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Unformatted text preview: Space IOE 202: lecture 4 outline Announcements Last time... More examples of optimization problems and linear programming models IOE 202: Operations Modeling, Fall 2009 Page 1 Space Last time
Optimization problems and (mathematical) optimization models
Optimization model components: decision variables, objective function, constraints Terminology: solution, feasible solution, optimal solution Linear Programming (LP) models — special kind of optimization model
Easy to represent and solve with computer packages, such as Excel solver If a problem can be modeled with an LP model, usually this is the model of choice! Pet food supplier’s problem Monet picture frame manufacturer problem Modeled and solved as LP models:
IOE 202: Operations Modeling, Fall 2009 Page 2 Space Example: problem of optimal resource allocation1 • The Monet company produces four types of picture frames, labeled A, B, C, and D. The table below lists the unit selling price Monet charges for each type of frame. • Each type requires a certain amount of skilled labor, metal, and glass, as shown in the table. For production during the coming week, Monet can purchase up to 4000 hours of skilled labor, 6000 ounces of metal, and 10,000 ounces of glass. The unit costs are also indicated in the table. • Also, market constraints are such that it is impossible to sell more than 1000 typeA frames, 2000 typeB frames, 500 typeC frames, and 1000 typeD frames. • How many frames of each type should Monet produce during the coming week to maximize its proﬁt?
Another problem of this type: Recreational Vehicle problem on p. 19 of Denardo.
IOE 202: Operations Modeling, Fall 2009 Page 3 1 Space Data (inputs) for the Monet production problem Frame type Frame A Frame B Frame C Frame D Max. amount of resource Resource Unit prices Skilled labor 2 1 3 2 4000 hours $8.00 per hour Metal 4 2 1 2 6000 oz $0.50 per 1 oz Glass 6 2 1 2 10,000 oz $0.75 per 1 oz Selling price $28.50 $12.50 $29.25 $21.50 Maximal production 1000 2000 500 1000 IOE 202: Operations Modeling, Fall 2009 Page 4 Space Mathematical model for Monet: Decision variables: xA , xB , xC , xD denote the number of frames A, B, C, and D to produce, respectively. Mathematical model: maximize 6xA + 2xB + 4xC + 3xD subject to 2xA + xB + 3xC + 2xD ≤ 4000 4xA + 2xB + xC + 2xD ≤ 6000 6xA + 2xB + xC + 2xD ≤ 10000 xA ≤ 1000 xB ≤ 2000 xC ≤ 500 xD ≤ 1000 xA , x B , xC , x D ≥ 0 Proﬁt objective Labor constraint Metal constraint Glass constraint Frame A sales constraint Frame B sales constraint Frame C sales constraint Frame D sales constraint Nonnegativity constraint This model is also a linear programming model, since all constraints are linear, and the objective function is a linear function.
IOE 202: Operations Modeling, Fall 2009 Page 5 Space Some comments An alternative (but equivalent) formulation can be constructed by also including variables to represent the amount of labor, metal, and glass purchased. Objective function and constraints could be expressed in terms of this variables. Both integer and noninteger levels of production were allowed in our formulation. (Is it a reasonable assumption?) Next time, we will discuss the changes in the models/solution methods when the last assumption is not reasonable; for now, let us allow variables to be noninteger. IOE 202: Operations Modeling, Fall 2009 Page 6 Space Optimal solution of the Monet problem Optimal solution: Optimal proﬁt: Which inequality constraints are “tight” (i.e., hold as equalities) at the optimal solution? An alternative way to characterize the optimal solution: “Do not make any frames of type D. Produce as many frames of type A as you can sell. Use up all available labor and metal.” IOE 202: Operations Modeling, Fall 2009 Page 7 Space Perturbation Theorem Above, we characterized the optimal solution to the Monet problem by identifying which constraints were tight, and which were slack at that optimal solution. This characterization comes in handy if the data of the problem turns out to be slightly diﬀerent than we assumed when we formulated the problem. Perturbation “Theorem”
