B-1 Operations Management Linear Programming Module B

B-2 Outline  What is Linear Programming (LP)?  Characteristics of LP.  Formulating LP Problems.  Graphical Solution to an LP Problem.  Formulation Examples.  Computer Solution.  Sensitivity Analysis.

B-3  Mathematical models designed to have optimal (best) solutions.  Linear and integer programming.  Nonlinear programming.  Mathematical model is a set of equations and inequalities that describe a system.  E = mc 2  Y = X Optimization Models

B-4  Mathematical technique to solve optimization models with linear objectives and constraints.  NOT computer programming!  Allocates scarce resources to achieve an objective.  Pioneered by George Dantzig in World War II. What is Linear Programming (LP)?

B-5  Routing school buses to minimize total distance traveled.  Allocating police patrols to high crime areas to minimize response time.  Scheduling tellers at banks to minimize total cost of labor.  Selecting the product mix in a factory with limited resources (labor, equipment, materials, etc.) to maximize profit. Examples of Successful LP Applications

B-6 Characteristics of an LP Problem 1 Deterministic (no probabilities). 2 Single Objective: maximize or minimize some quantity (the objective function). 3 Continuous decision variables (unknowns to be determined). 4 Constraints limit ability to achieve objective. 5 Objectives and constraints must be expressed as linear equations or inequalities.

B-7 4x 1 + 6x 2  9  4x 1 x 2 + 6x 2  9 3x - 4y + 5z = 8  3x - 4y 2 + 5z = 8 3x/4y = 8  3x/4y = 8y same as 3x - 32y = 0 4x 1 + 5x 3 =  4x = 8 Linear Equations and Inequalities

B-8 Formulating LP Problems Word Problem Mathematical Expressions Solution Output Formulation Solution via computer

B-9 Formulating LP Problems 1. Define decision variables. 2. Formulate objective. 3. Formulate constraints. 4. Nonnegativity (all variables  0).

B-10 Formulation #1 You wish to produce two products: (1) Walkman and (2) Watch-TV. Each Walkman takes 4 hours of electronic work and 2 hours of assembly time. Each Watch-TV takes 3 hours of electronic work and 1 hour of assembly time. There are 225 hours of electronic work time and 100 hours of assembly time available each month. The profit on each Walkman is $7; the profit on each Watch-TV is $5. Formulate a linear programming problem to determine how many of each product should be produced to maximize profit?

B-11 Formulation #1 You wish to produce two products…. Each Walkman takes 4 hours of electronic work and 2 hours of assembly time.… How many of each product should be produced to maximize profit? Producing 2 products from 2 materials. Objective: Maximize profit

B-12 Formulation #1 You wish to produce two products: (1) Walkman and (2) Watch-TV. Each Walkman takes 4 hours of electronic work and 2 hours of assembly time. Each Watch-TV takes 3 hours of electronic work and 1 hour of assembly time. There are 225 hours of electronic work time and 100 hours of assembly time available each month. The profit on each Walkman is $7; the profit on each Watch-TV is $5. Formulate a linear programming problem to determine how many of each product should be produced to maximize profit?

B-13 Formulation #1 - Objective.… The profit on each Walkman is $7; the profit on each Watch-TV is $5. Maximize profit:$7 per Walkman $5 per Watch-TV

B-14 Formulation #1 - Requirements... Each Walkman takes 4 hours of electronic work and 2 hours of assembly time. Each Watch-TV takes 3 hours of electronic work and 1 hour of assembly time.... Requirements: Walkman4 hrs elec. time2 hrs assembly time Watch-TV 3 hrs elec. time1 hr assembly time

B-15 Formulation #1 - Resources... There are 225 hours of electronic work time and 100 hours of assembly time available each month. … Available resources: electronic work time 225 hours assembly time 100 hours

B-16 Formulation #1 - Table Hours Required to Produce 1 Unit Department WalkmansWatch-TV’s Available Hours This Month Electronic43225 Assembly21100 Profit/unit$7$5

B-17 Formulation #1 - Decision Variables  What are we deciding? What do we control?  Number of products to make?  Amount of each resource to use?  Amount of each resource in each product?  Let:  x 1 = Number of Walkmans to produce each month.  x 2 = Number of Watch-TVs to produce each month.

