1. Optimization Models 14

2. Introduction
A wide variety of problems can be formulated as linear programming models, but there are some that cannot.
Some models require integer variables, or they are nonlinear in the decision variables.
Once the models are formulated, Solver can be used to solve them.
However, integer and nonlinear models are inherently more difficult to solve.
Solver must use more complex algorithms and is not always guaranteed to find the optimal solution.

3. Worker Scheduling Models
Many organizations use worker scheduling models to schedule employees to provide adequate service.
LP can be used to schedule employees on a daily basis, as shown in the next example.

4. Example 14.1: Worker Scheduling.xlsx (slide 1 of 2)
Objective: To develop an LP model that relates five-day shift schedules to daily numbers of employees available, and to use Solver on this model to find a schedule that uses the fewest number of employees and meets all daily workforce requirements.
Solution: The number of full-time employees that a post office needs each day is given in the table on the bottom left.
Union rules state that each full-time employee must work five consecutive days and then receive two days off.
The post office wants to meet its daily requirements using only full-time employees.
The variables and constraints for this problem appear in the table on the bottom right.

5. Example 14.1: Worker Scheduling.xlsx (slide 2 of 2)
Add an integer constraint in Solver to ensure that the number of employees starting work on some days is not a fraction.
The spreadsheet model with this integer constraint is shown below.
When you solve this problem, you might get a different schedule that is still optimal.
Such multiple optimal solutions are not at all uncommon and are good news for a manager, who can then choose among the optimal solutions.

6. Modeling Issues
The postal employee scheduling example is called a static scheduling model because we assume that the post office faces the same situation each week.
In reality, demands change over time. A dynamic scheduling model is necessary for such problems.
A scheduling model for a more complex organization has a larger number of decision variables, and optimization software such as Solver will have difficulty finding a solution.
Heuristic methods have been used to find solutions to these problems.
The scheduling model can be expanded to handle part-time employees, the use of overtime, and alternative objectives such as maximizing the number of weekend days off.

7. Blending Models
In many situations, various inputs must be blended to produce desired outputs.
In many of these situations, blending models can be used to find the optimal combination of outputs as well as the mix of inputs used to produce desired outputs.
Examples of blending problems:
A company using a blending model would run the model periodically (each day, for example) and set production on the basis of current inventory of inputs and the current forecasts of demands and prices.
Then the model would be run again to determine the next day’s production.

8. Example 14.2: Blending Oil.xlsx (slide 1 of 2)
Objective: To develop an LP model for finding the revenue-maximizing plan that meets quality constraints and stays within the limits on crude oil availabilities.
Solution: Chandler Oil has 5000 barrels of crude oil 1 and 10,000 barrels of crude oil 2 available.
Chandler sells gasoline and heating oil, which are produced by blending the two crude oils together.
Each barrel of crude oil 1 has a quality level of 10, and each barrel of crude oil 2 has a quality level of 5.
Gasoline must have an average quality level of at least 8, whereas heating oil must have an average quality level of at least 6.
Gasoline sells for $75 per barrel, and heating oil sells for $60 per barrel.
The variables and constraints required for this model are listed below.

9. Example 14.2: Blending Oil.xlsx (slide 2 of 2)
The spreadsheet model for this problem is shown below.

10. Logistics Models
In many situations a company produces products at locations called origins and ships these products to customer locations called destinations.
Each origin has a limited capacity that it can ship, and each destination must receive a required quantity of the product.
Logistic models can be used to determine the minimum-cost shipping method for satisfying customer demands.

11. Transportation Models
In a transportation problem, the only possible shipments are those directly from an origin to a destination.
A typical transportation problem requires three sets of numbers:
Capacities—indicate the most each plant can supply in a given amount of time under current operating conditions.
Customer demands—are typically estimated from some type of forecasting model.
Unit shipping costs—come from a transportation cost analysis.
The unit “shipping” cost can also include the unit production cost at each plant.

