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?

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

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 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

To Solve: Linear Programming Method

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”

• 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

Sensitivity Analysis

Infeasible formulations

• Result if some of the constraints are incompatible (check the direction of your constraints):

e.g. max A + C

subject toA 60C 50A + C 190Solver: “cannot find a feasible solutions”

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)

Wrenches Pliers AvailabilitySteel (lbs.) 1.5 1.0 27,000 lbs./dayMolding Machine (hours) 1.0 1.0 21,000

hours/dayAssembly Machine (hours) 0.3 0.5 9,000 hours/dayDemand Limit (tools/day) 15,000 16,000Contribution to Earnings ($/1,000 units)

$130 $100

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.1.How many wrenches and pliers should GTC produce per day in order to maximize the contribution to earnings?2.Which resources would be most critical in this plant?

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 MixA 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 Dkg 0.09 0.05 0.02 0.002

The ingredients have the following nutrient values and costs:

A B C D Cost/kgIngredient 1 0.1 0.08 0.04 0.01 $40Ingredient 2 0.2 0.15 0.02 - $60Filler - - - - $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 4000 3500 4000 4500 4000 3500

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

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 1

In-Class Exercise 2A 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.

Units per Ounce:Ingredient A B C Cost per OunceP 20 30 60 0.3Q 60 30 0 0.2R 10 50 30 0.15

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