B-18 Formulation #1 - Objective Hours Required to Produce 1 Unit Department WalkmansWatch-TV’s Available Hours This Month Electronic43225 Assembly21100 Profit/unit$7$5 x1x1 x2x2

B-19 Formulation #1 - Objective Hours Required to Produce 1 Unit Department WalkmansWatch-TV’s Available Hours This Month Electronic43225 Assembly21100 Profit/unit$7$5 x1x1 x2x2 Objective: Maximize: 7x 1 + 5x 2

B-20 Formulation #1 - 1st Constraint Hours Required to Produce 1 Unit Department WalkmansWatch-TV’s Available Hours This Month Electronic43225 Assembly21100 Profit/unit$7$5 x1x1 x2x2 Objective: Maximize: 7x 1 + 5x 2 Constraint 1: 4x 1 + 3x 2  225 (Electronic Time hrs)

B-21 Formulation #1 - 2nd Constraint Hours Required to Produce 1 Unit Department WalkmansWatch-TV’s Available Hours This Month Electronic43225 Assembly21100 Profit/unit$7$5 x1x1 x2x2 Objective: Maximize: 7x 1 + 5x 2 Constraint 1: 4x 1 + 3x 2  225 (Electronic Time hrs) Constraint 2: 2x 1 + x 2  100 (Assembly Time hrs)

B-22 Complete Formulation #1 (4 parts) Maximize: 7x 1 + 5x 2 4x 1 + 3x 2  225 2x 1 + x 2  100 x 1 = Number of Walkmans to produce each month. x 2 = Number of Watch-TVs to produce each month. x 1, x 2  0

B-23 Formulation #1 - Maximize Profit  Suppose instead of profit for each product, you are told: (1) the selling price of a Walkman is $60 and the selling price of a Watch-TV is $40; and (2) each hour of electronic time costs $10 and each hour of assembly time costs $8.  Calculate the per unit profit using equation:  Profit = Revenue - Cost

B-24 Calculate Profit = Revenue - Cost  Walkman: Revenue per unit = $60 Cost per unit = $10/hr  4 hr + $8/hr  2 hr = $56 Profit per unit = $60 - $56 = $4  Watch-TV: Revenue per unit = $40 Cost per unit = $10/hr  3 hr + $8/hr  1 hr = $38 Profit per unit = $40 - $38 = $2

B-25 Formulation #1 - Optimal Solution x 1 = 37.5 Walkmans produced each month. x 2 = 25 Watch-TVs produced each month. Profit = $387.5/month  Can you make 37.5??  Can you round to 38?? NO!! That requires 227 hrs of electronic time. 4   25 = 227 (> 225!)

B-26  Draw graph with vertical & horizontal axes.  (1st quadrant only since variables are  0 ).  Plot constraints as lines, then as planes.  Find feasible region.  Find optimal solution.  It will be at a corner point of feasible region! Graphical Solution Method - Only with 2 Variables!

B-27 Formulation #1 Graph x 1 +3x 2  225 (electronics) 2x 1 +x 2  100 (assembly) Number of Walkmans (X 1 ) Number of Watch-TVs (X 2 )

B-28 Feasible Region Feasible Region 4x 1 +3x 2  225 (electronics) 2x 1 +x 2  100 (assembly) Number of Walkmans (X 1 ) Number of Watch-TVs (X 2 )

B-29 Possible Solution Points X1X1 X 2 Feasible Region 4x 1 +3x 2  225 (electronics) 2x 1 +x 2  100 (assembly) Corner Point Solutions

B-30 C orner pt.x 1 x 2 Profit = 7 x x = = = = Optimal Solution X1X1 X 2 Feasible Region

B-31 Formulation #2 A company wants to develop a high energy snack food for athletes. It should provide at least 20 grams of protein, 40 grams of carbohydrates and 900 calories. The snack food is to be made from three ingredients, denoted A, B and C. Each ounce of ingredient A costs $0.20 and provides 8 grams of protein, 3 grams of carbohydrates and 150 calories. Each ounce of ingredient B costs $0.10 and provides 2 grams of protein, 7 grams of carbohydrates and 80 calories. Each ounce of ingredient C costs $0.15 and provides 5 grams of protein, 6 grams of carbohydrates and 100 calories. Formulate an LP to determine how much of each ingredient should be used to minimize the cost of the snack food.

