Case study: Nut supplier Supply Chain assignment 代写

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Case study: Nut supplier Supply Chain assignment 代写

Case study: Nut supplier

The firm makes four different products from almond nuts grown locally in

Australia: raw nuts (Raw), roasmixed nuts (Roasted), roasted and salted

mixed nuts (Salted), and chili-coated mixed nuts (Chili)The firm is barely able to

keep up with the increasing demand for these products. However, increasing raw

material prices and foreign competition are forcing the firm to watch its margins

to ensure it is operating in-the most efficient manner possible. To mee marketing demands for the coming week, the firm needs to produce at leas 1,500 kg of the Raw product, between 600 and 700 kg of the Roasted product, at

least 500 kg of the Salted product, and no more than 400 kg of the Chili product.

Each kilogram of the Raw, Roasted, Salted, and Chili product contains,

respectively, 100%, 70%, 55%, and 40% almond nuts with the remaining weight

made up of other nuts. The company has 3000 kg of almond nuts and 1000 kg of

othnuts for use in the next week. The various products are made using four

different machines that hull the nuts, roast the nuts, coat the nuts in chili (if

needed), and package the products. The following table summarises the required by each product on each machine. Each machine has 70 hours of time

available in the coming week.

Minutes required per kg

Machine Raw Roasted Salted Chili

Hulling 1.00 1.00 1.00 1.00

Roasting 0.00 1.00 1.75 1.50

Coating 0.00 0.00 0.00 1.00

Packaging 1.50 1.50 1.25 1.00

The controller recently presented management with the following financial

summary of the firm’s average weekly operations over the past quarter.

Product

Raw Roasted Salted Chili Total

Sales Revenue $5,412 $1,846 $623 $1,165 $9,046

Variable Costs

Direct materials $1,331 $560 $144 $320 $2,355

Direct labour $1,092 $400 $96 $130 $1,718

Manufacturing overhead $333 $140 $36 $90 $599

Selling & Administrative $540 $180 $62 $120 $902

Allocated Fixed Costs

Manufacturing overhead $688 $331 $99 $132 $1,250

Selling & Administrative $578 $278 $83 $111 $1,050

Units Sold 1040 500 150 200 1890

1. Formulate an LP model for this problem to maximise the total profit and

briefly describe each equation.

2Cree a spreadsheet model for this problem and solve it using Solver.

3. What is the optimal solution? You must report inputs, output, and

required resources.

4. Create a sensitivity report for this solution, and answer the following

questions:

a. If the firm wanted to decrease the production on any product,

which one would you recommend and why?

b. If the firm wanted to increase the production of any product,

which one would you recommend and why?

c. Which resources are preventing the firm from making more

money? If the fm could acquire more of this resource, how much

should it acquire and how much should it be willing to pay to

acquire this resource?

d. If the marketing department wanted to decrease the price of the

Roasted product by $1, would the optimal solution change and

why?

Supply Chain Analysis & Design

Topic 2

Linear Programming:

Concepts and Model

Formulation

An Introduction to Linear Programming

• Linear Programming Problem

• Problem Formulation

• A Simple Maximization Problem

• Application of LP models

• Revenue Management

Linear Programming

• Linear programming has nothing to do with computer

programming.

• The use of the word “programming” here means

“choosing a course of action.”

• Linear programming involves choosing a course of

action when the mathematical model of the problem

contains only linear functions.

Linear Programming (LP) Problem

• The maximization or minimization of some quantity is

the objective in all linear programming problems.

• All LP problems have constraints that limit the degree

to which the objective can be pursued.

• A feasible solution satisfies all the problem's

constraints.

• An optimal solution is a feasible solution that results in

the largest possible objective function value when

maximizing (or smallest when minimizing).

• A graphical solution method can be used to solve a

linear program with two variables.

Linear Programming (LP) Problem

• If both the objective function and the constraints are

linear, the problem is referred to as a linear

programming problem.

• Linear functions are functions in which each variable

appears in a separate term raised to the first power and

is multiplied by a constant (which could be 0).

• Linear constraints are linear functions that are

restricted to be "less than or equal to", "equal to", or

"greater than or equal to" a constant.

LP Model

0 ,

180 5

100 2 3

150 5 3

4 2

2 1



 

 

x x

to Subject

x x Z Max

Constraints

Objective Function

Non negativity constraints

Problem Formulation

• Problem formulation or modeling is the process of

translating a verbal statement of a problem into a

mathematical statement.

• Formulating models is an art that can only be mastered

with practice and experience.

• Every LP problems has some unique features, but most

problems also have common features.