12. Example 14.3: Transportation 1.xlsx (slide 1 of 4)
Objective: To develop an LP model for finding the least-cost way of shipping the automobiles from plants to regions, staying within plant capacities and meeting regional demands.
Solution: Grand Prix Automobile Company manufactures automobiles in three plants and then ships them to four regions of the country.
The plants can supply the amounts listed in the right column of the table below.
The customer demands by region are listed in the bottom row of the table.
The unit costs of shipping an automobile from each plant to each region are listed in the middle of the table.

13. Example 14.3: Transportation 1.xlsx (slide 2 of 4)
The variables and constraints for the problem are listed below.
A network diagram of the model is shown to the right.
A node, indicated by a circle, generally represents a geographical location. In this case, the nodes on the left correspond to plants, and the nodes on the right to regions.
An arc, indicated by an arrow, generally represents a route for getting a product from one node to another.
An arc pointed into a node is called an inflow, and an arrow pointed out of node is called an outflow.

14. Example 14.3: Transportation 1.xlsx (slide 3 of 4)
The spreadsheet model and a graphical representation of the optimal solution are shown below.

15. Example 14.3: Transportation 1.xlsx (slide 4 of 4)
It is often useful to model network problems by listing all of the arcs and their corresponding flows in one long list. Then constraints can be indicated by a separate section of the spreadsheet.
For each node in the network, there is a flow balance constraint.
These flow balance constraints for the basic transportation model are simply the supply and demand constraints, but they can be more general for other network models.
A more general network model of the Grand Prix problem is shown below.

16. Modeling Issues
The customer demands in typical transportation problems can be handled in one of two ways:
Think of forecasted demands as minimal requirements that must be sent to the customers.
Consider demands as maximal sales quantities, the most each region can sell.
If all the supplies and demands for a transportation problem are integers, the optimal Solver solution automatically has integer-valued shipments.
Explicit integer constraints are not required, so the “fast” simplex method can be used.
Shipping costs are often nonlinear due to quantity discounts.
There is a streamlined version of the simplex method, called the transportation simplex method, that is much more efficient than the ordinary simplex method for transportation problems.

17. Other Logistics Models (slide 1 of 2)
The general logistics model is like the transportation model, except for two possible differences:
Arc capacities are often imposed on some or all of the arcs.
They become simple upper-bound constraints in the model.
There can be inflows and outflows associated with any node.
An origin node is a location that starts with a certain supply.
A destination is the opposite; it requires a certain amount to end up there.
A transshipment point is a location where goods simply pass through.

18. Other Logistics Models (slide 2 of 2)
The best way to think of the node categories is in terms of net inflow and net outflow.
Net inflow for any node is defined as total inflow minus total outflow for that node.
Net outflow is the negative of this, total outflow minus total inflow.
An origin is a node with positive net outflow, a destination is a node with positive net inflow, and a transshipment point is a node with net outflow (and net inflow) equal to 0.
There are two types of constraints in logistics models:
Arc capacity constraints, which are simple upper bounds on the arc flows
Flow balance constraints, one for each node:
For an origin: Net Outflow = Capacity (or possibly Net Outflow ≤ Capacity)
For a destination: Net Inflow ≥ Demand (or possibly Net Inflow = Demand)
For a transshipment point: Net Inflow = 0 (equivalent to Net Outflow = 0)

19. Example 14.4: RedBrand Logistics1.xlsx (slide 1 of 3)
Objective: To develop an LP model for finding the minimum-cost way to ship the tomato product from suppliers to customers, possibly through warehouses, so that customer demands are met and supplier capacities are not exceeded.
Solution: RedBrand Company produces a tomato product at three plants. The cost of producing the product is the same at each plant.
The product can be shipped directly to the company’s two customers, or it can first be shipped to the company’s two warehouses and then to the customers.
The production capacity of each plant (in tons per year) and the demand of each customer are shown in the graphical representation below.
Nodes 1, 2, and 3 represent the plants (denoted by S for supplier).
Nodes 4 and 5 represent the warehouses (denoted by T for transshipment).
Nodes 6 and 7 represent the customers (denoted by D for destination).