B-32 Formulation #2 How many products? How many ingredients? How many attributes of products/ingredients?

B-33 Formulation #2 How many products? 1 How many ingredients? 3 How many attributes of products/ingredients? 3 Do we know how much of each ingredient (or resource) is in each product?

B-34 Formulation #2  40 Ingredient Cost ($/oz) Protein (g/oz) Calories (cal/oz) A Snack food Carbo. (g/oz) B C  20  900 Should carbohydrates and calorie constraints be  or =?

B-35 Formulation #2 Variables:: x i = Number of ounces of ingredient i used in snack food. i = 1 is A; i = 2 is B; i = 3 is C  40 Ingredient Cost ($/oz) Protein (g/oz) Calories (cal/oz) A Snack food Carbo. (g/oz) B C  20  900

B-36 Formulation #2 Minimize: 0.2x x x 3 8x 1 + 2x 2 + 5x 3  20 (protein) 3x 1 + 7x 2 + 6x 3  40 (carbs.) 150x x x 3  900 (calories) x 1, x 2, x 3  0 x i = Number of ounces of ingredient i used in snack food.

B-37 Formulation #2 - Additional Constraints x i = Number of ounces of ingredient i used in snack food. 1. At most 20% of the calories can come from ingredient A.

B-38 Formulation #2 - Additional Constraints x i = Number of ounces of ingredient i used in snack food. 1. At most 20% of the calories can come from ingredient A. calories from A = 150x 1 total calories = 150x x x 3

B-39 Formulation #2 - Additional Constraints x i = Number of ounces of ingredient i used in snack food. 2. The snack food must include at least 1 ounce of A and 2 ounces of B. 3. The snack food must include twice as much A as B.

B-40 Formulation #2 - Additional Constraints x i = Number of ounces of ingredient i used in snack food. 2. The snack food must include at least 1 ounce of A and 2 ounces of B. 3. The snack food must include twice as much A as B.

B-41 Formulation #2 - Additional Constraints x i = Number of ounces of ingredient i used in snack food. 4. The snack food must include twice as much A as B and C. 5. The snack food must include twice as much A and B as C.

B-42 Formulation #2 - Additional Constraints x i = Number of ounces of ingredient i used in snack food. 4. The snack food must include twice as much A as B and C. x 1 = 2x 2 x 1 = 2x 3 or x 1 = 2(x 2 + x 3 ) 5. The snack food must include twice as much A and B as C. x 1 = 2x 3 x 2 = 2x 3 or x 1 + x 2 = 2x 3

B-43 Formulation #3 4. NuTree makes two 2 types of paper (P1 and P2) from three grades of paper stock. Each stock has a different strength, color, cost and (maximum) availability as shown in the table below. Paper P1 must have a strength rating of at least 7 and a color rating of at least 6. Paper P2 must have a strength rating of at least 6 and a color rating of at least 5. Paper P1 sells for $200/ton and the maximum demand is 70 tons/week. Paper P2 sells for $100/ton and the maximum demand is 120 tons/week. NuTree would like to determine how to produce the two paper types to maximize profit. Paper Stock Strength Color Cost/Ton Availability R1 8 9 $ tons/week R2 6 7 $ tons/week R3 3 4 $ tons/week

B-44 Formulation #3 Paper Stock Strength Color Cost/Ton Availability R1 8 9 $ tons/week R2 6 7 $ tons/week R3 3 4 $ tons/week Paper Strength Color Price/Ton Max. Demand P1  7  6 $ tons/week P2  6  5 $ tons/week Produce 2 products (papers) from 3 ingredients (paper stocks) to maximize profit (= revenue - cost). Constraints: Availability(3); Demand(2); Strength(2); Color(2)

B-45 Formulation #3 - Decision Variables Produce 2 products (papers) from 3 ingredients (paper stocks). Do we know how much of each ingredient is in each product?

B-46 Formulation #3 - Decision Variables Produce 2 products (papers) from 3 ingredients (paper stocks). Do we know how much of each ingredient is in each product? NO!  6 variables for amount of each ingredient in each final product.

