EMGT 5414 Introduction to OR Linear Programming Dincer Konur Engineering Management and Systems Engineering

Outline Introduction to OR Introduction to Mathematical Programming Formulation Steps and An Example Introduction to Linear Programming Definition and A prototype Example Assumptions of Linear Programming Graphical Method, Terminology and Properties Examples Spreadsheet modeling Chapter 1-3

Outline Introduction to OR Introduction to Mathematical Programming Formulation Steps and An Example Introduction to Linear Programming Definition and A prototype Example Assumptions of Linear Programming Graphical Method, Terminology and Properties Examples Spreadsheet modeling

Introduction to OR What does OR do? Seeks solutions to problems that arise in complex organizations First used in the military during WW II Managing allocation of scarce resources Early 1950s: OR introduced into business, industry, and government Wikipedia definition: OR is a discipline that deals with the application of advanced analytical methods to help make better decisions. It is often considered to be a sub-field of mathematics. The terms management science and decision science are sometimes used as synonyms.

Introduction to OR How did OR grow? Improved techniques for solving problems Simplex method Computer revolution for running solution methods Fast solution of complex computational problems Increased competition and need for efficiency Need for better design, better manufacturing, better distribution, better service, better marketing, better return management, better better better better planning (management) of Strategies (long-term), tactics (mid-term), operations (short-term)

Introduction to OR How does OR process? Define Model Solve Carefully observe, define, and formulate the problem Gather data (know what your data means and tells) Construct a mathematical model Test whether the model represents the actual situation Modify the model as appropriate and validate again Analyze the properties of the model Characteristics of a good solution Solve the model to find the best solution(s) Or try to get good solutions (if finding the best is hard) Define Model Solve

Introduction to OR Defining a problem: For this course: Identify the appropriate objectives Identify constraints Identify interrelationships with other areas Identify alternative courses of action Define the time constraints For this course: READ THE QUESTIONS CAREFULLY!!!!

Introduction to OR Modeling a problem: Solving a problem: Mathematical programming techniques Solving a problem: Key to success: Attendance! Do not fall behind! On-time return of the assignments Topics build on one another The main objectives of this course

Outline Introduction to OR Introduction to Mathematical Programming Formulation Steps and An Example Introduction to Linear Programming Definition and A prototype Example Assumptions of Linear Programming Graphical Method, Terminology and Properties Examples Spreadsheet modeling

Mathematical Programming Mathematical programming is selecting the best option(s) from a set of alternatives mathematically Minimizing/maximizing a function where your alternatives are defined by functions Many industries use mathematical programming Supply chain, logistics, and transportation Health industry, energy industry, finance, airlines Manufacturing industry, agriculture industry, retail Education, Military

Formulation Steps There are four main steps to formulate a problem mathematically: STEP 0: Know the problem and gather your data Needed for problem understanding and as input into models Issue: too little data available Solution: build management information system to collect data Issue: too much data available Solution: data mining methods STEP 1: Identify your decision variables Decision variables are the things you control Represent the decisions to be made Examples: x1, x2, ….xn

Formulation Steps There are four main steps to formulate a problem mathematically: STEP 2: Define your objective function and objective Your objective function is the measure of performance as a result of your decisions. Example: profit, 𝑃=3𝑥1+2𝑥2+…5𝑥𝑛 Your objective is what you want to do with your objective function: Maximize it? Minimize it? STEP 3: Define your restrictions (constraints) There may be some restrictions which limit what you can do (hence, they define set of your alternatives) Mathematical expressions for the restrictions Often expressed as inequalities. Example: 𝑥1+3𝑥1𝑥2+2𝑥2≤10

An example Mathematical programming will save you! Suppose that you will be left in a deserted island You want to live as many days as you can to increase chance of rescue Fortunately (!), you can take a bag with you to the island What would you put into the bag? Note: This example is not in the book!

