Question

Question 1

CrystalClear supplies 75-centiliter glass bottles to beverage manufacturers all around Europe. The company’s best selling products are wine bottles and champagne bottles. Wine bottles are sold at a price of 3euro/unit, and champagne bottles are sold at 3.5 euro/unit.

These two products can be manufactured in three different production lines: PL1, PL2 and PL3. These production lines present a different variable cost per unit per product (see Table 1). They also have a limited capacity of 1000 bottles for PL1, 1500 bottles for PL2 and 2800 bottles for PL3.

Table 1: Variable production cost per unit (in euros/unit)

	wine bottle	champ bottle
PL1	0.89	0.58
PL2	0.45	0.43
PL3	0.32	0.91

There is also a limited availability of two important raw materials: 1000kg of silica sand (the major component in each bottle), and 300 kg of iron oxide (the component that gives the bottle a green color). Wine bottles and champagne bottles use different amounts of these raw materials (see Table 2).

Table 2: Raw material need per unit produced (in kg/unit)

	wine bottle	champ bottle
silica	0.2	0.25
iron	0.08	0.06

Assuming that demand is unlimited (all bottles produced are sold), please create and solve a model to decide which production plan would maximize profit.

What is the maximum profit CrystalClear could make under the conditions specified in this problem?

Question 2

CrystalClear is going to launch a new product: soda bottles. The estimated demand for this new product is 5000 units. The objective of the company is to produce these new type of bottles at the lowest cost. Soda bottles have a smaller size than 75-centiliter bottles and use specific additives. Therefore, although they can be produced in some of the existing production lines of the company, there is a set up cost associated with preparing the line to manufacture soda bottles. The company has three production lines (PL4, PL5 and PL6) with some spare capacity that could be used to manufacture soda bottles. But the operations manager has made very clear that there is a minimum production requirement, to make it worth the effort of adjusting the parameters of each machine. All the detailed information per production line can be found in Table 3.

Table 3: Costs (in euros) and min/max production requirements (in units) per production line.

	cost per unit	set up cost	min production	max production
PL4	0.52	500	1000	3000
PL5	0.41	700	500	4000
PL6	0.96	400	900	2500

Create and solve a model that helps you decide how much to produce in each line to minimize total production cost of soda bottles. (Note: Demand must be met, therefore you must plan on producing at least 5,000 units/month.)

What is the minimum production cost of soda bottles under the conditions specified in Question 2?

Question 3

The marketing department of CrystalClear has been doing some market analysis and they came to the conclusion that the demand for the new soda bottles is highly dependent on the price point. If the price is too low, customers will believe the product is of low quality, and if the price is too high, customers will look for other suppliers. They have created the following equation that relates demand with price:

\[ Demand = - 424 p^2 +1000p+5000\]

According to the marketing department, what would be the optimal price to maximize demand?

Answer 1

Let,

x11 = Number of wine bottles manufactured in production line 1
x12 = Number of champ bottles manufactured in production line 1
x21 = Number of wine bottles manufactured in production line 2
x22 = Number of champ bottles manufactured in production line 2
x31 = Number of wine bottles manufactured in production line 3
x32 = Number of champ bottles manufactured in production line 3

Objective is to maximize profit so objective function = Max 3x11+3x21+3x31 + 3.5x12+3.5x22+3.5x32 -(0.89x11+0.58x12+0.45x21+0.43x22+0.32x31+0.91x32)

=> Max 2.11x11+2.92x12+2.55x21+3.07x22+2.68x31+2.59x32

Subject to,

capacity constraints

x11+x12 <= 1000
x21+x22 <= 1500
x31+x32 <= 2800

0.2x11+0.2x21+0.2x31 +0.25x12+0.25x22+0.25x32<= 1000 (Silica constraint)

0.08x11+0.08x21+0.08x31 +0.06x12+0.06x22+0.06x32<= 300 (Iron constraint)

x11,x12,x21,x22,x31,x32 >= 0 (Non-negativity constraint)

solving in solver we get,

x11 = Number of wine bottles manufactured in production line 1 = 0
x12 = Number of champ bottles manufactured in production line 1 = 1000
x21 = Number of wine bottles manufactured in production line 2 = 0
x22 = Number of champ bottles manufactured in production line 2 = 1500
x31 = Number of wine bottles manufactured in production line 3 = 1875
x32 = Number of champ bottles manufactured in production line 3 = 0

Maximized profit = 12550

Solver screenshot

	x11	x12	x21	x22	x31	x32
	0	1000	0	1500	1875	0				Objective function	12550
Plant 1 - capacity	1	1					1000	<=	1000
Plant 2 - capacity			1	1			1500	<=	1500
Plant 3 - capacity					1	1	1875	<=	2800
Silica	0.2	0.25	0.2	0.25	0.2	0.25	1000	<=	1000
Iron	0.08	0.06	0.08	0.06	0.08	0.06	300	<=	300