On LT simulation Game ends at 8:45 p.m. Class breaks at 8:00 p.m. and re-gathers after the game ends. Meeting rooms available: Reg. 150, 151, 153, 154, 163, 165, 252, 356 Group report due before next class. 3 pages (1-sided/2-sided up to you) Submit using Bb – Drop Box – Group Report Proposal Discussions…during the break.

Product mix for The Furniture Company (TFC) TFC sells tables and chairs using a combination of small and large blocks. Help them maximize profit. A table sells for $57; a chair sells for $37. A table requires 3 labor hours; a chair requires 2 labor hours; 11 labor hours are available (assume no cost for labor) There are 8 small blocks and 6 large blocks available Small blocks cost $5; large blocks cost $8 There is no limit on market demand for tables and chairs

Consider the following changes… Assume everything else kept the same as before… A military contract: a table sells for $57; a chair sells for $51. What if you have … Chair still sells for $37 1 more unit of small block 1 more unit of large block 1 more labor hour How do your solutions and profits change? 3

Another Similar Example: Easy Rider Toys (ERT) manufactures and markets toy cars. This year ERT is planning to introduce several new product lines and wants to sell off existing inventories. These inventories consist of toy cars and toy trucks, and can be sold in two different sets. The Racer Set consists of seven cars and two trucks, and is sold for $34.99. The Construction Set consists of twelve trucks and three cars, and is sold for $43.99. Currently there are 10,000 cars and 12,000 trucks in inventory. How many Construction sets and Racer sets should ERT produce in order to max its profit?

Common Planning Problems in OM Production Planning Product mix Blending Workforce Scheduling Aggregated multi-period planning 5

Approach to solve Planning problems Step 1: identify the following for a given scenario: Inputs: deterministic and given, e.g. cost and revenue parameters Decision variables: how many to produce, how many to hire... so that Objective: optimize, usually max profits or min costs, subject to Constraints: available resources and requirements Step 2: formulate the problem mathematically Step 3: translate the problem into spreadsheet Step 4: obtain solutions using Solver in Excel Step 5: analyze the results and reports

Mathematical Formulation In the TFC original case…

To Solve: Linear Programming Method To optimally allocate existing resources LP Assumptions Linearity: the impact of decision variables is linear in constraints and objective function Divisibility: non-integer values of decision variables are acceptable: CAN buy 3.2 machines Certainty: values of parameters are known and constant Non-negativity: negative values of decision variables are unacceptable: CANNOT produce (-100) units 21

To Use Solver in Excel 2007 Use the function of “SumProduct” to set up Objective Function Constraints Add Solver to Excel: See previous Slide Data -> Analysis group -> Solver Target cell: Objective function cell Changing cell: Decision variable cells Constraints: Corresponding constraint cells Options: Check “assume Linear Model” and “Non-negative”

Sensitivity Analysis To obtain the result in Excel, click on the “Sensitivity” Report … Binding vs. Non-binding Constraints Shadow Prices on Constraints Change in the optimal objective function value as RHS of a constraint increased by one unit Marginal value: benefit from adding capacity Nonbinding constraint: shadow price = 0 21

Infeasible formulations Result if some of the constraints are incompatible (check the direction of your constraints): e.g. max A + C subject to A  60 C  50 A + C  190 Solver: “cannot find a feasible solutions” 11

Unbounded Formulations The formulation allows an infinitely high (low for min) value of the objective function (usually means that an important constraint has been omitted or min/max switches): e.g.: max A + C subject to A  60 Solver: “set cell values do not converge”

Product Mix Collection of products that can be sold Collection of resources needed to produce the products Each product has a corresponding Profit contribution rate Set of resource consumption rates Maximize profit without exceeding resource availability

