A firm produces perishable food at the cost of $25 per case and in batches of 10

Question

A firm produces perishable food at the cost of $25 per case and in batches of 100 cases. It sells the product to a grocery chain at $30 per case. The grocery chain buys the products in lots of 100, 200, or 300 cases every morning, but does not specify its requirements to the firm in time for the production run. If the demand is less than the production, the excess produce is lost. If the demand is more than the production, the firm has agreed to satisfy the excess demand with a special production run at a cost of $35 per case. The selling price, however, is always at $30 per case. The firm has estimated that the demand is for 100 cases 20% of the days, for 200 cases 50% of the days and for 300 cases the remaining 30% of the days.
(a) Set up the payoff table for this problem.
(b) What should be the firm's production decision if the objective is to maximize expected payoff?
(c) How much discount (per case) should the firm be willing to offer the grocery chain for specifying the demand in time for each day's production run? Assume that the distribution for demand remains unchanged at 0.2, 0.5, and 0.3 for 100, 200, and 300 cases respectively.
(d) Compose a short (one paragraph) memo for the CEO of the firm to explain why you should offer the grocery chain the per case discount determined in part (c).

Explanation / Answer

Cost                            $25 per case
Selling Price                 $30 per case
Profit                            $5 per case

Excess Demand cost     $35
Excess Demand profit   ($5)

            Cost       Revenue        Profit
100     $2,500       $3,000            $500
200     $5,000       $6,000         $1,000
300     $7,500       $9,000       $15,000

                               Pay off Table
______________________________________________________
Produced                            Demand
______________________________________________________
                              100            200                 300
                      _______________________________________
100                        $5001              02             ($500)3    
200                   ($2,000)4       $1,0005             $5006
300                   ($4,500)7      ($1,500)8         $1,5009
________________________________________________________________
1 (100 x $5) - (100 x $5)     = $500
2 (100 x $5) - (100 x $5)      = 0
3 (100 x $5) - (200 x $5)      = $500
4 ($5,000 - $3,000)             = $2,000
5 200 x $5                          = $1,000
6 (200 x $5) - (100 x $5)     = $500
7 ($7,500 - $3,000)             = $4,500
8 ($7,500 - $6,000)             = $1,500
9 300 x $5                          = $1,500

(b) What should be the firm's production decision if the objective is to maximize expected payoff?

Event 100 = ($500 x 0.2) + (0 x 0.5) + (-$500 x 0.3) = -$50
Event 200 = (-$2000 x 0.2) + ($1000 x 0.5) + ($500 x 0.3) = $250
Event 300 = (-$4500 x 0.2) + (-$1500 x 0.5) + ($1500 x 0.3) = -$1,200

Since the expected payoff is $250, the company should produce 200 units

c) How much discount (per case) should the firm be willing to offer the grocery chain for specifying the demand in time for each day's production run? Assume that the distribution for demand remains unchanged at 0.2, 0.5, and 0.3 for 100, 200, and 300 cases respectively.

Best situation with perfect
information (EPVI)           = 0.2(500) + 0.5(1000) + 0.3(1,500)
                                       =                  $1,050
Less : Expected payoff                            $250
Difference                                              $800

Average number of case = 100(0.2) + 200(0.5) + 300(0.3) = 210
Maximum discount per case = $800 / $210 = $3.80

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