you have a set of legos 8 small bricks 6 large bricks these are your “raw materials”. you have...

You have a set of legos

8 small bricks

6 large bricks

These are your “raw materials”.

You have to produce tables and chairs out of these legos. These are your “products”.

The Lego Production Problem

Weekly supply of raw materials:

6 Large Bricks8 Small Bricks

Products:

Chair TableProfit = 15 cents per Chair Profit = 20 cents per Table

The Lego Production Problem

X1 is the number of Chairs

X2 is the number of Tables

Large brick constraint

X1+2X2 6

Small brick constraint

2X1+2X2 8

Objective function is to Maximize

15X1+20 X2

X1 ≥ 0

X2 ≥ 0

Problem Formulation

We can make Product1 and Product2.

There are 3 resources; Resource1, Resource2, Resource3.

Product1 needs one hour of Resource1, nothing of Resource2, and three hours of resource3.

Product2 needs nothing from Resource1, two hours of Resource2, and two hours of resource3.

Available hours of resources 1, 2, 3 are 4, 12, 18, respectively.

Contribution Margin of product 1 and Product2 are $300 and $500, respectively.

Formulate the Problem

Solve the problem using solver in excel

Linear Programming

Objective Function Z = 3 x1 +5 x2

ConstraintsResource 1x1 4 Resource 2 2x2 12Resource 3 3 x1 + 2 x2 18Nonnegativityx1 0, x2 0

Problem Formulation

Given the following problem

Maximize Z = 3x1 + 5x2 Subject to: the following constraints

x1 ≤ 4 2x2 ≤ 123x1 + 2x2 ≤ 18

x1, x2 ≥ 0What combination of x1 and x2 could be the optimal solution?

A) x1 = 4, x2 = 4 B) x1 = -3, x2 = 6 C) x1 = 3, x2 = 4 D) x1 = 0, x2 = 7 E) x1 = 2, x2 = 6

Feasible, Infeasible, and Optimal Solution

Infeasible; Violates Constraint 3Infeasible; Violates nonnegativityFeasible; z = 3×3+ 5×4 = 29Infeasible; Violates Constraint 2

Feasible; z = 3×2+ 5×6 = 36 and Optimal

The Omega Manufacturing Co. has discontinued the production of a certain non-profitable product line. This act created considerable excess capacity. Management is considering devoting this excess capacity to one or more of three products. The hours required from each resource for each unit of product, the available capacity (hours per week) of the the three resources, as well as the profit of each unit of product are given below.

Optimal Product Mix

Sales department indicates that the sales potentials for products 1 and 2 exceeds maximum production rate, but the sales potential for product 3 is 20 units per week.

Formulate the problem and solve it using excel

Total hours avialableProduct1 Product2 Product3

9 3 5 5005 4 0 3503 0 2 150

$50 $20 $25 Profit

Hours used per unit

An appliance manufacturer produces two models of microwave ovens: H and W. Both models require fabrication and assembly work: each H uses four hours fabrication and two hours of assembly, and each W uses two hours fabrication and six hours of assembly. There are 600 fabrication hours this week and 450 hours of assembly. Each H contributes $40 to profit, and each W contributes $30 to profit.

a) Formulate the problem as a Linear Programming problem.

b) Solve it using excel.

c) What are the final values?

d) What is the optimal value of the objective function?

Practice (Page 304, Prob. 3)

A small candy shop is preparing for the holyday season. The owner must decide how many how many bags of deluxe mix how many bags of standard mix of Peanut/Raisin Delite to put up. The deluxe mix has 2/3 pound raisins and 1/3 pounds peanuts, and the standard mix has 1/2 pound raisins and 1/2 pounds peanuts per bag. The shop has 90 pounds of raisins and 60 pounds of peanuts to work with. Peanuts cost $0.60 per pounds and raisins cost $1.50 per pound. The deluxe mix will sell for 2.90 per pound and the standard mix will sell for 2.55 per pound. The owner estimates that no more than 110 bags of one type can be sold.





Practice (Page 304, Prob. 4)

The following table summarizes the key facts about two products, A and B, and the resources, Q, R, and S, required to produce them.

Assignment 6a:1 Due at the beginning of next class

Resource Usage per Unit Produced

Resource Product A Product B Amount of resource available

Q 2 1 2

R 1 2 2

S 3 3 4

Profit/Unit $3000 $2000





The Apex Television Company has to decide on the number of 27” and 20” sets to be produced at one of its factories. Market research indicates that at most 40 of the 27” sets and 10 of the 20” sets can be sold per month. The maximum number of work-hours available is 500 per month. A 27” set requires 20 work-hours and a 20” set requires 10 work-hours. Each 27” set sold produces a profit of $120 and each 20” set produces a profit of $80. A wholesaler has agreed to purchase all the television sets produced if the numbers do not exceed the maximum indicated by the market research.






Ralph Edmund loves steaks and potatoes. Therefore, he has decided to go on a steady diet of only these two foods (plus some liquids and vitamins supplements) for all his meals. Ralph realizes that this isn’t the healthiest diet, so he wants to make sure that he eats the right quantities of the two foods to satisfy some key nutritional requirements. He has obtained the following nutritional and cost information:


Grams of Ingredient per Serving

Ingredient Steak Potatoes Daily Requirements (grams)

Carbohydrates 5 15 ≥ 50

Protein 20 5 ≥ 40

Fat 15 2 ≤ 60

Cost per serving $4 $2

Ralph wishes to determine the number of daily servings (may be fractional of steak and potatoes that will meet these requirements at a minimum cost.

Formulate the problem as a Linear Programming problem. Solve it using excel. What are the final values? What is the optimal value of the objective function?


You are given the following linear programming model in algebraic form, where, X1 and X2 are the decision variables and Z is the value of the overall measure of performance.

Maximize Z = X1 +2 X2

Subject to

Constraints on resource 1: X1 + X2 ≤ 5 (amount available)

Constraints on resource 2: X1 + 3X2 ≤ 9 (amount available)

And

X1 , X2 ≥ 0


a) Identify the objective function, the functional constraints, and the non-negativity constraints in this model.

b) Incorporate this model into a spreadsheet.

c) Is (X1 ,X2) = (3,1) a feasible solution?

d) Is (X1 ,X2) = (1,3) a feasible solution?

e) Use the Excel Solver to solve this model.


You are given the following linear programming model in algebraic form, where, X1 and X2 are the decision variables and Z is the value of the overall measure of performance.

Maximize Z = 3X1 +2 X2

Subject to

Constraints on resource 1: 3X1 + X2 ≤ 9 (amount available)

Constraints on resource 2: X1 + 2X2 ≤ 8 (amount available)

And

X1 , X2 ≥ 0


a) Identify the objective function, the functional constraints, and the non-negativity constraints in this model.

b) Incorporate this model into a spreadsheet.

c) Is (X1 ,X2) = (2,1) a feasible solution?

d) Is (X1 ,X2) = (2,3) a feasible solution?

e) Is (X1 ,X2) = (0,5) a feasible solution?

f) Use the Excel Solver to solve this model.

Example

Purchased Part$5 / unit

RM1$20 per

unit

RM2$20 per

unit

RM3$25 per

unit

$90 / unit110 units / week

$100 / unit60 units / weekP: Q:

D10 min.

D5 min.

C10 min.

C5 min.

B25 min.

A15 min.

B10 min.

A10 min.

Time available at each work center: 2,400 minutes per week

Operating expenses per week: $6,000

A Production System Manufacturing Two Products, P and Q

you have a set of legos 8 small bricks 6 large bricks these are your “raw materials”. you have...

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