Gemstone Tool Company (7.2) It produces wrenches and pliers, made from steel, and the process involves molding the tools on a molding machine and then assembling the tools on an assembly machine. The below table list information on the amount of steel used in the production, the daily availability of steel, the machine utilization rates needed, the capacity of these machines, the daily market demand for these tools and their variable (per unit) contribution to earnings. How many wrenches and pliers should GTC produce per day in order to maximize the contribution to earnings? Which resources would be most critical in this plant? Wrenches Pliers Availability Steel (lbs.) 1.5 1.0 27,000 lbs./day Molding Machine (hours) 21,000 hours/day Assembly Machine (hours) 0.3 0.5 9,000 hours/day Demand Limit (tools/day) 15,000 16,000 Contribution to Earnings ($/1,000 units) $130 $100

Blending Problems Arise in the food, feed, metals and oil industries Collection of raw materials with associated attributes and costs Collection of finished products with associated requirements Minimize costs of the finished products while meeting requirements Many Wall Street firms uses the model to optimize its portfolios

Feed Mix A company produces feed mix for dairy cattle. The mix contains two active ingredients and a filler. One kg of feed mix must contain a minimum quantity of each of four nutrients below: Nutrient A B C D kg 0.09 0.05 0.02 0.002 The ingredients have the following nutrient values and costs: A B C D Cost/kg Ingredient 1 0.1 0.08 0.04 0.01 $40 Ingredient 2 0.2 0.15 0.02 - $60 Filler - - - - $1 What should be the amounts of active ingredients and filler in one kg of the feed mix?

Blending Problem Formulation Variables Objective function Constraints: content of each nutrient should be at least what is required:

Multi-Period Planning Problems Help manufacturers to plan production and inventory over multiple periods Decisions made in earlier periods partially determine the set of options available in future periods “Inventory” carried across periods: Inventory Balance Constraint: It-1+Pt-Dt=It Ending inventory of current period = Starting inventory of the next period If It>=0, unmet demands (backlogs) not allowed, all demands have to be satisfied each period

Multi-Period Planning Eg. Upton makes heavy-duty air compressors for home and light industrial use. We would like to plan production and inventory for next six months. Estimated demand is given by Month 1 2 3 4 5 6 Unit Production Cost $240 $250 $265 $285 $280 $260 Units Demanded 1000 4500 6000 5500 3500 4000 Maximum Production A maximum of 6000 units may be in inventory at the end of any month, but no less than 1500 as a safety buffer. To stabilize production, the minimum production needs to be half of the max capacity each month. Inventory carrying costs are $1 per unit per month, and we start with 2750 units in inventory at month 1.

Multi-Period Planning Formulation Decision Variables Pt = production in month t (It = inventory at the end of month t) determined by Pt Objective function Minimize the total production + inventory cost Constraints Keep production between min and max capacity Keep inventory between min and max capacity No need to write out explicit mathematical formulations

In-Class Exercise 1 A small construction firm specializes in building and selling single-family homes. The firm offers two basic types of houses, Model A and model B. Model A houses require 4000 labor hours, 2 tons of stone and 2000 board feet of lumber. Model B houses require 10000 hours of labor, 3 tons of stone and 2000 board feet of lumber. The firm has currently 400000 hours of labor, 150 tons of stone and 200000 board feet of lumber. Model A yields $1000 profit and model B yields $2000 profit. Formulate the LP mathematically Solve for solution using Excel

In-Class Exercise 2 A dietitian in a hospital is required to devise a recipe for a food which will provide at least the following amounts of vitamins: 500 units of vitamin A, 500 units of vitamin B and 700 units of vitamin C The dietitian may use three ingredients; P, Q, and R in the recipe which are described below. At least one ounce of R must be used in the recipe.

In-Class Exercise 3 A customer requires during the next 4 months respectively, 50, 65, 100 and 70 units of a commodity, which must be satisfied. Production costs are $5, $8, $4 and $7 per unit during those months. Storage cost per month is $2 per unit (based on the ending inventory). It is estimated that each unit inventory at the end of month 4 could be sold for $6. Determine how to minimize the net costs incurred in meeting the demands for the next 4 months. Constraint? # available on hand >= demand for each month Objective function? Min costs – resell value $6/unit: (- $6) holding cost/unit

Summary Solutions to in-class examples and exercises will be posted on Bb Readings: hand-out Install Solver on your PC Part of the take-home exam is about solving LP problems