Formulating LP Models

• Formulating linear programming models involves the

following steps:

1. Understand the problem thoroughly.

2. Determine the objective function.

3. Describe each constraint.

4. Define the decision variables (i.e. controllable

inputs).

5. Write the objective function in terms of the decision

variables.

6. Write the constraints in terms of the decision

variables.

Example 1

a : Giapetto’s Woodcarving

• Giapetto’s, Inc., manufactures wooden soldiers and trains.

1.Each soldier built:

• Sell for $36 and uses $19 worth of raw materials.

• Increase Giapetto’s variable labor/overhead costs by $14.

• Requires 2 hours of finishing labor.

• Requires 1 hour of carpentry labor.

2.Each train built:

• Sell for $21 and used $9 worth of raw materials.

• Increases Giapetto’s variable labor/overhead costs by $10.

• Requires 1 hour of finishing labor.

• Requires 1 hour of carpentry labor.

example from Operations Research Applications and Algorithms, Wayne L. Winston (2004), 4 th edition, Cengage Learning,

ISBN- 13: 978-0-534-38058-8

Ex. 1 - continued

• Each week Giapetto can obtain:

• All needed raw material.

• Only 100 finishing hours.

• Only 80 carpentry hours.

• Demand for the trains is unlimited.

• At most 40 soldiers are bought each week.

• Giapetto wants to maximize weekly profit (revenues –

costs).

• Formulate a mathematical model of Giapetto’s

situation that can be used maximize weekly profit.

Example 1: Solution

• The Giapetto solution model incorporates the

characteristics shared by all linear programming

problems.

• Decision variables should completely describe the

decisions to be made.

•x 1 = number of soldiers produced each week

•x 2 = number of trains produced each week

• The decision maker wants to maximize (usually

revenue or profit) or minimize (usually costs) some

function of the decision variables. This function to

be maximized or minimized is called the objective

function.

•For the Giapetto problem, fixed costs do not

depend upon the values of x 1 or x 2 .

12

Case study: Nut supplier Supply Chain assignment 代写

Ex. 1 - Solution continued

• Giapetto’s weekly profit can be expressed in terms of

the decision variables x 1 and x 2 :

Weekly profit =

weekly revenue – weekly raw material costs – the

weekly variable costs = 3x 1 + 2x 2

• Thus, Giapetto’s objective is to choose x 1 and x 2 to

maximize weekly profit. The variable z denotes the

objective function value of any LP.

• Giapetto’s objective function is

Maximize z = 3x 1 + 2x 2

• The coefficient of an objective function variable is

called an objective function coefficient.

Ex. 1 - Solution continued

• As x 1 and x 2 increase, Giapetto’s objective function

grows larger.

• For Giapetto, the values of x 1 and x 2 are limited by the

following three restrictions (often called constraints):

• Each week, no more than 100 hours of finishing time

may be used. (2 x 1 + x 2 ≤ 100)

• Each week, no more than 80 hours of carpentry time

may be used. (x 1 + x 2 ≤ 80)

• Because of limited demand, at most 40 soldiers

should be produced (x 1 ≤ 40)

• The coefficients of the decision variables in the constraints are

called the technological coefficients. The number on the right-

hand side of each constraint is called the constraint’s right-hand

side (or rhs).

• To complete the formulation of a linear programming problem,

the following question must be answered for each decision

variable.

• Can the decision variable only assume nonnegative values, or is the

decision variable allowed to assume both positive and negative

values?

• If the decision variable can assume only nonnegative values, the sign

restriction x i ≥ 0 is added.

• If the variable can assume both positive and negative values, the

decision variable x i is unrestricted in sign (often abbreviated urs).

Ex. 1 - Solution continued

• For the Giapetto problem model, combining the sign

restrictions x 1 ≥0 and x 2 ≥0 with the objective function

and constraints yields the following optimization

model:

Max z = 3x 1 + 2x 2 (objective function)

Subject to (s.t.)

2 x 1 + x 2 ≤ 100 (finishing constraint)

x 1 + x 2 ≤ 80 (carpentry constraint)

x 1 ≤ 40 (constraint on demand for soldiers)

x 1 ≥ 0 (sign restriction)

x 2 ≥ 0 (sign restriction)

Revenue Management

• Another LP application is revenue management.

• Revenue management involves managing the short-

term demand for a fixed perishable inventory in

order to maximize revenue potential.

• The methodology was first used to determine how

many airline seats to sell at an early-reservation

discount fare and how many to sell at a full fare.

• Application areas now include hotels, apartment

rentals, car rentals, cruise lines, and golf courses.

Question?

Case study: Nut supplier Supply Chain assignment 代写