If the data of the linear program are perturbed by small amounts, its optimal solution can change, but its tight constraints stay tight, and its slack constraints stay slack. IOE 202: Operations Modeling, Fall 2009 Page 8 Space x ✻2 Solving LPs graphically max 3x1 x1 +5x2 ≤4 ≤ 12 ≤ 18. 2x2 3x1 +2x2 x1 , x2 ≥ 0 ✲ x1
IOE 202: Operations Modeling, Fall 2009 Page 9 Space x ✻2 Demystifying the perturbation theorem max 3x1 x1 +5x2 ≤4 ≤ 12 ≤ 21 2x2 3x1 +2x2 x1 , x2 ≥ 0 ✲ x1
IOE 202: Operations Modeling, Fall 2009 Page 10 Space Shipping model2 A forest production company manufactures plywood in three plants located in diﬀerent timber zones, and ships it to four depots. Each plant has a monthly production capacity, and each depot has a monthly demand (these demands must be satisﬁed exactly). The table below speciﬁes the capacities, demands, and the unit shipping costs from each plant to each depot. The cost of producing plywood has been omitted since it is the same in each plant. What is the cheapest way to ship plywood to the depots each month? Depot 1 $4 $10 $3 2000 Depot 2 $7 $9 $6 3000 Depot 3 $3 $3 $4 2500 Depot 4 $5 $6 $4 1500 Capacity (units) 2500 4000 3500 Plant 1 Plant 2 Plant 3 Demand (units)
2 This is the problem considered in Section 4.5 of Denardo
Page 11 IOE 202: Operations Modeling, Fall 2009 Space Representation of the shipping model Plant
✓✏ ✓✏ ✲1 1 ❍ ✒✑ ✒✑ ❃ ✚ ✚✣ ❅ ❏ ❍❍ ✡ ✡ ❅ ❍❍✚✚ ✡ ❏ ❍ ❏ ❅ ✚ ❍❍ ✓✏ ✡ ✚ ❥ ❍ ❏❅ ✡✘ 2 ✿ ✚ ❅ ✘✘ ✘ ✒✑ ✡ ✚❏ ✓✏✘✘✘❅ ✒ ✚✘ ❏ ✡ ✘ 2 ❏ ✡❅ ✒✑ ❏ ❅ ✓✏ ✡ ❅ ❘ ✡ ③ 3 ❏ ✯ ✟ ❏ ✟ ✒✑ ✡ ✟✟ ❏ ✡ ✡ ✟✟✟ ❏ ✓✏ ✓✏ ✡✟ ❏ ❏ ⑦ ✟ ✲4 3 ✒✑ ✒✑
Page 12 Depot IOE 202: Operations Modeling, Fall 2009 Space Operational decisions in the shipping problem
What decisions do you need to make? What performance measure are you using to compare diﬀerent decisions? What constraints (restrictions) must your decisions satisfy? What assumptions are being made? IOE 202: Operations Modeling, Fall 2009 Page 13 Space Formulation of a mathematical model for the shipping problem Decision variables: represent decisions by variables. Objective function: express the performance criterion in terms of the decision variables; should it be minimized of maximized? IOE 202: Operations Modeling, Fall 2009 Page 14 Space Formulation of a model for the shipping problem – cont. Constraints: express all (explicit and implicit) constraints and restrictions on the values of the decision variables. Optimal solution:
IOE 202: Operations Modeling, Fall 2009 Page 15 Space A blending model3 After last week’s bakeoﬀ, you have a few ingredients left over (see table below), and you have decided to use them to make candy to sell to spectators at a minor league ballgame this weekend. You are considering producing two types of candies: “Easy Out” and “Slugger,” both of which consist solely of sugar, nuts, and chocolate. Ingredient Amount available Sugar 10,000 oz Nuts 2,000 oz Chocolate 3,000 oz The mixture used to make Easy Out must contain at least 20% nuts, while the mixture used to make Slugger must contain at least 10% nuts and 10% chocolate. Each ounce of Easy Out can be sold for $0.50, and each ounce of Slugger can be sold for $0.40 — how can you maximize revenue? Note that there is quite a bit of ﬂexibility in the recipes for the candy!