20. Example 14.4: RedBrand Logistics1.xlsx (slide 2 of 3)
The cost (in thousands of dollars) of shipping a ton of the product between each pair of locations is listed in the table below, where a blank indicates that RedBrand cannot ship along that arc.
The most that can be shipped between any two nodes is 200 tons. (This is the common arc capacity.)
The variables and constraints for RedBrand’s model are listed below.

21. Example 14.4: RedBrand Logistics1.xlsx (slide 3 of 3)
The spreadsheet model and a graphical representation of the optimal solution are shown below.

22. Modeling Issues
Solver uses the simplex method to solve logistics models, but the network simplex method is much more efficient and can solve large logistics problems.
If the given supplies and demands for the nodes are integers and all arc capacities are integers, the logistics model always has an optimal solution with all integer flows.
This integer benefit is guaranteed only for the basic logistics model.
When the model is modified in certain ways, such as by adding a shrinkage factor, the optimal solution is no longer guaranteed to be integer-valued.

23. Aggregate Planning Models
Models where we determine workforce levels and production schedules for a multiperiod time horizon are called aggregate planning models.
There are many variations, depending on the detailed assumptions made.
A number of inputs are required for this type of problem:
Initial inventory, holding costs, and demands
Data on the current number of workers, regular hours per worker per month, regular hourly wage rates, overtime hourly rate, and maximum number of overtime hours per worker per month
Costs for hiring and firing a worker
Unit production cost, which is a combination of two inputs:
Raw material cost per unit
Labor hours per unit

24. Example 14.5: Aggregate Planning 1.xlsx (slide 1 of 3)
Objective: To develop an LP spreadsheet model that relates workforce and production decisions to monthly costs, and to find the minimum-cost solution that meets forecasted demands on time and stays within limits on overtime hours and production capacity.
Solution: SureStep Company must meet (on time) the following demands for pairs of shoes: in month 1; 5000 in month 2; 2000 in month 3; and 1000 in month 4.
At the beginning of month 1, 500 pairs are on hand, and Sure Step has 100 workers.
A worker is paid $1500 per month
Each worker can work up to 160 hours per month before he or she receives overtime. A worker can work up to 20 hours of overtime per month and is paid $13 per hour for overtime labor.
It takes four hours of labor and $15 of raw material to produce a pair of shoes.
At the beginning of each month, workers can be hired or fired. Each hired worker costs $1600, and each fired worker costs $2000.
At the end of the month, a holding cost of $3 per pair of shoes left in inventory is incurred.
Production in a given month can be used to meet that month’s demand.

25. Example 14.5: Aggregate Planning 1.xlsx (slide 2 of 3)
The variables and constraints for this aggregate planning model are listed below.
The most difficult aspect of modeling this problem is knowing which variables the company gets to choose—the decision variables—and which variables are determined by these decisions.

26. Example 14.5: Aggregate Planning 1.xlsx (slide 3 of 3)
The spreadsheet model is shown below.

27. The Rolling Planning Horizon Approach
In reality, an aggregate planning model is usually implemented via a rolling planning horizon.
To implement the SureStep model in the rolling planning horizon context:
View the demands as forecasts and solve for a four-month model with these forecasts.
Implement only the month 1 production and work scheduling recommendation.
Next, observe month 1’s actual demand and replace the demands in the Demand range with updated forecasts for the next four months.
Rerun Solver and use the production levels and hiring and firing recommendations in column B as the production level and workforce policy for month 2.

28. Model with Backlogging Allowed
In many situations, backlogging of demand is allowed— that is, customer demand can be met at a later date.
There are several modeling approaches to the backlogging problem; the most natural approach for the SureStep model is shown below.
This is a nonlinear model using IF functions.
However, you should avoid these nonsmooth functions in optimization models if at all possible, because Solver often has trouble handling them.