An Example The bag can carry at most 50 lbs There are limited set of items you can carry Bread: You can survive for 2 days with 1 lb of bread Steak: You can survive for 5 days with 1 lb of steak (Memphis style grilled!) Chicken: You can survive for 3 days with 1 lb of chicken (southern style deep fried!) Chocolate: You can survive for 6 days with 1 lb of chocolate How much of each item to take with you?

An Example STEP 0: Know the problem and gather your data Your problem is to increase your chance of rescue by maximizing the number of days you survive You have 1 bag which can carry 50 lbs at most You can only put bread, steak, chicken and chocolate into your bag Each item enables you survive for a specific number of days for each pound you take Bread: 2 days/lb Steak: 5 days/lb Chicken: 3days/lb Chocolate: 6days/lb

An Example STEP 1: Identify your decision variables Decision variables are the things you control The amount of each item you will take with you 𝑥 1 :𝑡ℎ𝑒 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑏𝑟𝑒𝑎𝑑 (𝑙𝑏𝑠) 𝑥 2 :𝑡ℎ𝑒 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑠𝑡𝑒𝑎𝑘 (𝑙𝑏𝑠) 𝑥 3 :𝑡ℎ𝑒 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑐ℎ𝑖𝑐𝑘𝑒𝑛 (𝑙𝑏𝑠) 𝑥 4 :𝑡ℎ𝑒 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑐ℎ𝑜𝑐𝑜𝑙𝑎𝑡𝑒 𝑙𝑏𝑠 Warning: Always be careful with the metrics (try to use the same metrics)

An Example STEP 2: Define your objective function and objective Recall that you want to maximize the number of days you will survive in the island Express the performance of measure as a function of your decision variables to get the objective function If you take 𝑥 1 lbs of bread, you will survive for 2𝑥 1 days If you take 𝑥 2 lbs of steak, you will survive for 5𝑥 2 days If you take 𝑥 3 lbs of chicken, you will survive for 3𝑥 3 days If you take 𝑥 4 lbs of chocolate, you will survive for 6𝑥 4 days Then objective function in terms of decision variables: 2𝑥 1 + 5𝑥 2 + 3𝑥 3 + 6𝑥 4 Performance measure Objective 𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒

Limit on how much you can carry An Example STEP 3: Define your restrictions (constraints) There may be some restrictions which limit what you can do (hence, they define set of your alternatives) You can carry 1 bag and it can carry 50 lbs at most 𝑥 1 + 𝑥 2 + 𝑥 3 + 𝑥 4 ≤50 You cannot get negative amounts!!! 𝑥 1 ≥0, 𝑥 2 ≥0, 𝑥 3 ≥0, 𝑥 4 ≥0 Limit on how much you can carry Total amount you decide to carry

An Example Combine your objective, objective function, and constraints 2𝑥 1 + 5𝑥 2 + 3𝑥 3 + 6𝑥 4 𝑥 1 + 𝑥 2 + 𝑥 3 + 𝑥 4 ≤50 𝑥 1 ≥0, 𝑥 2 ≥0, 𝑥 3 ≥0, 𝑥 4 ≥0 This is your mathematical model 𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜

An Example Suppose that you can also take cheese Cheese: You can survive for 4 days with 1 lb of cheese 𝑥 5 :𝑡ℎ𝑒 𝑎𝑚𝑜𝑢𝑛𝑓 𝑜𝑓 𝑐ℎ𝑒𝑒𝑠𝑒 𝑙𝑏𝑠 We have a new decision variable: Update LP 2𝑥 1 + 5𝑥 2 + 3𝑥 3 + 6𝑥 4 + 4𝑥 5 𝑥 1 + 𝑥 2 + 𝑥 3 + 𝑥 4 + 𝑥 5 ≤50 𝑥 1 ≥0, 𝑥 2 ≥0, 𝑥 3 ≥0, 𝑥 4 ≥0, 𝑥 5 ≥0 𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜

