Question

Casa Migliore is a production company that manufactures high quality furniture. The

manager of Casa Migliore has realized the opportunity to increase the revenues by reducing the costs

related to production and inventory levels. He now wants to optimize the production planning of

three types of chairs, A, B, and C for the next 6 months. In Table 1, sales forecasts for each type of

chair are given.

Table 1. Demand for each type of chair for the next 6 months.

March    April    May    June    July    August

A       700       900       750       800       600       400

B    1500    1750    2500    2300    2200    1800

C    3200    3200    5200    6500    4700    4500

Due to the fluctuations in the prices, total material cost for each type of chair may vary each month

and expected material costs are given in Table 2.

Table 2. Expected material cost for each type of chair for the next 6 months.

March    April    May    June    July    August

A    20    20    20    40    40    40

B    16    22    22    26    26    26

C    11    11       8       8    16    16

For the production of the chairs type A, B, and C, 3, 2 and 1 worker hours are required, respectively.

There are currently 40 workers that produce these three types of chairs and each of them works 10

hours a day. The number of working days varies depending on the month and working days in each

month are given in Table 3.

Table 3. Number of working days for the next 6 months.

March    April    May    June    July    August

   21   22   20   21   22   20

At the beginning of each month, new workers are hired at a cost of 250 TL and existing workers can

be fired at a cost of 400 TL. Wages consist of the majority of the total cost and workers are paid 10 TL

per day including insurance, meal and transportation. Monthly wages may change from one month

to another month due to the number of working days. All demand must be satisfied on time. Assume

that there is no backorders and lost sales. The inventory holding costs are 10, 5, 2 TL per item per

month for products A, B and C, respectively. Note that initial inventory levels are 0.

In this project, you are required to develop an LP model to solve the production planning problem

of Casa Migliore and answer the following questions.

Question 1)

A. Write an LP model to minimize total cost.

- Define decision variables clearly

- Write explanations for each constraint

Question 2)

Rewrite the above model according to the following instructions. You have to adjust or

update the constraints and/or the objective function. It may be necessary to add new variables.

A. Assume that employees will not be fired in April or May since May 1 is the International Worker’s

Day.

B. An employer should inform his/her employee one month before firing. Otherwise, a

penalty equal to the employee’s salary must be paid by the employer. In other words, if an employee

is fired, he/she leaves the workforce after one month.

Answer 1

Let a_ij rightarrow Forecasted demand where, i rightarrow row no. j rightarrow column no. 1 lessthanorequalto i lessthanorequalto 3 1 lessthanorequalto j lessthanorequalto 6 Let, b_ig rightarrow material cost work hour required material: - Fc = 0 (Firing cost) HC = 0 (Firing cost) Work hour required material: - let, c_ij rightarrow Work hours required per unit Working hours required for j the month = sigma^3_i = 1 a_ij c_ij Let, d_j rightarrow working days available per month y_1 = 40 (working available in beginning) Initializing So, workers required per month (x/y) = working hrs req - per month/d_i times 10 = sigma^3_i = 1 a_ij c_ij/d_j times 10 Workers available currently = y_j It, x_j - y_j < 0(i.e. Row worker less than available) (f) Firing cost = (y_j - x_j)400 F(c) = F(c) + f If, x_j - y_j > 0 (i.e. req workers the then available) Firing cost of month