29. Financial Models
Optimization and other management science methods have also been applied successfully in a number of financial areas.
These include investment strategy and pension fund management.

30. Example 14.6: Investing.xlsx (slide 1 of 3)
Objective: To develop an LP model that relates investment decisions to total ending cash, and to use Solver to find the strategy that maximizes ending cash and invests no more than a given amount in any one investment.
Solution: At the beginning of year 1, Barney-Jones Investment Corporation has $100,000 to invest for the next four years. It wants to limit the amount put into any investment to $75,000.
There are five possible investments, labeled A through E.
Investment A: Invest at the beginning of year 1, and for every dollar invested, receive returns of $0.50 and $1.00 at the beginnings of years 2 and 3.
Investment B: Invest at the beginning of year 2, receive returns of $0.50 and $1.00 at the beginnings of years 3 and 4.
Investment C: Invest at the beginning of year 1, receive return of $1.20 at the beginning of year 2.
Investment D: Invest at the beginning of year 4, receive return of $1.90 at the beginning of year 5.
Investment E: Invest at the beginning of year 3, receive return of $1.50 at the beginning of year 4.

31. Example 14.6: Investing.xlsx (slide 2 of 3)
At the beginning of any year, Barney-Jones can invest only cash on hand, which includes returns from previous investments.
Any cash not invested in any year can be put in a short-term money market account that earns 3% annually.
The variables and constraints for this investment model are listed in the table below.

32. Example 14.6: Investing.xlsx (slide 3 of 3)
The spreadsheet model for this investment problem is shown below.

33. Example 14.7: Pension Fund Management.xlsx (slide 1 of 3)
Objective: To develop an LP model that relates initial allocation of money and bond purchases to future cash availabilities, and to minimize the initial allocation of money required to meet all future pension fund payments.
Solution: Armco Incorporated’s pension fund must be sufficient to make the payments listed in the table below.
Each payment must be made on the first day of the year, and the payments will be financed by purchasing bonds.

34. Example 14.7: Pension Fund Management.xlsx (slide 2 of 3)
It is currently the beginning of year 1, and three bonds are available for immediate purchase:
Bond 1 costs $980 and yields a $60 coupon in years 2 through 5 and a $1060 payment on maturity in year 6.
Bond 2 costs $970 and yields a $65 coupon in years 2 through 11 and a $1065 payment on maturity in year 12.
Bond 3 costs $1050 and yields a $75 coupon in years 2 through 14 and a $1075 payment on maturity in year 15.
Any excess cash on hand can earn an annual rate of 4% in a fixed-rate account.
The variables and constraints for this pension fund model are shown below.

35. Example 14.7: Pension Fund Management.xlsx (slide 3 of 3)
The optimal solution for this pension fund model is shown below.
In this solution, the numbers of bonds purchased have been constrained to integer values.

36. Integer Optimization Models
A 0-1 variable, or binary variable, is a variable that must equal 0 or 1.
When the variable equals 0, the activity is not undertaken; when it equals 1, the activity is undertaken.
Optimization models in which some or all of the variables must be integers are known as integer programming (IP) models.
Solver has a much harder time solving an IP problem than an LP problem, because IP problems are inherently difficult.
However, your ability to model complex problems increases tremendously when you are able to use IP, particularly with 0-1 variables.
Perhaps the simplest types of IP models are capital budgeting models.

37. Example 14.8: Capital Budgeting 1.xlsx (slide 1 of 3)
Objective: To use a binary IP model to find the set of investments that stays within budget and maximizes total NPV.
Solution: Tatham Company is considering seven investments.
The cash required for each investment and the net present value (NPV) each investment adds to the firm are listed in the table below.
Partial investments are not allowed.
The cash available for investment is $15,000.

