Lecture 3: Common Business Applications and Excel Solver

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1 Lecture 3: Common Business Applications and Excel Solver Common Business Applications Linear Programming (LP) can be used for many managerial decisions: - Product mix - Media selection - Marketing research - Portfolio selection - Shipping & transportation - Multi-period scheduling For a particular application we begin with the problem scenario and data, then: 1. Define the decision variables 2. Formulate the LP model using the decision variables - Write the objective function equation - Write each of the constraint equations 3. Implement the model in Excel Solver 4. Solve Common Business Applications Product Mix - Usually involve maximizing profit subject to: - Production resource constraints - Material Availability constraints - Standing orders - Quotas - Maximum or minimum proportions Example 1 Product Mix Imagine that you are managing a factory that is building three products: TV sets, stereos and speakers. Each product is assembled from parts in inventory, and there are five types of parts: Chassis, picture tubes, speaker cones, power supplies and electronics units. Your goal is to produce the mix of products which will maximize profits, given the inventory of products on hand. Assume that you can sell TV sets for a gross profit of $75 each, stereos for a profit of $50 each, and speaker for $35 each. To assemble a TV set, you need 1 chassis, 1 picture tube, 2 speaker cones, 1 power supply and 2 sets of electronics. To make a stereo, you need 1 chassis, 2 speaker cones, 1 power supply and 1 set of electronics. And to build a speaker, all you need is 1 speaker cone and 1 set of electronics. The parts you have on hand are 450 chassis, 250 picture tubes, 800 speaker cones, 450 power supplies and 600 sets of electronics. You can build only a limited number of products from the parts on hand. a) Formulate the LP model to Maximize the profit. b) Solve using Solver Page 1 of 17

Example 1 - Solution Step 1: Define the objective - Maximize the profit Step 2: Define the decision variables x 1 = number of TV sets assembled x 2 = the number of stereos assembled x 3 = the number

2 Example 1 - Solution Step 1: Define the objective - Maximize the profit Step 2: Define the decision variables x 1 = number of TV sets assembled x 2 = the number of stereos assembled x 3 = the number of speakers assembled Step 3: Write the mathematical objective function Maximize Z = 75 x x x 3 Step 4: Formulate the constraints 1 x x (Chassis) 1 x (Picture tubes) 2 x x x (Speaker cones) 1 x x (Power supplies) 2 x x x (Electronics) x 1, x 2, x 3 0 Step 5: Final Formulation Maximize Z = 75 x x x 3 S.t: 1 x x (Chassis) 1 x (Picture tubes) 2 x x x (Speaker cones) 1 x x (Power supplies) 2 x x x (Electronics) x 1, x 2, x 3 0 Some helpful notation x i = # of units of product i produced p i = profit per unit of product i r ij = amount of resource j needed to produce one unit of product i A j = amount of resource j available i= {1,2,3} and j={1,,5} Solver Solution Page 2 of 17

Investment Portfolio Usually involve maximizing return subject to Maximum risk constraints Maximum or minimum proportions in various asset classes OR Minimizing risk subject to Minimum return

3 Investment Portfolio Usually involve maximizing return subject to Maximum risk constraints Maximum or minimum proportions in various asset classes OR Minimizing risk subject to Minimum return constraints Maximum or minimum proportions in various asset classes Example 2a An Investment Example Welte Mutual Funds, located in New York City, just obtained $100,000 by converting industrial bonds to cash and is now looking for other investment opportunities for these funds. Based on Welte s current investments, the firm s top financial analyst recommended that all new investments be made in the oil industry, steel industry or in government bonds. Specifically, the analyst indentified five investment opportunities and projected their annual rates of return. The investments and rates of return are listed below. Management of Welte imposed the following investment guidelines: 1. Neither industry (oil or steel) should receive more than $50, Government bonds should be at least 25% of the steel industry investment. 3. The investment in Pacific Oil, the high-return but high risk investment, cannot be more than 60% of the total oil industry investment. What portfolio recommendations - investments and amounts, should be made with the available $100,000? Example 2a Solution Step 1: Define the objective Maximize the return Step 2: Define the decision variables A - Dollars invested in Atlantic Oil P - Dollars invested in Pacific Oil M - Dollars invested in Midwest Steel H - Dollars invested in Huber Steel G - Dollars invested in Government Bonds Step 3: Write the mathematical objective function Maximize Z = 0.073A+0.103P+0.064M+0.075H+0.045G Step 4: Formulate the constraints 1. Welte just obtained $100,000 by converting industrial bonds to cash and is now looking for other investment opportunities for these funds. A+P+M+H+G=100, Neither industry (oil or steel) should receive more than $50,000 A + P 50,000 M + H 50, Government bonds should be at least 25% of the steel industry investment. 4. The investment in Pacific Oil, the high return but high-risk investment, cannot be more than 60% of the total oil industry investment. Page 3 of 17

