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LP Modeling

LP Modeling. Examples discussed in class. Example: Basketball Team Selection.

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LP Modeling

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  1. LP Modeling Examples discussed in class

  2. Example: Basketball Team Selection Coach Jack is trying to choose the starting line-up for the basketball team. The team consists of seven players who have been rated (on a scale of 1=poor to 3=excellent) according to their ball-handling, shooting, rebounding, and defensive abilities. The positions that each player is allowed to play and the player’s abilities are listed in the following table: The five-player starting line-up must satisfy the following restrictions: • At least 4 members must be able to play guard, at least 2 members must be able to play forward, and at least 1 member must be able to play centre. • The average ball-handling, shooting, and rebounding level of the starting line-up must be at least 2. • If player 3 starts, then player 6 cannot start. • If player 1 starts, then player 4 and 5 must both start. • Either player 2 or player 3, or both, must start. • Given these constraints, Coach Jack wants to maximize the total defensive ability of the starting team. Formulate an IP that will help him choose his starting team.

  3. Uncontrollable factors (parameters) Independent (decision) variables Dependent variables Mathematical relationships The General Structure of a Mathematical Model Inputs Output

  4. Mathematical relationship: Decision variables: x1, x2 (What quantities of products 1 and 2 should be produced?) Maximize revenue Objective Dependent variable: R = 5x1+ 2x2 (total revenue) Subject to: x1+ x2 50 Constraint Uncontrollable variables: 5, 2, and 50 (market prices, marketing limitation) A Simplified Model of a Manufacturing Situation

  5. Example 1: LP Formulation • Papa Louis manufacturers wooden tables and chairs for small kids. Each "table" built: Sells for $27 and uses $10 worth of raw materials, increases Papa Louis’s variable labor/overhead costs by $14. Requires 2 hours of finishing labor and 1 hour of carpentry labor. Each "chair" built: Sells for $21 and uses $9 worth of raw materials, increase Papa Louis’s variable labor/overhead costs by $10. Requires 1 hours of finishing labor AND 1 hour of carpentry labor. Each week Papa Louis can obtain only 100 finishing hours and only 80 carpentry hours. Also demand for the chairs is unlimited. However, at most 40 tables are bought each week. Papa Louis wants to maximize weekly profit (revenues - expenses).

  6. Example 2: LP Formulation • Production Planning: A manufacturing firm has discontinued production of a certain unprofitable product line. This created considerable excess production capacity. Management is considering devoting this excess capacity to one or more of the three products: call them products 1, 2, and 3. The available capacity on the machines that might limit output is summarized in the following table: • The number of machine hours required for each unit of the respective products is given as: • The sales department indicates that the sales potential for products 1, and 2 exceeds the maximum production rate and that the sales potential for product 3 is 20 units per week. The unit profit would be $ 30.0, $20, and $ 15, respectively, on products 1, 2, and 3. Formulate the linear programming model for determining how much of each product to produce in order to maximize profits. Total Availability of resources <500 hr/week <350 hr/week <150 hr/week

  7. Example 3: Crude Petroleum • Shell Petroleum runs a small refinery on the Texas coast. The refinery distills crude petroleum from two sources, Saudi Arabia and Venezuela, into three main products: gasoline, jet fuel, and lubricants. The two crudes differ in chemical composition and thus yield different product mixes. Each barrel of Saudi crude yields 0.3 barrel of gasoline, 0.4 barrel of jet fuel, and 0.2 barrel of lubricants. On the other hand, each barrel of Venezuelan crude yields 0.4 barrel of gasoline but only 0.2 barrel of jet fuel and 0.3 barrel of lubricants. The remaining 10 of each barrel is lost to refining. • The crudes also differ in cost and availability, Shell can purchase up to 9000 barrels per day from Saudi Arabia at $20 per barrel. Up to 6000 barrels per day of Venezuelan petroleum are also available at the lower cost of $15 per barrel because of the shorter transportation distance. Shell's contracts with independent distributors require it to produce 2000 barrels per day of gasoline, 1500 barrels per day of jet fuel, and 500 barrels per day of lubricants. How can these requirements be fulfilled most efficiently?

  8. Example 4: LP Formulation • Manpower Planning: Certain types of facilities operate seven days each week and face the problem of allocating manpower during the week as staffing requirements change as a function of the day of the week. Perhaps the most fundamental staffing problem is the assignment of days off to full-time employees. In particular, it is regularly the case that each employee is entitled to two consecutive days off per week. If the number of employees required on each of the seven days of the week is given, then the problem is to find the minimum workforce size which allows these demands to be met and then to determine the days off for the people in this work-force. To be specific, let us study the problem faced by Grand River Transit Bus Company. The number of drivers required for each day of the week is as follows: • How many drivers should be scheduled to start a five-days stint on each day of the week? Formulate this problem as a linear program. • If daily pay is $50 per person on weekdays, $75 on Saturday, $90 on Sundays. Modify the LP formulation so that the objective is now to minimize the weekly payroll costs rather than to minimize the workforce size.

  9. Example 5: LP Formulation • Target Shirt Company makes three varieties of shirts: Collegiate, Traditional and European. These shirts are made from different combinations of cotton and polyester. The cost per yard of unblended cotton is $5 and for unblended polyester is $4. Target can receive up to 4,000 yards of raw cotton and 3,000 yards of raw polyester fabric weekly. The table shows below pertinent data concerning the manufacture of the shirts. (Meet weekly contracts or more; while not exceeding weekly demand). • Formulate and solve a linear program that would give a manufacturing policy for Target Shirt Company.

  10. Example 5: LP Formulation Finco has the following investments available: Investment A For each dollar invested at time 0, we receive $ 0.10 at time 1 and $1.30 at time 2. Time 0= now; time 1 = one year from now; and so on.) Investment B For each dollar invested at time 1, we receive $1.60 at time 2. Investment C For each dollar invested at time 2, we receive $1.20 at time 3. At any time, leftover cash may be invested in T-bills, which pay 10% per year. At time 0, we have $100. At most, $50 can be invested in each of investments A, B, and C. Formulate an LP that can be used to maximize Finco’s cash on hand at time 3.

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