Question

The gardening season is in full swing and many nurseries are planning how they will be able to meet customer demands over the next few months. PlantaLot is a nursery that specializes in cultivating starter tomato plants for distribution at local garden centres. They would like to develop a production schedule to determine the number of starter tomato plants that will need to be cultivated over the months of July, August, and September.
Based on the previous year’s sales, demand for July, August, and September are 6,000 starter plants, 8,000 starter plants, and 3,000 starter plants respectively. We can assume that the sales will be the same as estimated demands in this case. PlantaLot has a production capacity that allows them to produce up to 7,000 starter plants each month. Furthermore, PlantaLot has storage capacity for up to 1,000 starter plants at the end of every month. In June, PlantaLot produced 5500 starter plants, but only sold 5000. Thus, they have 500 starter plants in inventory at the start of July
In addition to the above information, it is desirable to have a production schedule that maintains the current number of employees and has the same production quantities every month. But given the fluctuations from month to month in terms of demand, this MAY NOT be possible. The nursery’s accounting department estimates that increasing the production by one starter plant from one month to the next will cost $0.50 per starter plant and decreasing production by one starter plant form one month to the next will cost $0.40 per starter plant.

Ignoring production and inventory costs, formulate the algebraic linear programming model that will help PlantaLot determine how many plants to produce in the months of July, August, and September such that the cost of changing the production level is minimized while PlantaLot also satisfies monthly demands.
ONLY FORMULATE – DO NOT SOLVE THE PROBLEM FOR AN OPTIMAL SOLUTION

Question 4 (continued)

Hint: The cost of changing production levels is the cost of the monthly increases in production plus the cost of monthly decreases in production.

You should use the following decision variables to formulate your model, thus there is no need to redefine or define more decision variables. Let Xi represent the number of tomato starter plants produced in month i Let Yi represent the number of tomato starter plants placed in storage at the end of month i Let ei represent the increase in the total production level in month i Let di represent the decrease in the total production level in month i where i = 1, 2, and 3 (1 is July; 2 is August, and 3 is September).

Answer 1

Decision Variables

Xi = number of tomato starter plants produced in month i

Yi = number of tomato starter plants placed in storage at the end of month i

ei = increase in the total production level in month i

di = decrease in the total production level in month i

where i = 1, 2, and 3 (1 is July; 2 is August, and 3 is September)

Our Objective Function will be the cost. We will try and minimize the cost of increase and decrease in production

Z = $\sum(0.5 ei + 0.4 di)_{}^{}$ where i = 1 to 3

Constraints

Xi <= 7000

Yi<=1000

ei = Xi+1 - Xi

di = Xi - Xi+1

X1 > = 6000 - 500

Y1 = (X1+500) - 6000

X2 >= 8000 - Y1

Y2 = (X2 +Y1)- 8000

X 3 > =3000 - Y2

Y 3 = (X3 + Y2) - 3000