3 For a similar model, see Section 4.3 of Denardo
Page 16 IOE 202: Operations Modeling, Fall 2009 Space Representation of the blending model 10,000 oz Sugar Sllugger 2,000 oz Nuts Easy Out 3,000 oz Choc olate IOE 202: Operations Modeling, Fall 2009 Page 17 Space Operational decisions in the candymaking business
What decisions do you need to make? What performance measure are you using to compare diﬀerent decisions? What constraints (restrictions) must your decisions satisfy? What assumptions are being made? IOE 202: Operations Modeling, Fall 2009 Page 18 Space Formulation of a mathematical model for candymaking Decision variables: represent decisions by variables. Objective function: express the performance criterion in terms of the decision variables; should it be minimized of maximized? IOE 202: Operations Modeling, Fall 2009 Page 19 Space Formulation of a model for candymaking – cont. Constraints: express all (explicit and implicit) constraints and restrictions on the values of the decision variables. Optimal solution:
IOE 202: Operations Modeling, Fall 2009 Page 20 Space Postal employee scheduling4 A post oﬃce requires diﬀerent numbers of employees on diﬀerent days of the week. The number of employees required is as follows: Day Min. required Mon 17 Tue 13 Wed 15 Thu 19 Fri 14 Sat 16 Sun 11 Union rules state that each employee must work 5 consecutive days and then receive 2 days oﬀ. For example, an employee might work Wednesday through Sunday, and be oﬀ Monday and Tuesday. The post oﬃce wants to minimize the number of employees it needs to hire while meeting the daily requirements. Note: Each employee hired needs to be assigned to one of the seven schedules, or “tours.” 4 Similar to the problem discussed in Section 4.8 of Denardo
Page 21 IOE 202: Operations Modeling, Fall 2009 Space Operational decisions in the scheduling problem
What decisions do you need to make? What performance measure are you using to compare diﬀerent decisions? What constraints (restrictions) must your decisions satisfy? What assumptions are being made? IOE 202: Operations Modeling, Fall 2009 Page 22 Space Formulation of a mathematical model for the scheduling problem Decision variables: represent decisions by variables. Objective function: express the performance criterion in terms of the decision variables; should it be minimized of maximized? IOE 202: Operations Modeling, Fall 2009 Page 23 Space Formulation of a model for the scheduling problem – cont. Constraints: express all (explicit and implicit) constraints and restrictions on the values of the decision variables. Starting day of the schedule Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Optimal solution: ≥ ≥ ≥ ≥ ≥ ≥ ≥ IOE 202: Operations Modeling, Fall 2009 Page 24 Space An investment model5 Presently, you have $1,000 to invest. Cash ﬂows associated with 5 available investments are shown in the table; you can put no more than $500 in any investment. In addition to these investments, you can invest as much money as you want into 12month CDs, which pay 6% interest. How should you invest to maximize your cash at hand at the end of year 3? Investment A B C D E Now $1.00 $1.00 $1.00 Year 1 +$1.15 +$1.28 $1.00 $1.00 +$1.15 +$1.32 Year 2 Year 3 +$1.40 Note: each year, you can only invest cash available on hand!
Although presented in a diﬀerent setting, the model in this problem turns out to be similar to the Activity Analysis models of Section 4.10 of Denardo
IOE 202: Operations Modeling, Fall 2009 Page 25 5 Space Operational decisions in the investment problem
What decisions do you need to make? What performance measure are you using to compare diﬀerent decisions? What constraints (restrictions) must your decisions satisfy? What assumptions are being made? IOE 202: Operations Modeling, Fall 2009 Page 26 Space Formulation of a mathematical model for the investment problem Decision variables: represent decisions by variables. Objective function: express the performance criterion in terms of the decision variables; should it be minimized of maximized? IOE 202: Operations Modeling, Fall 2009 Page 27 Space Formulation of a model for the investment problem – cont. Constraints: express all (explicit and implicit) constraints and restrictions on the values of the decision variables. Optimal solution:
IOE 202: Operations Modeling, Fall 2009 Page 28 ...
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This note was uploaded on 03/17/2010 for the course IOE 202 taught by Professor Marinaepelman during the Fall '09 term at University of MichiganDearborn.
 Fall '09
 MarinaEpelman

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