An Example Furthermore, suppose you have a budgetary limit 1 lb of bread costs you $3 1 lb of steak costs you $6 1 lb of chicken costs you $7 1 lb of chocolate costs you $15 1 lb of cheese costs you $8 You can spend at most $150 Write the new restriction: 3 𝑥 1 + 6𝑥 2 +7 𝑥 3 +15 𝑥 4 +8 𝑥 5 ≤150 Limit on how much you can spend Total money you decide to spend

An Example 2𝑥 1 + 5𝑥 2 + 3𝑥 3 + 6𝑥 4 + 4𝑥 5 𝑥 1 + 𝑥 2 + 𝑥 3 + 𝑥 4 + 𝑥 5 ≤50 3 𝑥 1 + 6𝑥 2 +7 𝑥 3 +15 𝑥 4 +8 𝑥 5 ≤150 𝑥 1 ≥0, 𝑥 2 ≥0, 𝑥 3 ≥0, 𝑥 4 ≥0, 𝑥 5 ≥0 You cannot get more meat (chicken+steak) than bread 𝑥 2 + 𝑥 3 ≤ 𝑥 1 For each lb of cheese, you need at least 2 lbs of bread 2𝑥 5 ≤ 𝑥 1 𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜

Summary of Formulation Steps Gather all of your data, know what they mean Then Identify your decision variables Identify your objective and objective function Identify your constraints Express your objective function and constraints in terms of your decision variables Steps 2 and 3 can change order

Outline Introduction to OR Introduction to Mathematical Programming Formulation Steps and An Example Introduction to Linear Programming Definition and A prototype Example Assumptions of Linear Programming Graphical Method, Terminology and Properties Examples Spreadsheet modeling Chapter 1-3

Linear Programming Recall that “Mathematical programming is selecting the best option(s) from a set of alternatives mathematically” Minimizing/maximizing a function where your alternatives are defined by functions Linear Programming model (LP) is a mathematical programming model where all of your functions are linear

Definition A function is linear when The variables have power of 1 and the variables are not multiplied with each other 𝑓 𝑥 =3𝑥+5  linear 𝑓 𝑥 =3 𝑥 2 +5  not linear 𝑓 𝑥 1 , 𝑥 2 =3 𝑥 1 +5 𝑥 2 +7  linear 𝑓 𝑥 1 , 𝑥 2 =3 𝑥 1 𝑥 2 +7  not linear It is assumed that our decision variables are continuous They can take fractional values 1.1, 1.11, 1.111…. 𝑓 𝑥 𝑥

A Prototype Example Wyndor Glass Co. produces windows and glass doors Plant 1 makes aluminum frames and hardware Plant 2 makes wood frames Plant 3 produces glass and assembles products Company introducing two new products Product 1: 8 ft. glass door with aluminum frame Product 2: 4 x 6 ft. double-hung, wood-framed window Problem: What mix of products would be most profitable? Assuming company could sell as much of either product as could be produced

A Prototype Example Products are produced in batches of 20 Data needed: Number of hours of production time available per week in each plant for new products Production time used in each plant for each batch of each new product Profit per batch of each new product

A Prototype Example Recall our steps for formulating the problem Step 0: Gather data (we have it in Table 3.1) Step 1: Decision variables x1 = number of batches of product 1 produced per week x2 = number of batches of product 2 produced per week Step 2: Objective function We want to maximize the weekly profit Z = total profit per week (thousands of dollars) from producing these two products From bottom row of Table 3.1 𝑍 = 3 𝑥1+5𝑥2

A Prototype Example Step 3: Constraints 𝑥1≤4 2𝑥2≤12 3𝑥1+2𝑥2≤18 We have limited time available per week in each plant For plant 1: the time used from plant 1 per week should be less than or equal to 4, so we have 𝑥1≤4 For plant 2: the time used from plant 2 per week should be less than or equal to 12, so we have 2𝑥2≤12 For plant 3: the time used from plant 3 per week should be less than or equal to 18, so we have 3𝑥1+2𝑥2≤18 We cannot produce negative number of batches!!! 𝑥1≥0, 𝑥2≥0