B-47 Formulation #3 - Key! Produce 2 products (papers: P1 and P2) from 3 ingredients (paper stocks: R1, R2 and R3). x ij = Number of tons of stock i in paper j; i=1,2,3 j=1,2 P1 P2 R1 R2 R3 x 11 x 12 x 21 x 22 x 31 x 32

B-48 Formulation #3 x ij = Number of tons of stock i in paper j; i=1,2,3 j=1,2 Tons of stock R1 used = x 11 + x 12 Tons of stock R2 used = x 21 + x 22 Tons of stock R3 used = x 31 + x 32 P1 P2 R1 R2 R3 x 11 x 12 x 21 x 22 x 31 x 32

B-49 Formulation #3 x ij = Number of tons of stock i in paper j; i=1,2,3 j=1,2 : Amount of paper P1 produced = x 11 + x 21 + x 31 Amount of paper P2 produced = x 12 + x 22 + x 32 P1 P2 R1 R2 R3 x 11 x 12 x 21 x 22 x 31 x 32

B-50 Paper Stock Strength Color Cost/Ton Availability R1 8 9 $ tons/week R2 6 7 $ tons/week R3 3 4 $ tons/week Formulation #3 - Objective Maximize Profit = Revenue - Cost Cost = 150(tons of R1) + 110(tons of R2) + 50(tons of R3) Cost = 150(x 11 + x 12 ) + 110(x 21 + x 22 ) + 50(x 31 + x 32 )

B-51 Formulation #3 - Objective Revenue = 200(tons of P1 made) + 100(tons of P2made) Revenue = 200(x 11 + x 21 + x 31 ) + 100(x 12 + x 22 + x 32 ) Paper Strength Color Price/Ton Max. Demand P1  7  6 $ tons/week P2  6  5 $ tons/week

B-52 Formulation #3 - Objective Maximize profit = Revenue - Cost Revenue = 200(x 11 + x 21 + x 31 ) + 100(x 12 + x 22 + x 32 ) Cost = 150(x 11 + x 12 ) + 110(x 21 + x 22 ) + 50(x 31 + x 32 ) Maximize 50x x x x x x 32

B-53 Paper Stock Strength Color Cost/Ton Availability R1 8 9 $ tons/week R2 6 7 $ tons/week R3 3 4 $ tons/week Formulation #3 - Constraints Availability of each ingredient x 11 + x 12  40 (R1) x 21 + x 22  60 (R2) x 31 + x 32  100 (R3)

B-54 Formulation #3 - Constraints Paper Strength Color Price/Ton Max. Demand P1  7  6 $ tons/week P2  6  5 $ tons/week Demand for each product x 11 + x 21 + x 31  70 (P1) x 12 + x 22 + x 32  120 (P2)

B-55 Paper Stock Strength Color Cost/Ton Availability R1 8 9 $ tons/week R2 6 7 $ tons/week R3 3 4 $ tons/week Formulation #3 - Constraints Paper Strength Color Price/Ton Max. Demand P1  7  6 $ tons/week P2  6  5 $ tons/week Strength and color are weighted averages, where weights are tons of each ingredient used. 1 ton of R1 + 1 ton of R2 = 2 tons with Strength = 7 2 tons of R1 + 1 ton of R2 = 3 tons with Strength = 7.333

B-56 Paper Stock Strength Color Cost/Ton Availability R1 8 9 $ tons/week R2 6 7 $ tons/week R3 3 4 $ tons/week Formulation #3 - Constraints Paper Strength Color Price/Ton Max. Demand P1  7  6 $ tons/week P2  6  5 $ tons/week Strength P1: average strength from ingredients of P1  7

B-57 Formulation #3 Maximize 50x x x x x x 32 x ij = Number of tons of stock i in paper j; i=1,2,3 j=1,2 x 11, x 12, x 13, x 21, x 22, x 23  0 x 11 + x 12  40 x 21 + x 22  60 x 31 + x 32  100 x 11 + x 21 + x 31  70 x 12 + x 22 + x 32  120 (strength P1) x 11 - x x 31  0 (strength P2) 2x x 32  0 (color P1) 3x 11 + x x 31  0 (color P2) 4x x 12 - x 32  0