Step 5: Final Formulation Solver Solution Diet Problems Usually involve minimizing cost of diet subject to Minimum and maximum

Lifegym dietitian wants to develop a breakfast that will be high in calories, calcium, protein and fiber, which are especially

4 Step 5: Final Formulation Solver Solution Diet Problems Usually involve minimizing cost of diet subject to Minimum and maximum nutritional requirements Example 3 Diet Problem Lifegym, a health and fitness center, operates a morning fitness program for senior citizens. The program includes aerobic exercise, either swimming or step exercise, followed by a health breakfast in the dining room. Lifegym dietitian wants to develop a breakfast that will be high in calories, calcium, protein and fiber, which are especially important to seniors, but low in fat and cholesterol. She also wants to minimize cost. She has selected the following possible food items, whose individual nutrient contributions and cost from which to develop a standard breakfast menu are shown in the slide. Page 4 of 17

5 Diet Problem Decision Variables x 1 = cups of bran cereal x 2 = cups of dry cereal x 3 = cups of oatmeal x 4 = cups of oat bran x 5 = eggs x 6 = slices of bacon x 7 = oranges x 8 = cups of milk x 9 = cups of orange juice x 10 = slices of wheat toast Diet Problem Formulation Formulation - Excel Page 5 of 17

4a - A Blend Example Formulation Since we have the selling price we will maximize profit.

6 Diet Problem Solution Blending Problems May be similar to diet problems in that we may minimize the cost of formulating subject to Minimum and maximum component requirements Alternatively we could be maximizing margin or profit earned Example 4a - A Blend Example Formulation Since we have the selling price we will maximize profit. Note that the selling price was given in liters, whereas the input costs were in barrels Second, we have variables for each input used in each product Page 6 of 17

Decision Variables LS = Light Sweet Crude used in Standard Oil LP =

Green Oil MS = Med Alta Crude used in Standard Oil MP = Med Alta Crude

Recycled Oil used in Standard Oil RP = Recycled Oil used in Premium

7 Decision Variables LS = Light Sweet Crude used in Standard Oil LP = Light Sweet Crude used in Premium Oil LG = Light Sweet Crude used in Green Oil MS = Med Alta Crude used in Standard Oil MP = Med Alta Crude used in Premium Oil MG = Med Alta Crude used in Green Oil RS = Recycled Oil used in Standard Oil RP = Recycled Oil used in Premium Oil RG = Recycled Oil used in Green Oil Objective Function Constraints Page 7 of 17

8 Formulation Page 8 of 17

Example 5 - A Blending Problem: The Agri- Pro Company Defining the Decision Variables X 1 = pounds of feed 1 to use in the mix X 2 = pounds of feed 2 to use in the mix X 3 = pounds of feed 3 to use

9 Example 5 - A Blending Problem: The Agri- Pro Company Defining the Decision Variables X 1 = pounds of feed 1 to use in the mix X 2 = pounds of feed 2 to use in the mix X 3 = pounds of feed 3 to use in the mix X 4 = pounds of feed 4 to use in the mix Defining the Objective Function Minimize the total cost of filling the order. MIN: Z=0.25X X X X 4 Defining the Constraints - Produce 8,000 pounds of feed X 1 + X 2 + X 3 + X 4 = 8,000 - Mix consists of at least 20% corn (0.3X X X X 4 )/ Mix consists of at least 15% grain (0.1X X X X 4 )/ Mix consists of at least 15% minerals (0.2X X X X 4 )/ Non-negativity conditions X 1, X 2, X 3, X 4 0 A Comment About Scaling Notice the coefficient for X 2 in the corn constraint is 0.05/8000 = As Solver runs, intermediate calculations are made that make coefficients larger or smaller. Storage problems may force the computer to use approximations of the actual numbers. Such scaling problems sometimes prevents Solver from being able to solve the problem accurately. Most problems can be formulated in a way to minimize scaling errors... Re-Defining the Decision Variables X 1 = thousands of pounds of feed 1 to use in the mix X 2 = thousands of pounds of feed 2 to use in the mix X 3 = thousands of pounds of feed 3 to use in the mix X 4 = thousands of pounds of feed 4 to use in the mix Page 9 of 17

