A fertile field for the use of LP is in planning for the optimal mix of products to manufacture. A company must meet a myriad of constraints, ranging from financial concerns to sales demand to material contracts to union labor demands. Its primary goal is to generate the largest profit possible.

Fifth Avenue Industries, a nationally known manufacturer of menswear, produces four varieties of ties. One is an expensive, all-silk tie; one is an all-polyester tie; one is a combination of polyester and cotton; and one is a combination of silk and cotton. The following table illustrates the cost and availability (per monthly production planning period) of the three materials used in the production process:

The firm has fixed contracts with several major department store chains to supply ties. The contracts require that Fifth Avenue Industries supply a minimum quantity of each tie but allow for a larger demand if Fifth Avenue chooses to meet that demand. (Most of the ties are not shipped with the name Fifth Avenue on their label, incidentally, but with “private stock” labels supplied by the stores.) Table 8.1 summarizes the contract demand for each of the four styles of ties, the selling price per tie, and the fabric requirements of each variety. Fifth Avenue’s goal is to maximize its monthly profit. It must decide upon a policy for product mix.

In formulating this problem, the objective is to maximize profit. There are three constraints (one for each material) indicating that the amount of silk, polyester, and cotton cannot exceed the amount that is available. There are four constraints (one for each type of tie) that specify that the number of all-silk ties, all-polyester ties, poly–cotton ties, and silk–cotton ties produced must be at least the minimum contract amount. There are four more constraints (one for each type of tie) that indicate that the number of each of these ties produced cannot exceed the monthly demand. The variables are defined as

But first the firm must establish the profit per tie:

1. Each all-silk tie $\left(X_1\right)$ requires 0.125 yard of silk, at a cost of $24.00 per yard. Therefore, the material cost per tie is $3.00. The selling price is $19.24, leaving a net profit of $16.24 per silk tie.

2. Each all-polyester tie $\left(X_2\right)$ requires 0.08 yard of polyester, at a cost of $6 per yard. Therefore, the material cost per tie is $0.48. The selling price is $8.70, leaving a net profit of $8.22 per polyester tie.

3. Each poly–cotton (combination) tie $\left(X_3\right)$ requires 0.05 yard of polyester, at a cost of $6 per yard, and 0.05 yard of cotton, at $9 per yard, for a cost of $\$0.30+\$0.45=\$0.75$ per tie. The selling price is $9.52, leaving a net profit of $8.77 per poly–cotton tie.

4. Performing similar calculations will show that each silk–cotton (combination) tie $\left(X_4\right)$ has a material cost per tie of $1.98 and a profit of $8.66.

The objective function may now be stated as

Using Excel and its Solver command, the computer-generated solution is to produce 5,112 all-silk ties each month; 14,000 all-polyester ties; 16,000 poly–cotton combination ties; and 8,500 silk–cotton combination ties. This produces a profit of $412,028 per production period. See Program 8.3 for details.

Setting a low-cost production schedule over a period of weeks or months is a difficult and important management problem in most plants. The production manager has to consider many factors: labor capacity, inventory and storage costs, space limitations, product demand, and labor relations. Because most companies produce more than one product, the scheduling process is often quite complex.

Basically, the problem resembles the product mix model for each period in the future. The objective is either to maximize the profit from or to minimize the total cost (production plus inventory) of carrying out the task.

Production scheduling is amenable to solution by LP because it is a problem that must be solved on a regular basis. When the objective function and constraints for a firm are established, the inputs can easily be changed each month to provide an updated schedule.

Greenberg Motors, Inc., manufactures two different electrical motors for sale under contract to Drexel Corp., a well-known producer of small kitchen appliances. Its model GM3A is found in many Drexel food processors, and its model GM3B is used in the assembly of blenders.

Three times each year, the procurement officer at Drexel contacts Irwin Greenberg, the founder of Greenberg Motors, to place a monthly order for each of the coming 4 months. Drexel’s demand for motors varies each month based on its own sales forecasts, production capacity, and financial position. Greenberg has just received the January–April order and must begin his own 4-month production plan. The demand for motors is shown in Table 8.2.

When setting up the production schedule, Irwin Greenberg must consider several factors:

1. The company must meet the demand for each of the two products in each of the four months (see Table 8.2). Also, the company would like to have 450 units of the GM3A and 300 units of the GM3B in inventory at the end of April, as demand in May is expected to be somewhat higher than demand in the previous months.

2. There is a carrying, or holding, cost for any inventory left at the end of the month. So producing too many extra units of either product may not be desirable. The carrying cost assigned to the GM3A is $0.36 per unit per month, while the carrying cost for the GM3B is $0.26 per unit per month.

3. The company has been able to maintain a no-layoff policy, and it would like to continue with this. This is easier if the labor hours used do not fluctuate too much from month to month. Maintaining a production schedule that would require from 2,240 to 2,560 labor hours per month is desired. The GM3A requires 1.3 labor hours per unit, while the GM3B requires only 0.9 labor hour.

4. Warehouse limitations cannot be exceeded without great additional costs. There is room at the end of the month for only 3,300 units of the GM3A and GM3B combined.

Although these factors sometimes conflict, Greenberg has found that linear programming is an effective tool in setting up a production schedule that will minimize total cost. Production costs are currently $20 per unit for the GM3A and $15 per unit for the GM3B. However, each of these is due to increase by 10% on March 1 as a new labor agreement goes into effect.

In formulating this as a linear program, it is important to understand how all the important factors are related, how the costs are calculated, how the labor hours per month are calculated, and how demand is met with both production and inventory on hand. To help with understanding this, try to determine the number of labor hours used, the number of units left in inventory at the end of each month for each product, and the total cost if exactly 1,000 of the GM3A and exactly 1,200 of the GM3B were produced each month.

To begin formulating the linear program for the Greenberg production problem, the objective and the constraints are:

Objective:

Constraints:

The decisions involve determining how many units of each of 2 products to produce in each of 4 months, so there will be 8 variables. But, since the objective is to minimize cost and there are costs associated not only with the units produced each month but also with the number of units of each left in inventory, it would be best to define variables for these also. Let

The objective function in the LP model is

In setting up the constraints, we must recognize the relationship among last month’s ending inventory, the current month’s production, and the sales to Drexel this month. The inventory at the end of a month is

While the constraints could be written in this form, inputting the problem into the computer requires all variables to be on the left-hand side of the constraint. Rearranging the terms to do this results in

Using this, the demand constraints are

The constraints for the minimum and maximum number of labor hours each month are

The storage capacity constraints are

The solution was obtained using Solver in Excel 2016, as shown in Program 8.4. Some of the variables are not integers, but this is not a problem because work in process can be carried over from one month to the next. Table 8.3 summarizes the solution with the values rounded. The total cost is about $169,295. Greenberg can use this model to develop production schedules again in the future by letting the subscripts on the variables represent new months and making minor changes to the problem. The only things in the model that would have to be changed are the right-hand-side values for the demand constraints (and inventory desired at the end of the fourth month) and the objective function coefficients (costs) if they should change.