method Examples of standard LP computer programs that use
the simplex method are:
IBM’s Optimization Solutions Library
POM Software Library
GAMS
MPL for Windows
Solver -- available within spreadsheet packages such as MicrosoftExcel, Lotus 1-2-3, and Quattro Pro
What’s Best! and Premium Solver -- add-ons to common spreadsheet packages

24. Example: Excel/Solver Solution
Partial Spreadsheet Showing Problem Data

25. Example: Excel/Solver Solution
Partial Spreadsheet Showing Formulas

26. Example: Excel/Solver Solution
Partial Spreadsheet Showing Solution

27. Transportation Method
This method can solve a special form of LP problem, including the classical transportation problem, with these typical characteristics:
m sources and n destinations
number of variables is m x n
number of constraints is m + n (constraints are for source capacity and destination demand)
costs appear only in objective function (objective is to minimize total cost of shipping)
coefficients of decision variables in the constraints are either 0 or 1

28. Example: Transportation LP
National Packaging has plants in Tulsa, Memphis, and Detroit that ship to the firm’s warehouses in San Diego, Norfolk, and Pensacola. The three warehouses require at least 4,000, 2,500, and 2,500 pounds of cardboard per week, respectively. The plants each have 3,000 pounds of cardboard per week available for shipment.
The shipping cost per pound from each plant to each warehouse is shown on the next slide. Formulate and solve an LP to determine the shipping arrangements (source, destination, and quantity) that will minimize the total weekly shipping cost.

29. Example: Transportation LP
Destination
Source San Diego Norfolk Pensacola
Tulsa $ $ $ 5
Memphis
Detroit

30. Example: Transportation LP
Define the objective
Minimize the total weekly shipping cost
Define the decision variables
There are m x n = 3 x 3 = 9 decision variables.
xij represents the number of pounds of cardboard to be shipped from plant i to warehouse j.
San Diego Norfolk Pensacola
Tulsa x x x13
Memphis x x x23
Detroit x x x33

31. Example: Transportation LP
Write the mathematical function for the objective
Min Z = 12x11 + 6x12 + 5x x x22
+ 9x x x x33

32. Example: Transportation LP
Write the constraints
(1) x11 + x12 + x13 < (Plant 1 capacity in pounds)
(2) x21 + x22 + x23 < (Plant 2 capacity in pounds)
(3) x31 + x32 + x33 < (Plant 3 capacity in pounds)
(4) x11 + x21 + x31 > (Wareh. 1 demand in pounds)
(5) x12 + x22 + x32 > (Wareh. 2 demand in pounds)
(6) x13 + x23 + x33 > (Wareh. 3 demand in pounds)

33. Assignment Method
This method can solve a special form of LP problem, including the classical assignment problem, with these typical characteristics:
is a special case of a transportation problem
the right-hand sides of constraints are all 1
the signs of the constraints are = rather than < or >
the value of all decision variables is either 0 or 1

34. Example: Assignment LP
A sprinkler system installation company has three residential projects to complete and three work teams available and capable of completing the projects. The work teams are not equally efficient at completing a particular project.
Shown on the next slide are the estimated labor-hours required for each team to complete each project. How should the work teams be assigned in order to minimize total labor-hours?

35. Example: Assignment LP
Projects
Work Team A B C
Alice, Ted
Gary, Marv
Tina, Sam

36. Real LP Problems Real-world LP problems often involve:
Hundreds or thousands of constraints
Large quantities of data
Many products and/or services
Many time periods
Numerous decision alternatives
… and other complications

37. Computer Solutions of LP Problems
Computer output typically contains:
Original-problem formulation
Basis variables (variables in the final solution) and their values
Slack-variable values -- slack represents the amount of a scarce resource that is not used by the decision variables
Shadow prices -- which indicate the impact on Z if the constraints’ right-hand sides are changed
… more

38. Computer Solutions of LP Problems
Computer output typically contains:
Ranges of right-hand side values for which the shadow prices are valid
For each nonbasic variable: amount Z is reduced (in a MAX problem) or increased (in a MIN problem) for one unit of x in the solution

39. Wrap-Up: World-Class Practice
Managers at all levels use LP to solve complex problems and aid in decision making
Some companies establish formal operations research, management science, or operations analysis departments
Other companies employ consultants
LP is applied to long, medium, and short range decision making

40. End of Chapter 8