10 Re-Defining the Objective Function Minimize the total cost of filling the order. MIN: 250X X X X 4 Re-Defining the Constraints Produce 8,000 pounds of feed X 1 + X 2 + X 3 + X 4 = 8 Mix consists of at least 20% corn (0.3X X X X 4 )/8 >= 0.2 Mix consists of at least 15% grain (0.1X X X X 4 )/8 >= 0.15 Mix consists of at least 15% minerals (0.2X X X X 4 )/8 >= 0.15 Non-negativity conditions X 1, X 2, X 3, X 4 >= 0 Scaling: Before and After Before: Largest constraint coefficient was 8,000 Smallest constraint coefficient was: 0.05/8000 = After: Largest constraint coefficient is 8 Smallest constraint coefficient is: 0.05/8 = The problem is now more evenly scaled! Implementing the Model See file Agri-Pro Time Related Models Up until now all the LP examples have been static, or one-period, models. Linear programming can also be used to determine optimal decisions in multi-period, or dynamic, models. Multi-Period Scheduling These applications optimize the scheduling of resources through time Minimizing cost Examples Aggregate Planning Production Scheduling Workforce Scheduling Cash Budgeting Inventory Planning Page 10 of 17

11 Defining the Decision Variables P i = number of units to produce in month i, i=1 to 6 B i = beginning inventory month i, i=1 to 6 Defining the Objective Function Page 11 of 17

constraints are not listed as constraints in Solver (they

12 Implementing the Model UPTON MANUFACTURING Note: only the Pi are variables in the Excel model flow balance constraints are not listed as constraints in Solver (they are handled directly by the spreadsheet) More Examples Page 12 of 17

invested in treasury bills ($) x 4 = amount invested in growth stock fund($) Step 3: Write the mathematical

13 Example 2b Solution Step 1: Define the objective Maximize the return Step 2: Define the decision variables x 1 = amount invested in municipal bonds ($) x 2 = amount invested in certificates of deposit ($) x 3 = amount invested in treasury bills ($) x 4 = amount invested in growth stock fund($) Step 3: Write the mathematical objective function Maximize Z = 0.085x x x x 4 Step 4: Formulate the constraints Page 13 of 17

Step 5: Final Formulation Maximize Z = 0.085x 1 + 0.

14 Step 5: Final Formulation Maximize Z = 0.085x x x x 4 S.t: Solver Solution Page 14 of 17

The maximum quantities available of each component and their cost per barrel are as follows: To ensure the appropriate blend, each grade has certain general specifications.

15 A Blend Example A Petroleum company produced three grades of motor oil Super, Premium and Extra. From three components. The company wants to determine the optimal mix of the three components in each grade of motor oil that will maximize profit. The maximum quantities available of each component and their cost per barrel are as follows: To ensure the appropriate blend, each grade has certain general specifications. Each grade must have minimum amount of component 1 plus combination of other components, as follows: The company wants to produce at least 3000 barrels of each grade of motor oil. Solution Step 1: Define the objective Maximize Profit Step 2: Define the decision variables x ij = barrels of component i used in motor oil grade j per day, where i = 1, 2, 3 and j = s (super), p (premium), and e (extra). Step 3: Write the mathematical objective function Maximize Z = $23(x1s+x2s+x3s)+ 20(x1p+x2p +x3p) +18(x1e+x2e+x3e) 12(x1s+x1p+x1e) - 10(x2s+x2p +x2e) 14(x3s+x3p +x3e) Page 15 of 17

Super contain at least 50% of component 1: Step 4: Formulate the constraints Super not more then 30% of

16 Step 4: Formulate the constraints 1. The first set of constraints reflects the limited amount of each component available on daily basis: Step 4: Formulate the constraints 2. The next set of constraints is for blend specifications for each grade of motor oil. Super contain at least 50% of component 1: Step 4: Formulate the constraints Super not more then 30% of component 2: 3. The two blend specifications for premium motor oil is (use the same technique): 4. The two blend specifications for Extra motor oil is (use the same technique): 5. The last set of constraints reflects the requirement that at least 3,000 barrels of each grade be produced. Page 16 of 17

17 Solution Step 5: Final Formulation Example 4 Solution Page 17 of 17

Solving Examples of Linear Programming Models

Solving Examples of Linear Programming Models Chapter 4 Copyright 2013 Pearson Education 4-1 Chapter Topics 1. A Product Mix Example 2. A Diet Example 3. An Investment Example 4. A Marketing Example 5.