A Prototype Example Combining everything, we get our linear programming model (LP) for Wyndor: Functional constraints Nonnegativity constraints

Assumptions of LP Proportionality: The contribution of each activity to the value of the objective function (or left-hand side of a functional constraint) is proportional to the level of the activity If assumption does not hold, one must use nonlinear programming Additivity: Every function in a linear programming model is the sum of the individual contributions of the activities Divisibility: Decision variables in a linear programming model may have any values (including non-integer values) Assumes that activities can be run at fractional values Certainty: Value assigned to each parameter of a linear programming model is assumed to be a known constant First three assumptions indicate that the functions we have are continuous linear functions

Graphical Method So, we know how to formulate LP models Modeling is the start, we need to solve them If you have two decision variables, a graphical method can be used to solve the LP model Two dimensional graph with x1 and x2 as the axes Identify values of x1 and x2 permitted by the restrictions Pick a point in the feasible region that maximizes or minimizes (depending on your objective) value of Z

Graphical Method Example not in the book! Maximize your profit Decide how many to produce product types 1,2 To be produced, each product requires different amount of 2 resources per unit Each resource is limited

Graphical Method You can formulate the following LP model 𝑥 1 :𝑡ℎ𝑒 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 type 1 produced (units) 𝑥 2 :𝑡ℎ𝑒 𝑎𝑚𝑜𝑢𝑛𝑑 𝑜𝑓𝑝𝑟𝑜𝑑𝑢𝑐𝑡 type 2 produced (units) Total profit Resource A limit Resource B limit Non-negativity

Graphical Method First, draw your decision variables as the axes of your graph

Graphical Method Start with the easy constraints Non-negativity constraints! Assume equality of the constraint ( 𝑥 1 =0 or 𝑥 2 =0 ) Then, find the region defined by the constraint

Graphical Method Resource A limit constraint Draw 𝑥 1 + 𝑥 2 =5 Find the region where 𝑥 1 + 𝑥 2 ≤5 Hint: take a point in each of one of the regions and see if this point satisfies the constraint. The region including the point satisfying your constraint is the region defined by your constraint (you can take the origin as your point)

Graphical Method Resource B limit constraint Draw 3𝑥 1 + 2𝑥 2 =12 Find the region where 3𝑥 1 + 2𝑥 2 ≤12

Graphical Method Define the feasible region It is the region defined by the intersection of all of your constraints It includes all of the feasible solutions to your LP

Improvement Direction Graphical Method Improvement Direction

Graphical Method Optimum solution will be one of the corner solutions!! Graph the constraints Define your feasible region Evaluate the corner points (or draw iso-lines until you leave feasible region) Choose the best corner point

Graphical Method Consider the Wyndor Co. problem x1 and x2 are the axes

Graphical Method Consider the Wyndor Co. problem

Graphical Method Consider the Wyndor Co. problem

Graphical Method Summary of the steps: Draw the constraint boundary line for each constraint. Use the origin (or any point not on the line) to determine which side of the line is permitted by the constraint. Find the feasible region by determining where all constraints are satisfied simultaneously. Determine the slope of one objective function line (iso-line). All other objective function lines will have the same slope. Move a straight edge with this slope through the feasible region in the direction of improving values of the objective function. Stop at the last instant that the straight edge still passes through a point in the feasible region. This line given by the straight edge is the optimal objective function line. A feasible point on the optimal objective function line is an optimal solution.

