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  Scheduling 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. Examples of Successful LP Applications

B-6 Examples of Successful LP Applications - continued  Blending raw materials in feed mills to maximize profit while producing animal feed.  Selecting the product mix in a factory to make best use of available machine- and labor-hours available while maximizing profit.  Allocating space for tenants in a shopping mall to maximize revenues to the leasing company.

B-7 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-8 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-9 Formulating LP Problems Word Problem Mathematical Expressions Solution Formulation Computer

B-10 Formulating LP Problems 1. Define decision variables. 2. Formulate objective. 3. Formulate constraints. 4. Nonnegativity (all variables  0).

B-11 Formulation Example 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-12 Formulation Example 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-13 Formulation Example 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-14 Formulation Example - 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-15 Formulation Example - 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-16 Formulation Example - 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-17 Formulation Example - Table Hours Required to Produce 1 Unit Department WalkmansWatch-TV’s Available Hours This Month Electronic43225 Assembly21100 Profit/unit$7$5

B-18 Formulation Example - 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-19 Formulation Example - Objective Hours Required to Produce 1 Unit Department WalkmansWatch-TV’s Available Hours This Month Electronic43225 Assembly21100 Profit/unit$7$5 x1x1 x2x2

B-20 Formulation Example - 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-21 Formulation Example - 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-22 Formulation Example - 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-23 Complete Formulation (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-24 Formulation Example - Max Profit  Suppose you are not given the profit for each product, but are given:  The selling price of a Walkman is $60 and the selling price of a Watch-TV is $40.  Each hour of electronic time costs $10 and each hour of assembly time costs $8.  Profit = Revenue - Cost Walkman profit = $60 - ($10/hr  4 hr + $8/hr  2 hr) = $4 Watch-TV profit = $40 - ($10/hr  3 hr + $8/hr  1 hr) = $2

B-25 Formulation Example - 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).  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 Example 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 Profit = 7 x x 2 1. x 1 = 0, x 2 = 0 profit = 0 2. x 1 = 0, x 2 = 75 profit = x 1 = 50, x 2 = 0 profit = x 1 = 37.5, x 2 = 25 profit = Opitmal Solution X1X1 X 2 Feasible Region

B-31 Formulation #1 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 #1 How many products? How many ingredients? How many attributes of products/ingredients?

B-33 Formulation #1 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 #1  40 Ingredient cost protein calories A $0.2/oz 8 2 Snack food carbo. B C $0.1/oz $0.15/oz  20  900

B-35 Formulation #1 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 protein calories A $0.2/oz 8 2 Snack food carbo. B C $0.1/oz $0.15/oz  20  900

B-36 Formulation #1 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 #1 - 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 #1 - 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 #1 - 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 #1 - 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 #1 - 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 #1 - 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 #2 2. Plant fertilizers consist of three active ingredients, Nitrogen, Phosphate and Potash, along with inert ingredients. Fertilizers are defined by three numbers representing the percentages of Nitrogen, Phosphate, Potash. For example a fertilizer includes 20% Nitrogen, 10% Phosphate and 40% Potash. NuGrow makes three different fertilizers, packaged in 40 lb. bags: , and The fertilizer sells for $8/bag and at least 3000 bags must be produced next month. The fertilizer sells for $4/bag. The fertilizer sells for $6/bag and at least 4000 bags must be produced next month. The cost and availability of the fertilizer ingredients is as follows:

B-44 Formulation #2 - continued Ingredient Amount Available (tons/month) Cost ($/ton) Nitrogen (N) Phosphate (Ph) Potash (Po) Inert (In) unlimited 100 Formulate an LP to determine how many bags of each type of fertilizer NuGrow should make next month to maximize profit.

B-45 Formulation #2 Produce 3 products (fertilizers) from 4 ingredients. Do we know how much of each ingredient (or resource) is in each product? If ‘YES’, variables are probably amount of each product to produce. If ‘NO’, variables are probably amount of each ingredient (or resource) to use in each product.

B-46 Formulation #2 - continued Product Minimum req’d (bags) N Ph Po In Price ($/bag) lbs. of ingredient per bag x i = Number of bags of fertilizer type i to make next month. i=1: i=2: i=3:

B-47 Formulation #2 - Constraints Produce 3 products (fertilizers) from 4 ingredients. 3 variables. How many constraints? Usually: - one (or two) for each ingredient - one (or two) for each final product - others?

B-48 Formulation #2 - Constraints Produce 3 products (fertilizers) from 4 ingredients. 3 variables. How many constraints? Usually: - one for each ingredient (3, no constraint for Inert) - one for each final product (2, no constraint for type 2) - others? (no) 3 variables, 5 constraints

B-49 Formulation #2 - Objective x i = Number of bags of fertilizer type i to make next month. : Maximize Profit = Revenue - Cost Revenue = 8x 1 + 4x 2 + 6x 3 Cost = (cost per bag of type 1) x 1 + (cost per bag of type 2) x 2 + (cost per bag of type 3) x 3 Cost per bag is cost of all ingredients in a bag.

B-50 Formulation #2 - Costs Cost for one bag of type 1 ( ) = cost for N 8  0.15 ($300/ton=$0.15/lb) + cost for Ph 4  0.10 ($200/ton=$0.10/lb) + cost for Po 16  0.20 ($400/ton=$0.20/lb) + cost for In 12  0.05 ($100/ton=$0.05/lb) = $5.4 x i = Number of bags of fertilizer type i to make next month. Similarly: Cost for one bag of type 2 ( ) = $3.2 Cost for one bag of type 3 ( ) = $4.4

B-51 Formulation #2 - Objective x i = Number of bags of fertilizer type i to make next month. : Maximize Profit = Revenue - Cost Revenue = 8x 1 + 4x 2 + 6x 3 Cost = 5.4x x x 3 Maximize 2.6x x x 3

B-52 Formulation #2 : Maximize: 2.6x x x 3 x i = Number of bags of fertilizer type i to make next month. 8x 1 + 4x x 3  (N) 4x 1 + 4x x 3  (Ph) 16x 1 + 4x 2 + 4x 3  (Po) x 1, x 2, x 3  0 x 1  3000 ( ) x 3  4000 ( )

B-53 Formulation #2 - Additional Constraints 1. NuGrow can produce at most 4000 lbs. of fertilizer next month. 2. The fertilizer should be at least 50% of the total production. x i = Number of bags of fertilizer type i to make next month.

B-54 Formulation #2 - Additional Constraints 1. NuGrow can produce at most 4000 lbs. of fertilizer next month. 2. The fertilizer should be at least 50% of the total production. x i = Number of bags of fertilizer type i to make next month.

B-55 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-56 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-57 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-58 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-59 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-60 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-61 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-62 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-63 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-64 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-65 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-66 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-67 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-68 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-69 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