A manufacturing company has three plants, designated A, B, and C. The capacities and manufacturing costs of these plants are as shown in Table 1.

                                                       Manufacturing Cost

Plant              Capacity (units)                 ($/unit)

   A                        2000                          $25.50

   B                        3000                          $24.90

   C                        2500                          $25.30

Table 1.

The demand requirements for the next year in the four marketing regions are as shown in Table 2.

Marketing Region                Requirements (units)

         1                                          2100

         2                                           900

         3                                          2400

         4                                           900

Table 2.

The shipping costs per unit from plant to marketing region are as shown in Table 3.

From                      To Marketing Region

Plant              1                     2                     3                     4

   A           $1.50              $2.10              $4.20              $3.75

   B           $2.35              $3.55              $2.10              $2.85

   C           $4.20              $3.10              $0.90              $1.70

Table 3.

The manufacturing company's objective is to minimize costs for the year while satisfying demand in all four marketing regions.

The decision variables represent the number of units shipped from a plant to a marketing region (e.g. A2 is the number of units shipped from plant A to marketing region 2). Thus, for three plants and four regions, there are 3x4 = 12 variables

I have formulated the problem as a linear program and solved it using LINDO. The output is reproduced on the last page. Each of the following questions refers to the original problem formulation and solution. The questions are not cumulative, i.e., they are to be considered independent of each other.

1.     What is the optimal shipping plan?

2.     What is the annual cost of this shipping plan?

3.     Which plant (if any) has excess capacity?

4.     Consider the effect of losing 100 units of production capacity at plant A: How would that affect the total operating cost? Would the production and shipping plan change?

5.     Consider the effect of a 50 cent increase in the unit transportation cost between plant A and region 2: How would that affect the total operating cost? Would the production and shipping plan change?

6.     Suppose that the firm had to ship at least 10 units (of the 900 units required by region 2) between plant B and region 2 (notice, currently   B2 has a value of 0 indicating that nothing should be shipped from plant B to region 2). How would that affect the total operating cost? Would the production and shipping plan change?

(Hint: this question refers to reduced cost of B2, which is $.60/unit. If you are forced to ship between plant B and region 2, B2 is forced to be B2= 10. Every extra unit if B2 hurts cost by $.60 each).

7.     The requirements shown in Table 2 correspond to contracts for which prices and quantities have long been settled and are not negotiable. As you are preparing a production and shipping plan, the marketing department approaches you with a potential new contract for 80 units to be shipped to region 1 at a delivered price (manufacturing company's revenue) of $27.50 per unit. Would additional revenue cover the associated manufacturing and transportation costs? Why?

(Hint: look at dual price, highlighted in blue, for Rgn1 to determine the impact of additional 80 units shipped there)

8.     The marketing department believes that a modification to the above contract (80 units at $27.50) could be negotiated. For a total rebate of $50 to the client, the client should be willing to receive the shipment in its region 3 branch instead of region 1.Would that be a more attractive alternative to the manufacturing company? Why?

(Hint: dual price for Rgn3 indicates that it is cheaper to ship to Rgn 3 , only $26.95 per unit compared to shipping to Rgn 1, which is $27.25 per unit. Thus, this question asks if the savings are enough to be able to pay a rebate)

9.     The planning department is considering the option of increasing the capacity of one of the plants by the addition of an extra production line. Two types of lines are available:                                 

A small line has a capacity of 500 units and involves a yearly fixed cost of $400. A large line has a capacity of 800 units and involves a yearly fixed cost of $550. Unit manufacturing costs would not change; they would depend on the location as given in Table 1. Should capacity be increased by the addition of one line? Which of the two types of lines is more attractive to the manufacturing company? In what plant should it be installed?

(Hint: in question 3 you determined which plant has excess capacity, thus it would not make sense to add any more capacity to that plant. Consider adding capacity to those plans where there is no slack. Look at capacity dual prices to see if the fixed cost can be recovered by adding units of capacity)