38. Example 14.8: Capital Budgeting 1.xlsx (slide 2 of 3)
The variables and constraints for this model are listed below.
The most important part is that the decision variables must be binary, where a 1 means an investment is undertaken and 0 means that it is not.

39. Example 14.8: Capital Budgeting 1.xlsx (slide 3 of 3)
The spreadsheet model is shown below.

40. Modeling Issues
Capital budgeting models with multiple periods can also be handled. The figure below shows one possibility.
If a company can choose a fractional amount of an investment, then its NPV can be maximized by deleting the binary constraint.
Any IP involving binary variables with only one constraint is called a knapsack problem.

41. Fixed-Cost Models
Fixed-cost models are used when a fixed cost is incurred if an activity is undertaken at any positive level.
The cost is independent of the level of the activity and is known as a fixed cost (or fixed charge).
The fixed cost feature makes the problem inherently nonlinear, which means that a straightforward application of LP is not possible.
However, binary variables are often used to transform a nonlinear model into a linear (integer) model.

42. Example 14.9: Fixed Cost Manufacturing.xlsx (slide 1 of 3)
Objective: To develop a linear model with binary variables that can be used to maximize the company’s profit, correctly accounting for fixed costs and staying within resource availabilities.
Solution: Great Threads Company is capable of manufacturing shirts, shorts, pants, skirts, and jackets.
Each type of clothing requires Great Threads to acquire the appropriate type of machinery.
The machinery needed to manufacture each type of clothing must be rented at the weekly rates shown in the table below. This table also lists the amounts of cloth and labor required per unit of clothing, as well as the selling price and the unit variable cost for each type of clothing.
There are 4000 labor hours and 4500 square yards of cloth available in a given week. The fixed costs are the given rental rates for the machinery.

43. Example 14.9: Fixed Cost Manufacturing.xlsx (slide 2 of 3)
The variables and constraints for this model are listed in the table below.
The cost structure violates the proportionality assumption that is needed for a linear model, so binary variables must be used to obtain a linear model.
The tricky part of the model is that if any of a given type of clothing is produced, then its binary variable must equal 1.
Because Solver is unable to deal with IF functions predictably, the fixed-cost constraints are modeled as follows:

44. Example 14.9: Fixed Cost Manufacturing.xlsx (slide 3 of 3)
The spreadsheet model is shown below.

45. Set-Covering Models
In a set-covering model, each member of a given set (set 1) must be “covered” by an acceptable number of another set (set 2).
The objective in a set-covering problem is to minimize the number of members in set 2 necessary to cover all the members in set 1.
Set-covering models have been applied to areas as diverse as airline crew scheduling, truck dispatching, political redistricting, and capital investment.

46. Example 14.10: Locating Hubs1.xlsx (slide 1 of 3)
Objective: To develop a binary model to find the minimum number of hub locations that can cover all cities.
Solution: Western Airlines wants to design a hub system that connects flights to and from cities within 1000 miles of each hub.
The table below lists the cities that are within 1000 miles of other cities.

47. Example 14.10: Locating Hubs1.xlsx (slide 2 of 3)
The variables and constraints for this set-covering model are listed in the table below.
There is a binary variable for each city to indicate whether a hub is located there.
Then the number of hubs that cover each city is constrained to be at least one.

48. Example 14.10: Locating Hubs1.xlsx (slide 3 of 3)
The spreadsheet model for Western and a graphical representation of the optimal solution are shown below.

49. Nonlinear Optimization Models
In many optimization models, the objective and/or the constraints are nonlinear functions of the decision variables.
Such an optimization model is called a nonlinear programming (NLP) model.
When you solve an NLP model, it is very possible that Solver will obtain a suboptimal solution.
This is because a nonlinear function can have local optimal solutions that are not the global optimal solution.
A local optimal solution is one that is better than all nearby points.
The global optimum is the one that beats all points in the entire feasible region.
There are mathematical conditions that guarantee that the Solver solution is the global optimum, but these are difficult to understand.
A simpler way is to run Solver several times, each time with different starting values in the changing cells.