Terminology and Properties A solution is assignment of values to the decision variables. For instance, (x1,x2,x3)=(-2,5,7) is a solution if x1,x2, and x3 are your only decision variables. Feasible solution Solution for which all constraints are satisfied Might not exist for a given problem Infeasible solution Solution for which at least one constraint is violated Optimal solution Has most favorable value of objective function It is the best feasible solution

Terminology and Properties An LP problem can have: Infeasibility No feasible solutions! Unique optimal solution Only one feasible solution is optimum Multiple optimal solutions (alternative optima) More than one feasible solutions that are optimum Unboundedness There is always a better feasible solution

Terminology and Properties Infeasibility example (not in book): Consider the following LP Practice: draw the feasible region The feasible region is empty That is, no solution satisfies the constraints (i.e., there is no feasible solution) So, no optimal solution exists

Terminology and Properties Wyndor example modified: infeasible

Terminology and Properties Unique optimum example (not in book): Consider the following LP The feasible region is not empty All of the points in the feasible region are feasible solutions But we have a single optimal solution

Terminology and Properties Alternative optima example (not in book) Consider the following LP problem The feasible region is not empty There are many feasible solutions Also, there are many optimal solutions but still there are corner points that are also optimum!

Terminology and Properties Wyndor example modified: alternative optima

Terminology and Properties Unboundedness example (not in book) Consider the following LP problem The feasible region is unbounded and You can increase the objective function value as much as you can while you are still in the feasible region Unbounded feasible region does not mean unboundedness of LP

Terminology and Properties Wyndor example modified: unbounded

Terminology and Properties Standard form Other legitimate forms Minimizing (rather than maximizing) the objective function Functional constraints with greater-than-or-equal-to inequality Some functional constraints in equation form Some decision variables may be negative  Later, we will see how to convert any given model into a standard form

Terminology and Properties Given that the constraints do not change Maximizing 3A+4B = Minimizing -3A-4B Maximization is opposite of minimization  Maximizing 3A+4B = Maximizing 100+3A+4B Adding a constant to the objective function does not change the optimum solution, you can just ignore it for optimization Maximizing 3A+4B = Maximizing 15A+20B Multiplying all of the coefficients of the objective function with the same positive constant does not change the optimization On another note, you can multiply both sides of a constraint with the same non-zero constant and it will not change the feasible region, i.e., A+B<= 5 defines the same region with 2A+2B<=10

Terminology and Properties Corner-point feasible (CPF) solution Solution that lies at the corner of the feasible region Linear programming problem with feasible solution and bounded feasible region Must have CPF solutions and optimal solution(s) Best CPF solution must be an optimal solution

Examples General problem terminology and examples Resources: money, particular types of machines, vehicles, or personnel Activities: investing in particular projects, advertising in particular media, or shipping from a particular source Problem involves choosing levels of activities to maximize overall measure of performance

Examples Example 1: Design of radiation therapy for Mary’s cancer treatment Goal: select best combination of beams and their intensities to generate best possible dose distribution Dose is measured in kilorads

Examples Example 1: Design of radiation therapy x1 and x2 represent the dose (in kilorads) at the entry point for beam 1 and beam 2, respectively

Examples Example 1: Design of radiation therapy Linear programming model Using data from Table 3.7 Feasible region is a line segment

Examples Example 2: Reclaiming solid waste SAVE-IT company collects and treats four types of solid waste materials Materials amalgamated into salable products Three different grades of product possible Fixed treatment cost covered by grants Objective: maximize the net weekly profit Determine amount of each product grade Determine mix of materials to be used for each grade

Examples Example 2: Reclaiming solid waste

Examples Example 2: Reclaiming solid waste Decision variables: Then, for material 1, for instance, we will have: This is not linear!!!!!

Examples Example 2: Reclaiming solid waste There is another way to define decision variables: 𝑥𝑖𝑗=𝑧𝑖𝑗𝑦𝑖 (for i = A, B, C; j = 1,2,3,4) number of pounds of material j allocated to product grade i per week

Examples Example 2: Reclaiming solid waste First specification for product A: But we can do something about it

Examples Example 2: Reclaiming solid waste

Examples Example 2: Reclaiming solid waste

Spreadsheet Modeling Recall the Wyndor Glass Co. problem First, put the data you have into Excel

Spreadsheet Modeling Similar to formulation steps: Dedicate cells to your decision variables Changing cells: Cells containing the decisions to be made Write your functions (objective and constraints) Hours used in Plant 1 =x1 Hours used in Plant 2 =2x2 Hours used in Plant 3 =3x1+2x2 Total profit =3x1+5x2 x1 x2

Spreadsheet Modeling And your spreadsheet model is Sumproduct function is helpful!!

Spreadsheet Modeling First, you need to add Excel Solver Open an Excel file

Excel Solver Add-In Click on File button on left top

Excel Solver Add-In Then go to Options and click

Excel Solver Add-In A new window will open: go to Add-ins and click

Excel Solver Add-In A new screen will come up: click Go…

Excel Solver Add-In A new window will show Check Solver Add-in Then click Ok

Excel Solver Add-In Now you have Solver under Data tool

Solving with Excel Solver Pick the cell where your objective function is defined Go to Data bar and click on Solver Pick your objective Click here to add a constraint.. A new window will pop up Pick the cells where your decision variables are defined

Solving with Excel Solver Adding a constraint Enter the cell where left hand side of your constraint is defined Enter the cell where right hand side of your constraint is defined Choose the inequality type you have

Solving with Excel Solver Solve the LP Non-negativity constraints by default Pick Simplex LP method Solve it!!!

Solver Output Solver output

Case: Capital Budgeting Capital Budgeting Case: Financial planning Think-Big Development Co. is a major investor in commercial real-estate development projects. They are considering three large construction projects Construct a high-rise office building. Construct a hotel. Construct a shopping center. Each project requires each partner to make four investments: a down payment now, and additional capital after one, two, and three years. Question: At what fraction should Think-Big invest in each of the three projects?

Case: Capital Budgeting The company currently has $25M and $20M, $20M and $15M will become available after 1, 2, and 3 years, respectively. Funds that are not used in a year will be available next year Therefore, we should consider the cumulative availability at the end of each period!!

Case: Capital Budgeting Cumulative Capital Required Cumulative Capital Available 25 45 65 80 Office Shopping Building Hotel Center Now 40 80 90 End of Year 1 100 160 140 End of Year 2 190 240 End of Year 3 200 310 220 Office Shopping Building Hotel Center Net Present Value 45 70 50 Decision variables: OB = Participation share in the office building H = Participation share in the hotel, SC = Participation share in the shopping center.

Case: Capital Budgeting The LP model: Maximize NPV = 45OB + 70H + 50SC subject to 40OB + 80H + 90SC ≤ 25 100OB + 160H + 140SC ≤ 45 190OB + 240H + 160SC ≤ 65 200OB + 310H + 220SC ≤ 80 OB ≥ 0, H ≥ 0, SC ≥ 0. Net present value Total invested now Total invested within 1 year Total invested within 2 years Total invested within 3 years Non-negativity

Case: Capital Budgeting The Excel formulation and solution

Case: Personnel Scheduling Union Airways is adding more flights to and from its hub airport and so needs to hire additional customer service agents The five authorized eight-hour shifts are Shift 1: 6:00 AM to 2:00 PM Shift 2: 8:00 AM to 4:00 PM Shift 3: Noon to 8:00 PM Shift 4: 4:00 PM to midnight Shift 5: 10:00 PM to 6:00 AM Question: How many agents should be assigned to each shift?