50. Example 14.11: Peak-Load Pricing.xlsx (slide 1 of 2)
Objective: To use a nonlinear model to determine prices and capacity when there are two different daily usage patterns, peak-load and off-peak.
Solution: Florida Power and Light must determine the price per kilowatt hour (kwh) to charge during both peak-load and off-peak periods.
The daily demand for power during each period (in kwh) is related to price as follows:
Here, Dp and Pp are demand and price during peak-load times, and D0 and P0 are demand and price during off-peak times.
Assume that it costs FPL $10 per day to maintain one kwh of capacity.
The variables and constraints for this model are shown below.

51. Example 14.11: Peak-Load Pricing.xlsx (slide 2 of 2)
The spreadsheet model is shown below.

52. Portfolio Optimization Models
Portfolio optimization models are used to determine the percentage of assets to invest in stocks, gold, and Treasury bills.
To do any work with investments, the following formulas must be understood:
All investors want to choose portfolios with high return (measured by the expected value in the first equation), but they also want portfolios with low risk (usually measured by the variance in the second equation).
The second equation can be rewritten slightly by using covariances instead of correlations, as shown below.
This makes calculating the portfolio variance very easy with Excel’s® matrix function.

53. Matrix Functions in Excel
Two built-in Excel matrix functions, MMULT and TRANSPOSE, simplify the calculation for the variance of portfolio return.
The matrix is a rectangular array of numbers.
A matrix is an i x j matrix if it consists of i rows and j columns.
An example of a 2 x 3 matrix is:
If matrix A has the same number of columns as matrix B has rows, it is possible to calculate the matrix product of A and B, denoted AB.
The Excel MMULT function performs matrix multiplications in a single step.

54. The Portfolio Selection Model
Most investors have two objectives in forming portfolios: to obtain a large expected return and to obtain a small variance (to minimize risk).
The problem is inherently nonlinear because the portfolio variance is nonlinear in the investment amounts.
The most common way of handling this two-objective problem is to specify a minimal required expected return and then minimize the variance subject to the constraint on the expected return.
Financial analysts typically estimate the required means, standard deviations, and correlations for stock returns from historical data.
However, there is no guarantee that these estimates are relevant for future returns.

55. Example 14.12: Portfolio Selection.xlsx (slide 1 of 3)
Objective: To use NLP to find the portfolio that minimizes the risk, measured by portfolio variance, subject to achieving an expected return of at least 12%.
Solution: Perlman & Brothers intends to invest a given amount of money in three stocks.
The means and standard deviations of annual returns have been estimated as shown in the first table below.
The correlations between the annual returns on the stocks are listed in the second table.

56. Example 14.12: Portfolio Selection.xlsx (slide 2 of 3)
The company wants to find a minimum-variance portfolio that yields an expected annual return of at least 0.12 (that is 12%).
The variables and constraints for this model are presented below.
One interesting aspect of this model is that it is not necessary to specify the amount of money invested.

57. Example 14.12: Portfolio Selection.xlsx (slide 3 of 3)
The portfolio optimization model is shown below.

58. Modeling Issues
Typical real-world portfolio selection problems involve a large number of potential investments. However, the basic model does not change at all.
If a company is allowed to short a stock, the fraction invested in that stock is allowed to be negative.
To implement this, eliminate the nonnegativity constraints on the changing cells.
An alternative objective might be to minimize the probability that the portfolio loses money.

59. Keys to Solving Most Optimization Problems
Determine the changing cells.
Set up the model so that you can easily calculate what you wish to maximize or minimize.
Set up the model so that the relationships between the cells in the spreadsheet and the constraints of the problem are readily apparent.
Optimization models do not always fall into ready- made categories.
A model might involve a combination of the ideas discussed in the production scheduling, blending, and aggregate planning examples.