Case: Personnel Scheduling

Case: Personnel Scheduling Decision variables: Si = Number working shift i (for i = 1 to 5) Minimize Cost = $170S1 + $160S2 + $175S3 + $180S4 + $195S5 subject to S1 ≥ 48 S1 + S2 ≥ 79 S1 + S2 ≥ 65 S1 + S2 + S3 ≥ 87 S2 + S3 ≥ 64 S3 + S4 ≥ 73 S3 + S4 ≥ 82 S4 ≥ 43 S4 + S5 ≥ 52 S5 ≥ 15 Si ≥ 0 (for i = 1 to 5) See also 3.4-10 from practice problems

Case: Personnel Scheduling Excel model…

Case: Big M Company The Big M Company produces a variety of heavy duty machinery at two factories. One of its products is a large turret lathe. Orders have been received from three customers for the turret lathe. Question: How many lathes should be shipped from each factory to each customer?

Case: Big M Company Decision variables: Sij = Number of lathes to ship from i to j

Case: Big M Company Minimize Cost = $700SF1-C1 + $900SF1-C2 + $800SF1-C3 + $800SF2-C1 + $900SF2-C2 + $700SF2-C3 subject to Factory 1: SF1-C1 + SF1-C2 + SF1-C3 = 12 Factory 2: SF2-C1 + SF2-C2 + SF2-C3 = 15 Customer 1: SF1-C1 + SF2-C1 = 10 Customer 2: SF1-C2 + SF2-C2 = 8 Customer 3: SF1-C3 + SF2-C3 = 9

Case: Big M Company Big M Company on Excel

Example: 3.4-10 from the Book Larry Edison is the director of the Computer Center for Buckly College. He now needs to schedule the staffing of the center. It is open from 8 A.M. until midnight. Larry has monitored the usage of the center at various times of the day, and determined that the following number of computer consultants are required:

Example: 3.4-10 Two types of computer consultants can be hired: full-time and part-time. The full-time consultants work for 8 consecutive hours in any of the following shifts: morning (8 A.M.–4 P.M.), afternoon (noon–8 P.M.), and evening (4 P.M.–midnight). Full-time consultants are paid $40 per hour. Part-time consultants can be hired to work any of the four shifts listed in the above table. Part-time consultants are paid $30 per hour. An additional requirement is that during every time period, there must be at least 2 full-time consultants on duty for every part-time consultant on duty. Larry would like to determine how many full-time and how many part-time workers should work each shift to meet the above requirements at the minimum possible cost.

Example: 3.4-10

Example: 3.4-15 from the Book Oxbridge University maintains a powerful mainframe computer for research use by its faculty, Ph.D. students, and research associates. During all working hours, an operator must be available to operate and maintain the computer, as well as to perform some programming services. Beryl Ingram, the director of the computer facility, oversees the operation. It is now the beginning of the fall semester, and Beryl is confronted with the problem of assigning different working hours to her operators. Because all the operators are currently enrolled in the university, they are available to work only a limited number of hours each day, as shown in the following table. There are six operators (four undergraduate students and two graduate students). They all have different wage rates because of differences in their experience with computers and in their programming ability. The above table shows their wage rates, along with the maximum number of hours that each can work each day. Each operator is guaranteed a certain minimum number of hours per week that will maintain an adequate knowledge of the operation. This level is set arbitrarily at 8 hours per week for the undergraduate students (K. C., D. H., H. B., and S. C.) and 7 hours per week for the graduate students (K. S. and N. K.).

Example: 3.4-15 from the Book The computer facility is to be open for operation from 8 A.M. to 10 P.M. Monday through Friday with exactly one operator on duty during these hours. On Saturdays and Sundays, the computer is to be operated by other staff. Because of a tight budget, Beryl has to minimize cost. She wishes to determine the number of hours she should assign to each operator on each day.

Example: 3.4-15 from the Book fdsa

Next time Actual linear programming models Modeling language Can have hundreds or thousands of functional constraints Number of decision variables may also be very large Modeling language Used to formulate very large models in practice Expedites model management tasks Modeling language examples AMPL, MPL, OPL, GAMS, and LINGO Most of them use Simplex method!!!!